2 Review of the construction of the reciprocal dual $\widehat {{\mathcal {A}}}$

In this section, we review the construction of the reciprocal dual $\widehat {{\mathcal {A}}}$ of a unital Kirchberg algebra ${\mathcal {A}}$ following [Reference Matsumoto and Sogabe18]. Throughout the section, we denote by $\mathcal {K}(H)$ the $C^*$ -algebra of compact operators on a separable infinite dimensional Hilbert space $H$ , written in short as $\mathcal {K}$ . We refer to Blackadar’s textbook [Reference Blackadar2] for the basic facts and notation related to $\mathrm {K}\!\mathrm {K}$ -groups.

We write $\mathcal {S} = C_0(0,1)$ and $\mathcal {S}^k = \mathcal {S}^{\otimes k}$ . For a separable unital nuclear $C^*$ -algebra ${\mathcal {A}}$ , the mapping cone algebra $C_{\mathcal {A}}$ is defined by $C_{\mathcal {A}} =\{a(t) \in C_0(0,1] \otimes {\mathcal {A}}\mid a(1) \in {\mathbb {C}}1_{{\mathcal {A}}}\}$ . Following Skandalis [Reference Skandalis26], we write

$$ \begin{align*} \mathrm{Ext}_{\mathrm{s}}^i({\mathcal{A}}) = \mathrm{K}\!\mathrm{K}(C_{\mathcal{A}}, \mathcal{S}^{i+1}), \quad i=0,1. \end{align*} $$

As in [Reference Blackadar2], the group $\mathrm {Ext}_{\mathrm {w}}^i({\mathcal {A}})$ is given by $\mathrm {K}\!\mathrm {K}({\mathcal {A}}, \mathcal {S}^i)$ for $i=0,1$ . It is well known that the group $\mathrm {Ext}_{\mathrm {s}}^1({\mathcal {A}})$ (respectively $\mathrm {Ext}_{\mathrm {w}}^1({\mathcal {A}})$ ) is realized as the strong (respectively weak) extension group $\mathrm {Ext}_{\mathrm {s}}({\mathcal {A}})$ (respectively $\mathrm {Ext}_{\mathrm {w}}({\mathcal {A}})$ ), which is the abelian group of strong (respectively weak) unitary equivalence classes of unital Busby invariants (see [Reference Blackadar2, Reference Skandalis26]). We write $\mathcal {S}\otimes {\mathcal {A}}$ as $\mathcal {S} {\mathcal {A}}$ . Applying the $\mathrm {K}$ -homology functor $\mathrm {K}\!\mathrm {K}(\, - \, , \mathbb {C})$ to the natural short exact sequence $0\rightarrow \mathcal {S}{\mathcal {A}} \rightarrow C_{\mathcal {A}} \rightarrow \mathbb {C} \rightarrow 0$ , we have a cyclic six-term exact sequence of ${{\mathrm {Ext}}}_*$ -groups:

(2-1)

The class of $\iota _{\mathcal {A}}(1)$ in $\mathrm {Ext}_{\mathrm {s}}^1({\mathcal {A}})$ is denoted by $[\iota _{\mathcal {A}}(1)]_s$ , which corresponds to a strong unitary equivalence class of a Busby invariant $\mathrm {Ad}u\circ \rho $ with a unital trivial extension $\rho $ and a unitary $u$ in the Calkin algebra ${\mathcal {B}}(H)/\mathcal {K}(H)$ with the Fredholm index $1$ .

Throughout the rest of this section, we fix a Kirchberg algebra ${\mathcal {A}}$ . Let ${\mathcal {F}}$ be a separable unital nuclear UCT $C^*$ -algebra containing $\mathcal {K}(H)$ as an essential ideal of ${\mathcal {F}}$ such that

(2-2) $$ \begin{align} (\mathrm{K}_0({\mathcal{F}}), [f]_0, \mathrm{K}_1({\mathcal{F}})) \cong (\mathrm{Ext}_{\mathrm{s}}^1({\mathcal{A}}), [\iota_{\mathcal{A}}(1)]_s, \mathrm{Ext}_{\mathrm{s}}^0({\mathcal{A}})), \end{align} $$

where $[f]_0$ is the class in $\mathrm {K}_0({\mathcal {F}})$ of a fixed minimal projection $f\in \mathcal {K}(H)$ so that $f \in \mathcal {K}(H) \subset {\mathcal {F}}$ . Such a separable unital $C^*$ -algebra ${\mathcal {F}}$ for a given unital Kirchberg algebra ${\mathcal {A}}$ always exists by [Reference Pennig and Sogabe22, Theorem 1.2]. Since $\mathcal {K}(H)$ is an essential ideal of ${\mathcal {F}}$ , there exists an injective $*$ -homomorphism $\pi _{\mathcal {F}}:{\mathcal {F}}\rightarrow {\mathcal {B}}(H)$ . Let us denote by $e_k, k \in \mathbb {N}$ the canonical basis of $\ell ^2(\mathbb {N})$ defined by $e_k(m) = \delta _{k,m}, \, k,m \in \mathbb {N}.$ Define the Hilbert $C^*$ -bimodule $H_{\mathcal {F}}$ over ${\mathcal {F}}$ by setting

$$ \begin{align*} H_{\mathcal{F}}: =H\otimes_{\mathbb{C}} \ell^2(\mathbb{N})\otimes_{\mathbb{C}} {\mathcal{F}}, \end{align*} $$

where the ${\mathcal {F}}$ -valued inner product $\langle \mid \rangle _{\mathcal {F}}$ on $H_{\mathcal {F}}$ is defined by

$$ \begin{align*} \langle \xi\otimes e_k \otimes x \mid \eta\otimes e_l\otimes y\rangle _{\mathcal{F}} =\langle \xi \mid \eta\rangle _H \delta_{k,l} x^* y \in {\mathcal{F}} \end{align*} $$

for $\xi , \eta \in H$ , $e_k, e_l \in \ell ^2(\mathbb {N})$ , and $x, y \in {\mathcal {F}}.$ The left action of ${\mathcal {F}}$ on $H_{\mathcal {F}}$ to the algebra ${\mathcal {L}}(H_{\mathcal {F}})$ of adjointable bounded module maps on $H_{\mathcal {F}}$ is given by

$$ \begin{align*} \varphi_{\mathcal{F}}:= \pi_{\mathcal{F}}\otimes 1_{\ell^2(\mathbb{N})} \otimes 1_{\mathcal{F}}: {\mathcal{F}}\otimes \mathbb{C} 1_{\ell^2(\mathbb{N})}\otimes \mathbb{C}1_{\mathcal{F}} \rightarrow {\mathcal{L}}(H_{\mathcal{F}}). \end{align*} $$

Take Pimsner’s Toeplitz algebra $\mathcal {T}_{H_{\mathcal {F}}}$ (see [Reference Pimsner24]), which is $\mathrm {K}\!\mathrm {K}$ -equivalent to ${\mathcal {F}}$ via the unital embedding $ {\mathcal {F}} \hookrightarrow \mathcal {T}_{H_{\mathcal {F}}}.$ Since ${\mathcal {F}}\otimes \mathbb {C} 1_{\ell ^2(\mathbb {N})}\otimes \mathbb {C}1_{\mathcal {F}} \cap \mathcal {K}(H_{\mathcal {F}}) = \{ 0 \},$ Kumjian’s results [Reference Kumujian14, Proposition 2.1] and [Reference Kumujian14, Theorem 3.1], together with [Reference Pimsner24, Corollary 4.5] tell us that the Cuntz–Pimsner algebra ${\mathcal {O}}_{H_{\mathcal {F}}}$ is a separable unital simple purely infinite nuclear UCT $C^*$ -algebra, and hence a Kirchberg algebra such that ${\mathcal {O}}_{H_{\mathcal {F}}}$ and $\mathcal {T}_{H_{\mathcal {F}}}$ are canonically isomorphic, and the natural unital embedding ${\mathcal {F}}\hookrightarrow \mathcal {T}_{H_{\mathcal {F}}}$ yields a $\mathrm {K}\!\mathrm {K}$ -equivalence. We identify the minimal projection $f$ in $\mathcal {K}(H)$ with the projection in ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ through the embedding and the identification:

(2-3) $$ \begin{align} f\in \mathcal{K}(H) \subset {\mathcal{F}} \hookrightarrow \mathcal{T}_{H_{\mathcal{F}}} \cong {\mathcal{O}}_{H_{\mathcal{F}}} \ni f. \end{align} $$

We then have

$$ \begin{align*} (\mathrm{K}_0({\mathcal{O}}_{H_{\mathcal{F}}}), [f]_0, \mathrm{K}_1({\mathcal{O}}_{H_{\mathcal{F}}})) \cong (\mathrm{K}_0(\mathcal{T}_{H_{\mathcal{F}}}), [f]_0, \mathrm{K}_1(\mathcal{T}_{H_{\mathcal{F}}})) \cong (\mathrm{K}_0({\mathcal{F}}), [f]_0, \mathrm{K}_1({\mathcal{F}})). \end{align*} $$

The reciprocal duality (see [Reference Matsumoto and Sogabe17, Proposition 3.7]) tells us

$$ \begin{align*} (\mathrm{Ext}_{\mathrm{s}}^1({\mathcal{A}}), [\iota_{\mathcal{A}}(1)]_s, \mathrm{Ext}_{\mathrm{s}}^0({\mathcal{A}})) \cong (\mathrm{K}_0(\widehat{{\mathcal{A}}}), [1_{\widehat{{\mathcal{A}}}}]_0, \mathrm{K}_1(\widehat{{\mathcal{A}}})), \end{align*} $$

so that by the hypothesis (2-2), we obtain that

$$ \begin{align*} (\mathrm{K}_0({\mathcal{O}}_{H_{\mathcal{F}}}), [f]_0, \mathrm{K}_1({\mathcal{O}}_{H_{\mathcal{F}}})) \cong (\mathrm{K}_0(\widehat{{\mathcal{A}}}), [1_{\widehat{{\mathcal{A}}}}]_0, \mathrm{K}_1(\widehat{{\mathcal{A}}})), \end{align*} $$

showing that

(2-4) $$ \begin{align} f {\mathcal{O}}_{H_{\mathcal{F}}} f \cong \widehat{{\mathcal{A}}}, \end{align} $$

by the Kirchberg–Philips classification theorem of Kirchberg algebras [Reference Kirchberg13, Reference Phillips23].

3 Universality of the Cuntz–Pimsner algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$

Let ${\mathcal {F}}$ be a separable $C^*$ -algebra with unit $1_{\mathcal {F}}$ containing $\mathcal {K}(H)$ as an essential ideal. Take a minimal projection $f$ in $\mathcal {K}(H)$ . In this section, we begin with only $f \in \mathcal {K}(H) \triangleleft {\mathcal {F}}$ from which one may construct the Cuntz–Pimsner algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ . We need not provide a Kirchberg algebra ${\mathcal {A}}$ in this section, so we do not assume the hypothesis (2-2).

Lemma 3.1. The $C^*$ -algebra ${\mathcal {O}}_{H_{\mathcal {F}}}$ is generated by ${\mathcal {F}}$ and isometries $S_i, i \in \mathbb {N}$ satisfying

(3-1) $$ \begin{align} S_i^* S_i = 1_{{\mathcal{O}}_{H_{\mathcal{F}}}}, \quad i \in \mathbb{N} \quad \text{and} \quad \sum_{j=1}^m S_j S_j^* < f, \quad m \in \mathbb{N}. \end{align} $$

Proof. We follow the previous construction of the $C^*$ -algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ in detail. Take a unit vector $\xi _0 \in H$ such that $ f H= \mathbb {C}\xi _0$ . The Hilbert $C^*$ -bimodule $H_{\mathcal {F}} = H \otimes _{\mathbb {C}} \ell ^2(\mathbb {N}) \otimes _{\mathbb {C}} {\mathcal {F}}$ has a left action $\varphi _{\mathcal {F}}: {\mathcal {F}} \rightarrow {\mathcal {L}}(H_{\mathcal {F}})$ defined by $\varphi _{\mathcal {F}}(a) = \pi _{\mathcal {F}}(a) \otimes 1_{\ell ^2(\mathbb {N})} \otimes 1_{\mathcal {F}}$ for $a \in {\mathcal {F}}$ . We write $\pi _{\mathcal {F}}(a)$ as $a$ for short. The action $\varphi _{\mathcal {F}}$ is denoted by $\varphi $ . Consider the Fock space ${\mathcal {E}}_{\mathcal {F}}$ of $H_{\mathcal {F}}$ defined by

$$ \begin{align*} {\mathcal{E}}_{\mathcal{F}} :={\mathcal{F}} \oplus \bigoplus_{n=1}^\infty H_{\mathcal{F}}^{\otimes n}, \end{align*} $$

where $H_{\mathcal {F}}^{\otimes n}$ is the $n$ -times relative tensor product $\overbrace {H_{\mathcal {F}} \otimes _{\mathcal {F}} H_{\mathcal {F}} \otimes _{\mathcal {F}} \cdots \otimes _{\mathcal {F}} H_F}^{n\ \text {times}}$ over ${\mathcal {F}}$ . The creation operator $T_{\xi \otimes e_k \otimes y}$ for $\xi \otimes e_k \otimes y \in H_{\mathcal {F}}$ is defined by

$$ \begin{align*} T_{\xi\otimes e_k \otimes y} (a) &= \xi\otimes e_k \otimes ya, \quad a \in {\mathcal{F}},\\ T_{\xi\otimes e_k \otimes y} (\zeta_1\otimes \zeta_2\otimes \cdots \otimes \zeta_n) &= (\xi\otimes e_k \otimes y)\otimes \zeta_1\otimes \zeta_2\otimes \cdots \otimes \zeta_n \in H_{\mathcal{F}}^{\otimes n+1} \end{align*} $$

for $\zeta _1\otimes \zeta _2\otimes \cdots \otimes \zeta _n \in H_{\mathcal {F}}^{\otimes n}$ . The Toeplitz algebra $\mathcal {T}_{H_{\mathcal {F}}}$ for the $C^*$ -bimodule $H_{\mathcal {F}}$ is defined by the $C^*$ -algebra $C^*(T_{\xi \otimes e_k \otimes y}\mid \xi \in H,\, k \in \mathbb {N},\, y \in {\mathcal {F}})$ generated by the creation operators $T_{\xi \otimes e_k \otimes y} \in {\mathcal {L}}({\mathcal {E}}_{\mathcal {F}})$ for $\xi \otimes e_k \otimes y \in H_{\mathcal {F}}$ . Define $\varphi _+: x \in {\mathcal {F}} \rightarrow \varphi _+(x) \in {\mathcal {L}}({\mathcal {E}}_{\mathcal {F}})$ by

$$ \begin{align*} \varphi_+(x) (a) &= xa, \quad a \in {\mathcal{F}}, \\ \varphi_+(x)(\zeta_1\otimes \zeta_2\otimes \cdots \otimes \zeta_n) &=(\varphi(x)\zeta_1)\otimes \zeta_2\otimes \cdots \otimes \zeta_n, \end{align*} $$

$\zeta _1\otimes \zeta _2\otimes \cdots \otimes \zeta _n \in H_{\mathcal {F}}^{\otimes n}.$ We then have

(3-2) $$ \begin{align} T_{\xi\otimes e_k \otimes y}^* T_{\eta\otimes e_l \otimes z} = \varphi_+(\langle \xi\otimes e_k \otimes y \mid \eta\otimes e_l \otimes z \rangle ) = \varphi_+(\langle \xi\mid \eta\rangle _H \delta_{k,l} y^*z) \end{align} $$

for $\xi \otimes e_k \otimes y, \, \eta \otimes e_l \otimes z \in H_{\mathcal {F}}$ , and

$$ \begin{align*} \varphi_+(x) T_{\xi_0\otimes e_k \otimes 1_{\mathcal{F}}} \varphi_+(y) =T_{\varphi(x) \xi_0\otimes e_k \otimes y}. \end{align*} $$

Hence, the $C^*$ -algebra $\mathcal {T}_{H_{\mathcal {F}}}$ is generated by $\varphi _+(x), x \in {\mathcal {F}}$ and $T_{\xi _0\otimes e_k\otimes 1_{\mathcal {F}}}, k \in \mathbb {N}$ . We write

$$ \begin{align*} S_k := T_{\xi_0\otimes e_k\otimes 1_{\mathcal{F}}} \end{align*} $$

for $k \in \mathbb {N}$ . By (3-2), we have $S_k^* S_l = \delta _{k,l}.$ It is easy to see that

$$ \begin{align*} S_k S_k^* (\zeta_1\otimes \zeta_2\otimes \cdots \otimes \zeta_n) = {\begin{cases} \zeta_1\otimes \zeta_2\otimes \cdots \otimes \zeta_n & \text{ if } \zeta_1 \in \mathbb{C}\xi_0\otimes \mathbb{C}e_k \otimes {\mathcal{F}}, \\ 0 & \text{ otherwise,} \end{cases} } \end{align*} $$

so that $S_k S_k^*$ is the projection onto the subspace spanned by

$$ \begin{align*} (\xi_0\otimes e_k \otimes {\mathcal{F}}) \oplus \bigoplus_{n=1}^\infty (\xi_0\otimes e_k \otimes {\mathcal{F}})\otimes H_{\mathcal{F}}^{\otimes n}. \end{align*} $$

The projection onto the subspace

$$ \begin{align*} f {\mathcal{F}} \oplus (\xi_0\otimes \ell^2(\mathbb{N}) \otimes {\mathcal{F}}) \oplus \bigoplus_{n=1}^\infty (\xi_0\otimes \ell^2(\mathbb{N}) \otimes {\mathcal{F}})\otimes H_{\mathcal{F}}^{\otimes n} \end{align*} $$

corresponds to the minimal projection $f$ in $\mathcal {K}(H)$ by identifying $f$ with $\varphi _+(f).$ We may regard $f$ as an element of ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ through the correspondence (2-3) under the natural identification between $\mathcal {T}_{H_{\mathcal {F}}}$ and ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ . Since $S_k S_k^* < f$ for all $k \in \mathbb {N}$ and $\{S_kS_k^*\}_{k\in \mathbb {N}}$ are mutually orthogonal,

$$ \begin{align*} \sum_{j=1}^m S_j S_j^* < f \quad \text{for all } m \in \mathbb{N}. \end{align*} $$

We may write $\varphi _+(x) T_{\xi \otimes e_k \otimes 1}\varphi _+(y)$ as $x S_k y$ for short, so that the $C^*$ -algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ is generated by ${\mathcal {F}}$ and $S_k, k\in \mathbb {N}$ satisfying (3-1).

We note that the algebraic structure of the Cuntz–Pimsner algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ is determined by the following two relations (see [Reference Brown and Ozawa3, Definition 4.6.14]):

$$ \begin{align*} \begin{cases} \bullet \quad T_{\xi\otimes e_k \otimes y}^* T_{\eta\otimes e_l \otimes z} = \langle \xi\mid \eta\rangle _H \delta_{k,l} \varphi_+( y^*z), \\ \bullet \quad \varphi_+(x) T_{\xi\otimes e_k \otimes 1}\varphi_+(y) = T_{x\xi\otimes e_k \otimes y} \end{cases} \end{align*} $$

for $\xi , \eta \in H_{\mathcal {F}}, \, k,l \in \mathbb {N}, \, x,y \in {\mathcal {F}}.$

Lemma 3.2. Keep the unital $C^*$ -algebra ${\mathcal {F}}$ containing $\mathcal {K}(H)$ as an essential ideal and fix a minimal projection $f \in \mathcal {K}(H)$ . Suppose that there exist a unital representation $\pi : {\mathcal {F}} \rightarrow {\mathcal {B}}$ of ${\mathcal {F}}$ to a unital $C^*$ -algebra ${\mathcal {B}}$ and a family of isometries $s_j \in {\mathcal {B}}, j \in \mathbb {N}$ satisfying

(3-3) $$ \begin{align} s_i^* s_i = 1_{\mathcal{B}} \quad \text{for } i\in \mathbb{N} \quad \text{and} \quad \sum_{j=1}^m s_j s_j^* < \pi(f) \quad \text{for } m\in \mathbb{N}. \end{align} $$

Then, there exists a linear map $\tau : H_{\mathcal {F}}\rightarrow {\mathcal {B}}$ such that

$$ \begin{align*} \begin{cases} \bullet \quad \tau(a u b) = \pi(a) \tau(u) \pi(b) \quad \text{for } u \in H_{\mathcal{F}} \ \text{and} \ a,b \in {\mathcal{F}}, \\ \bullet \quad \tau(u)^* \tau(v) = \pi(\langle u\mid v\rangle _{\mathcal{F}}) \quad \text{for } u, v \in H_{\mathcal{F}}. \end{cases} \end{align*} $$

Hence, the pair $(\pi ,\tau )$ gives rise to a covariant representation of the Hilbert $C^*$ -bimodule $H_{\mathcal {F}}$ such that the $C^*$ -algebra $C^*(\tau (u)\mid u \in H_{\mathcal {F}})$ generated by $\tau (u), u \in H_{\mathcal {F}}$ coincides with the $C^*$ -algebra $C^*(\pi ({\mathcal {F}}), \{ s_j\}_{j \in \mathbb {N}})$ generated by $\pi ({\mathcal {F}})$ and $\{s_j\}_{j \in \mathbb {N}}.$

Proof. Take a unit vector $\xi _0 \in H$ such that $f H = \mathbb {C} \xi _0.$ For $\xi \in H_{\mathcal {F}}$ , let us denote by $t_{\xi , \xi _0}\in \mathcal {K}(H)$ the operator on $H$ defined by $t_{\xi ,\xi _0}(\zeta ) = \xi \langle \xi _0\mid \zeta \rangle _H, \, \zeta \in H_{\mathcal {F}}.$ As $\mathcal {K}(H) \subset {\mathcal {F}},$ it belongs to ${\mathcal {F}}.$ For $u = \xi \otimes e_k\otimes y \in H_{\mathcal {F}},$ define

$$ \begin{align*} \tau(u) = \pi(t_{\xi,\xi_0}) s_k \pi(y) \end{align*} $$

and extend it linearly to $H_{\mathcal {F}}$ in a natural way. As $t_{\xi _0, \xi _0} =f$ and $s_k s_k^* < \pi (f),$ we have for $a,b \in {\mathcal {F}}$ , $u = \xi \otimes e_k \otimes y, \, v = \eta \otimes e_l\otimes z \in H_{\mathcal {F}}$ ,

$$ \begin{align*} \tau(aub) = \pi(t_{a\xi, \xi_0}) s_k \pi(yb) = \pi(a) \pi(t_{\xi, \xi_0}) s_k \pi(y) \pi(b) = \pi(a) \tau(u) \pi(b) \ \end{align*} $$

and

$$ \begin{align*} \tau(u)^* \tau(v) = \pi(y)^* s_k^* \pi(\langle \xi \mid \eta\rangle _H f) s_l \pi(z) = \pi(\langle u \mid v\rangle _{H_{\mathcal{F}}}). \end{align*} $$

As $\varphi ({\mathcal {F}}) \cap \mathcal {K}(H_{\mathcal {F}}) = \{ 0 \}$ , [Reference Kumujian14, Proposition 2.1] shows that the representation $(\pi ,\tau )$ is covariant by [Reference Brown and Ozawa3, Definition 4.6.9]. For $k \in \mathbb {N}$ and $a \in {\mathcal {F}}$ , we have $s_k = \tau (\xi _0\otimes e_k \otimes 1)$ and

$$ \begin{align*} \pi(a) & = \pi(a) s_k^* s_k = \tau(a\langle \xi_0\otimes e_k \otimes 1_{\ell^2(\mathbb{N})} \mid \xi_0\otimes e_k \otimes 1_{\ell^2(\mathbb{N})} \rangle _{\mathcal{F}}) \\ & = \tau(\langle \xi_0\otimes e_k \otimes a^* \mid \xi_0\otimes e_k \otimes 1 \rangle _{\mathcal{F}}), \end{align*} $$

showing that $C^*(\tau (u)\mid u \in H_{\mathcal {F}})= C^*(\pi ({\mathcal {F}}), \{ s_j\}_{j \in \mathbb {N}}).$

Therefore, we have the following proposition.

Proposition 3.3. Keep the unital $C^*$ -algebra ${\mathcal {F}}$ containing $\mathcal {K}(H)$ as an essential ideal and fix a minimal projection $f \in \mathcal {K}(H)$ . Suppose that there exist a unital representation $\pi : {\mathcal {F}} \rightarrow {\mathcal {B}}$ of ${\mathcal {F}}$ to a unital $C^*$ -algebra ${\mathcal {B}}$ and a family of isometries $s_j \in {\mathcal {B}}, j \in \mathbb {N}$ satisfying (3-3). Then, there exists a surjective $*$ -isomorphism $\Phi : {{\mathcal {O}}_{H_{\mathcal {F}}}} \rightarrow C^*(\pi ({\mathcal {F}}), \{s_j\}_{j\in \mathbb {N}})$ such that

(3-4) $$ \begin{align} \Phi(S_i) = s_i \quad \text{for } i \in \mathbb{N} \quad \text{and} \quad \Phi(\varphi_+(x)) =\pi(x) \quad \text{for } x \in {\mathcal{F}}. \end{align} $$

Hence, we have ${{\mathcal {O}}_{H_{\mathcal {F}}}} \cong C^*(\pi ({\mathcal {F}}), \{s_j\}_{j\in \mathbb {N}}).$

Proof. As in [Reference Brown and Ozawa3, Definition 4.6.14], the Cuntz–Pimsner algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ has universality for a covariant representation such that there exists a surjective $*$ -homomorphism $\Phi : {{\mathcal {O}}_{H_{\mathcal {F}}}} \rightarrow C^*(\tau (u)\mid u\in H_{\mathcal {F}})=C^*(\pi ({\mathcal {F}}), \{s_j\}_{j\in \mathbb {N}})$ satisfying

$$ \begin{align*} \Phi(T_u) = \tau(u) \quad \text{for } u \in H_{\mathcal{F}} \quad \text{and} \quad \Phi(\varphi_+(x)) =\pi(x) \quad \text{for } x \in {\mathcal{F}}. \end{align*} $$

As $ \tau (\xi _0\otimes e_k \otimes 1) =\pi (t_{\xi _0,\xi _0}) s_k \pi (1) = s_k, $ we obtain (3-4). As in [Reference Kumujian14, Theorem 2.8], the $C^*$ -algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ is simple, so that $\Phi : {{\mathcal {O}}_{H_{\mathcal {F}}}} \rightarrow C^*(\pi ({\mathcal {F}}), \{s_j\}_{j\in \mathbb {N}})$ is isomorphic.

The proposition above implies the following theorem.

Theorem 3.5. Let ${\mathcal {F}}$ be a separable unital $C^*$ -algebra containing $\mathcal {K}(H)$ as an essential ideal, and $f \in \mathcal {K}(H)$ a fixed minimal projection. The $C^*$ -algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ is the universal unique unital $C^*$ -algebra $C^*_{\text {univ}}({\mathcal {F}}, \{S_j\}_{j \in \mathbb {N}})$ generated by ${\mathcal {F}}$ and a family $\{S_j\}_{j \in \mathbb {N}} $ of isometries subject to the relations:

(3-5) $$ \begin{align} S_i^* S_i = 1_{\mathcal{F}} \quad \text{for } i\in \mathbb{N} \quad \text{and} \quad \sum_{j=1}^m S_j S_j^* < f \quad \text{for } m\in \mathbb{N}. \end{align} $$

4 Universality of the reciprocal algebra ${\widehat {{\mathcal {A}}}}$

Keep the situation that ${\mathcal {F}}$ is a separable unital $C^*$ -algebra containing $\mathcal {K}(H)$ as an essential ideal and fix a minimal projection $f \in \mathcal {K}(H)$ . Let ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ be the $C^*$ -algebra $C^*_{\text {univ}}({\mathcal {F}}, \{S_j\}_{j \in \mathbb {N}})$ as in Theorem 3.5. The unit $1_{\mathcal {F}}$ of ${\mathcal {F}}$ coincides with the unit $1_{{\mathcal {O}}_{H_{\mathcal {F}}}}$ of ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ . We define a $C^*$ -algebra $\mathcal {T}$ and a projection $e \in \mathcal {T}$ by

(4-1) $$ \begin{align} \mathcal{T} := S_1 {\mathcal{F}} S_1^*, \quad e:= S_1 f S_1^*, \end{align} $$

and a family $\{ T_j \}_{j \in \mathbb {N}}$ of partial isometries in $C^*_{\text {univ}}({\mathcal {F}}, \{S_j\}_{j \in \mathbb {N}})$ by

(4-2) $$ \begin{align} T_1 := S_1 f \quad \text{and} \quad T_j := S_j S_1^* \quad \text{for }\ j\ge 2. \end{align} $$

Hence, the unit $1_{\mathcal {T}}$ of the algebra $\mathcal {T}$ is $S_1 S_1^*$ . Note that $\mathcal {T}\not \subset \mathcal {F}$ .

Proof. (i) ( $T$ 1). For $j \ge 2$ , we have $ T_j^* T_j = S_1 S_j^* S_j S_1^* = S_1 1_{\mathcal {F}} S_1^* = 1_{\mathcal {T}}. $

( $T$ 2). We have $T_1 T_1^* = S_1 f S_1^* = e. $ For $m \ge 2$ ,

$$ \begin{align*} 1_{\mathcal{T}} + \sum_{j=2}^m T_j T_j^* = S_1 S_1^* + \sum_{j=2}^m S_jS_1^* S_1 S_j^* = \sum_{j=1}^m S_j S_j^* < f = f S_1^* S_1 f = T_1^* T_1. \end{align*} $$

The identities $ T_1 T_1^* = S_1 f S_1^* = V T_1^*, \, \, T_1^* T_1 = f = S_1^* S_1 f = V^* T_1, $ and $V V^* = S_1 S_1^* = S_1 1_{\mathcal {F}} S_1^* = 1_{\mathcal {T}}, \, \, V^* V = S_1^* S_1 = 1_{\mathcal {F}} (= 1_{{{\mathcal {O}}_{H_{\mathcal {F}}}}}) $ show us Identities ( $T$ 3) and ( $T$ 4).

(ii) It suffices to show that $ {{\mathcal {O}}_{H_{\mathcal {F}}}} = C^*(S_1{\mathcal {F}} S_1^*, \, S_1 f, \, \{S_j S_1^*\}_{2\le j \in \mathbb {N}}, \, S_1 ). $ Since ${{\mathcal {O}}_{H_{\mathcal {F}}}} = C^*({\mathcal {F}}, \{S_j \}_{j \in \mathbb {N}})$ , the inclusion relation ${{\mathcal {O}}_{H_{\mathcal {F}}}} \supset C^*(S_1{\mathcal {F}} S_1^*, \, S_1 f, \, \{S_j S_1^*\}_{2\le j \in \mathbb {N}}, \, S_1 )$ is obvious. Conversely, for $a \in {\mathcal {F}}$ and $ 2\le j \in \mathbb {N}$ ,

$$ \begin{align*} a = S_1^* S_1 a S_1^* S_1= S_1^*\cdot S_1 a S_1^* \cdot S_1, \quad S_j = S_j S_1^* S_1 = S_j S_1^*\cdot S_1, \end{align*} $$

so that ${{\mathcal {O}}_{H_{\mathcal {F}}}} \subset C^*(S_1{\mathcal {F}} S_1^*, \, S_1 f, \, \{S_j S_1^*\}_{2\le j \in \mathbb {N}}, \, S_1 ).$

Lemma 4.2. The corner $f {\mathcal {O}}_{H_{\mathcal {F}}} f$ of the $C^*$ -algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ is generated by

(4-3) $$ \begin{align} S_n {\mathcal{F}} S_n^*, \quad S_n f, \quad S_n S_i S_n^*, \quad n, i\in \mathbb{N}. \end{align} $$

Proof. We first note that the identity $f = (S_1 f)^* S_1 f$ holds. Since $\sum _{j=1}^m S_j S_j^* < f$ for all $m \in \mathbb {N}$ , we have $f S_n = S_n$ and $S_n^* f = S_n$ for all $n \in \mathbb {N}.$ Hence, it is obvious to see that all the elements written in (4-3) belong to $f {\mathcal {O}}_{H_{\mathcal {F}}} f.$ Let us denote $\mathbb {N}^* = \bigcup _{m=0}^\infty \mathbb {N}^m$ , where $\mathbb {N}^0=\{\emptyset \}, \mathbb {N}^m = \{(\mu _1,\ldots , \mu _m) \mid \mu _i \in \mathbb {N}\}$ . For $\mu =(\mu _1,\ldots , \mu _m) \in \mathbb {N}^m$ , we write $S_\mu = S_{\mu _1}\cdots S_{\mu _m}$ and $S_\mu =1_{{{\mathcal {O}}_{H_{\mathcal {F}}}}}$ for $\mu = \emptyset $ . Since $S_j^*S_i = \delta _{j,i}1_{{{\mathcal {O}}_{H_{\mathcal {F}}}}}$ , the linear span of elements of the form $f a_1 a_2 \cdots a_k f$ for $a_i \in {\mathcal {F}}$ or $a_i = S_\mu S_\nu ^*$ with $\mu , \nu \in \mathbb {N}^*$ is dense in $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ . As $1_{{{\mathcal {O}}_{H_{\mathcal {F}}}}} = 1_{\mathcal {F}} \in {\mathcal {F}}$ , one may assume that the element $f a_1 a_2 \cdots a_k f$ satisfies $a_1, a_3 \in {\mathcal {F}}$ and $a_2 = S_\mu S_\nu ^*$ with $\mu =(\mu _1,\ldots , \mu _m), \nu = (\nu _1,\ldots ,\nu _n) \in \mathbb {N}^*$ .

We have three cases.

Case 1: $\, m \ge 1$ and $n \ge 1.$ As $S_j = f S_j$ and $S_j^* = S_j^* f$ ,

$$ \begin{align*} & f a_1 a_2 a_3 \cdots a_k f \\ & \quad= f \cdot S_1^* S_1\cdot a_1\cdot S_1^* S_1 \cdot f S_\mu S_\nu^* f a_3 \cdots a_k f \\ & \quad= (S_1 f)^* S_1 a_1 S_1^*\cdot S_1 f \cdot S_{\mu_1} S_{\mu_2} \cdots S_{\mu_m}S_{\nu_n}^*\cdots S_{\nu_1}^* f a_3 \cdots a_k f, \end{align*} $$

where

$$ \begin{align*} S_{\mu_1} S_{\mu_2}\cdots S_{\mu_m}S_{\nu_n}^*\cdots S_{\nu_1}^* = S_{\mu_1} S_{\mu_2}S_{\mu_1}^*\cdot S_{\mu_1}S_{\mu_3} S_{\mu_1}^* \cdots S_{\mu_1}S_{\mu_m}S_{\mu_1}^*\cdot ( S_{\mu_1} S_{\nu_n}^*\cdots S_{\nu_1}^* f) \end{align*} $$

and

$$ \begin{align*} S_{\mu_1} S_{\nu_n}^*\cdots S_{\nu_1}^* f = S_{\mu_1} S_{\nu_n}^*S_{\mu_1}^*\cdot S_{\mu_1} S_{\nu_{n-1}} S_{\mu_1}^* \cdot S_{\mu_1}\cdots S_{\mu_1} S_{\nu_1}^* S_{\mu_1}^* \cdot S_{\mu_1} f. \end{align*} $$

Hence, the word $f a_1 a_2(= f a_1 f S_{\mu _1} S_{\mu _2}\cdots S_{\mu _m}S_{\nu _n}^*\cdots S_{\nu _1}^*) $ can be written in words of elements of the form of (4-3).

Case 2. $\, m=0$ and $n \ge 1.$ We then have

$$ \begin{align*} & f a_1 a_2 a_3 \cdots a_k f \\ & \quad= f a_1 S_\nu^* f a_3 \cdots a_k f \\ & \quad= f \cdot S_1^* S_1\cdot a_1 \cdot S_1^* S_1\cdot S_{\nu_n}^* \cdot S_1^* S_1 \cdots S_1^* S_1\cdot S_{\nu_1}^*\cdot S_1^* S_1 \cdot f a_3 \cdots a_k f \\ & \quad= (S_1 f)^* \cdot S_1 a_1 S_1^* \cdot (S_1 S_{\nu_n} S_1^*)^* \cdots (S_1 S_{\nu_1} S_1^*)^* \cdot S_1 f a_3 \cdots a_k f. \end{align*} $$

Hence, the word $f a_1 a_2(=f a_1 S_\nu ^*)$ can be written in words of elements of the form of (4-3).

Case 3. $\, m \ge 1$ and $n =0.$ We then have

$$ \begin{align*} & f a_1 a_2 a_3 \cdots a_k f \\ & \quad= f S_1^* S_1 a_1 S_1^* S_1 S_{\mu_1} S_1^* S_1 \cdots S_1^* S_1 S_{\mu_m}S_1^* S_1 a_3 S_1^* S_1 a_4 \cdots a_k f \\ & \quad= (S_1 f)^* \cdot S_1 a_1 S_1^* \cdot S_1 S_{\mu_1} S_1^* \cdot S_1 \cdots S_1^* \cdot S_1 S_{\mu_m}S_1^* \cdot S_1 a_3 S_1^* \cdot f S_1 a_4 \cdots a_k f. \end{align*} $$

Hence, the word $f a_1 S_{\mu _1}\cdots S_{\mu _m} S_1^* S_1 a_3 S_1^*$ can be written in words of elements of the form of (4-3).

Since $f S_1 f = f\cdot S_1 f$ , $ f S_1 S_j f = f S_1 S_jS_1^*\cdot S_1 f, $ and $ f S_1 S_j^* f = f S_1 S_j^* S_1^*\cdot S_1 f, $ by induction on the length $k$ of the word $f a_1 a_2 a_3 \cdots a_k f$ , we may conclude that the word $ f a_1 a_2 a_3 \cdots a_k f $ can be written in words of elements of the form of (4-3).

Lemma 4.3. The $C^*$ -algebra $f {\mathcal {O}}_{H_{\mathcal {F}}} f$ is generated by

$$ \begin{align*} S_1 {\mathcal{F}} S_1^*, \quad S_1 f, \quad S_1 S_i S_1^*, \quad i\in \mathbb{N}. \end{align*} $$

Proof. We set

$$ \begin{align*} {\mathcal{A}}_1 & := C^*(S_1 {\mathcal{F}} S_1, \, \, S_1 f,\, \, S_1 S_i S_1^* \mid i \in \mathbb{N}), \\ {\mathcal{A}}_* & := C^*(S_n {\mathcal{F}} S_n, \, \, S_n f, \, \, S_n S_i S_n^* \mid n, i \in \mathbb{N}) \end{align*} $$

and $ \mathcal {T}: = S_1 {\mathcal {F}} S_1^*, \, \, S:= S_1 f, \, \, R_i :=S_1 S_i S_1^*, \, i \in \mathbb {N}. $ By the preceding lemma, we know that $f {\mathcal {O}}_{H_{\mathcal {F}}} f ={\mathcal {A}}_*$ , and ${\mathcal {A}}_1 \subset {\mathcal {A}}_*$ is clear. We show that ${\mathcal {A}}_1\supset {\mathcal {A}}_*$ . For $S_i S_1^*$ with $i \in \mathbb {N}$ ,

$$ \begin{align*} S_i S_1^* = f S_i S_1^* = f S_1^* S_1 S_i S_1^* =S^*\cdot R_i \end{align*} $$

so that $S_i S_1^* \in {\mathcal {A}}_1$ . For $n \in \mathbb {N}$ and $a \in {\mathcal {F}}$ , as $S_1^* S_1 =1_{{\mathcal {O}}_{H_{\mathcal {F}}}}$ ,

$$ \begin{align*} S_n a S_n^* = S_n \cdot S_1^*S_1\cdot a \cdot S_1^*S_1\cdot S_n^* = S^* R_n \cdot S_1 a S_1^* \cdot R_n^* S \end{align*} $$

so that $S_n a S_n^* \in {\mathcal {A}}_1$ for all $n \in \mathbb {N}$ , and similarly the identity $ S_n S_i S_n^* = S^* R_n \cdot R_i \cdot R_n^* S $ holds so that $S_n S_i S_n^* \in {\mathcal {A}}_1$ for all $n, i \in \mathbb {N}$ . We also have

$$ \begin{align*} S_n f = S_n \cdot S_1^*S_1 \cdot f = S^* R_n \cdot S \end{align*} $$

so that $S_n f \in {\mathcal {A}}_1$ for all $n \in \mathbb {N}$ , showing that ${\mathcal {A}}_* \subset {\mathcal {A}}_1$ and, hence, ${\mathcal {A}}_* = {\mathcal {A}}_1.$

Recall that the $C^*$ -subalgebra $\mathcal {T}$ of ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ and a family $\{ T_j \}_{j \in \mathbb {N}}$ of partial isometries in ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ are defined by (4-1) and (4-2), respectively. Hence, we have the following lemma.

Lemma 4.4. The $C^*$ -algebra $f {\mathcal {O}}_{H_{\mathcal {F}}} f$ is generated by $\mathcal {T}$ and $T_j, j \in \mathbb {N}$ :

$$ \begin{align*} f {{\mathcal{O}}_{H_{\mathcal{F}}}} f = C^*(\mathcal{T}, \{T_j\}_{j\in \mathbb{N}}). \end{align*} $$

Proof. We have $ S_1 S_j S_1^* = S_1 f S_j S_1^* = T_1 T_j $ for $j \ge 2$ , and

$$ \begin{align*} S_1 S_1 S_1^* = S_1 f S_1 S_1^* =S_1 f S_1 S_2^* S_2 S_1^* = T_1 T_2^*T_2 \quad\text{for } j = 1. \end{align*} $$

However, we have $ T_j =(S_1 f)^* S_1 S_j S_1^* $ for $j \ge 2$ . By Lemma 4.3, we conclude that $ f {{\mathcal {O}}_{H_{\mathcal {F}}}} f = C^*(\mathcal {T}, \{T_j\}_{j \in \mathbb {N}}). $

Recall that the projection $e \in \mathcal {T}$ is defined by $e= S_1 f S_1^*\in \mathcal {T}$ .

Lemma 4.5. Assume that there exist a degenerate representation $\pi : \mathcal {T} \rightarrow {\mathcal {B}}({\frak {H}})$ and a family $t_j \in {\mathcal {B}}({\frak {H}}), j \in \mathbb {N}$ of partial isometries on a Hilbert space ${\frak {H}}$ satisfying

$$ \begin{align*} \left\{\begin{array}{@{}ll} (t1) \quad t_1^* t_1 = 1_{{\mathcal{B}}({\frak{H}})}, \quad t_j ^* t_j = \pi(1_{\mathcal{T}}) &\text{ for } j\ge 2, \\ (t2) \quad t_1 t_1^* = \pi(e), \quad \pi(1_{\mathcal{T}}) + \sum_{j=2}^m t_j t_j^* < 1_{{\mathcal{B}}({\frak{H}})} & \text{ for } 2 \le m \in \mathbb{N}. \end{array}\right. \end{align*} $$

Then, there exist a Hilbert space ${\widetilde {\frak {H}}} = {\frak {H}}\oplus {\frak {K}}$ for some Hilbert space ${\frak {K}}$ and an isometry $v \in {\mathcal {B}}({\widetilde {\frak {H}}})$ such that

$$ \begin{align*} \left\{\begin{array}{@{}ll} (t3) \quad t_1 t_1^* = v t_1^*, & t_1^* t_1 = v^* t_1, \\ (t4) \quad v v^* = \pi(1_{\mathcal{T}}), & v^* v = 1_{{\mathcal{B}}({\widetilde{\frak{H}}})}, \end{array}\right. \end{align*} $$

where ${\mathcal {B}}({\frak {H}}) = {\mathcal {B}}({\frak {H}}\oplus 0) \subset {\mathcal {B}}({\widetilde {\frak {H}}})$ .

Proof. Put ${\frak {K}} : = \pi (1_{\mathcal {T}} - e){\frak {H}}$ and ${\widetilde {\frak {H}}} = {\frak {H}} \oplus {\frak {K}}$ . Represent $C^*(\pi (\mathcal {T}), \{ t_j \}_{j \in \mathbb {N}})$ on ${\widetilde {\frak {H}}}$ by

$$ \begin{align*} x (h, k) := (x(h), 0) \quad \text{for } x \in C^*(\pi(\mathcal{T}), \{t_j\}_{j \in \mathbb{N}}),\, (h, k)\in{\frak{H}}\oplus{\frak{K}}. \end{align*} $$

Hence, $x \in C^*(\pi (\mathcal {T}), \{t_j\}_{j \in \mathbb {N}})$ is written as $x = \begin {bmatrix} x & 0 \\ 0 & 0 \end {bmatrix} \text { on } {\frak {H}} \oplus {\frak {K}}. $ Define

$$ \begin{align*} v := \begin{bmatrix} t_1 & \pi(1_{\mathcal{T}} -e) \\ 0 & 0 \end{bmatrix} \quad \text{on } {\frak{H}} \oplus {\frak{K}}. \end{align*} $$

Since

$$ \begin{align*} \pi(1_{\mathcal{T}} -e) t_1 =\pi(1_{\mathcal{T}} -e) t_1t_1^* t_1 = \pi(1_{\mathcal{T}} -e) \pi(e) t_1 =0, \end{align*} $$

it is direct to see that $v$ is an isometry on ${\widetilde {\frak {H}}}$ satisfying Conditions $(t3)$ and $(t4)$ .

Proposition 4.7. Keep the assumption of Lemma 4.5. Take the Hilbert space ${\widetilde {\frak {H}}}$ and the isometry $v$ on ${\widetilde {\frak {H}}}$ satisfying Conditions $(t3)$ and $(t4)$ in Lemma 4.5. Define the partial isometries $\tilde {s}_j, \, j \in \mathbb {N}$ on ${\widetilde {\frak {H}}}$ and a representation $\tilde {\pi }: {\mathcal {F}} \rightarrow {\mathcal {B}}({\widetilde {\frak {H}}})$ by setting

$$ \begin{align*} \tilde{s}_j := \begin{cases} v^* t_1 v & \text{ for } j=1, \\ v^* t_1 t_j v & \text{ for } j \ge 2, \end{cases} \quad \text{and} \quad \tilde{\pi}(a) : = v^* \pi(S_1a S_1^*)v \quad \text{for } a \in {\mathcal{F}}. \end{align*} $$

Then, we have the following properties.

1. (i) $\tilde {\pi }: {\mathcal {F}} \rightarrow {\mathcal {B}}({\widetilde {\frak {H}}})$ is a unital representation such that

   $$ \begin{align*} \tilde{s}_i^* \tilde{s}_i = 1_{{\mathcal{B}}({\widetilde{\frak{H}}})} \quad \text{for } i \in \mathbb{N} \quad \text{and} \quad \sum_{j=1}^m \tilde{s}_j \tilde{s}_j^* < \tilde{\pi}(f) \quad \text{for } m \in \mathbb{N}. \end{align*} $$

2. (ii) There exists an injective $*$ -homomorphism $\Phi : {{\mathcal {O}}_{H_{\mathcal {F}}}}\rightarrow {\mathcal {B}}({\widetilde {\frak {H}}})$ such that:

   1. (a) $ \Phi (S_i) = \tilde {s}_i \text { for } i \in \mathbb {N} $ and $ \Phi (a) = \tilde {\pi }(a) \text { for } a \in {\mathcal {F}}, $ and hence,

      $$ \begin{align*} {{\mathcal{O}}_{H_{\mathcal{F}}}} \cong C^*(\tilde{\pi}({\mathcal{F}}), \{\tilde{s}_j\}_{j\in \mathbb{N}}); \end{align*} $$

   2. (b) $\Phi (f {{\mathcal {O}}_{H_{\mathcal {F}}}} f) = C^*(\pi (\mathcal {T}), \{ t_j \}_{j \in \mathbb {N}})$ and, hence,

      $$ \begin{align*} f {{\mathcal{O}}_{H_{\mathcal{F}}}} f \cong C^*(\pi(\mathcal{T}), \{ t_j \}_{j \in \mathbb{N}}). \end{align*} $$

Proof. (i) The equalities

$$ \begin{align*} \tilde{\pi}(1_{\mathcal{F}}) = v^* \pi(S_1 1_{\mathcal{F}} S_1^*) v = v^*\pi(1_{\mathcal{T}}) v = v^* v v^* v = 1_{{\mathcal{B}}({\widetilde{\frak{H}}})} \end{align*} $$

show that $\pi : {\mathcal {F}} \rightarrow {\mathcal {B}}({\widetilde {\frak {H}}})$ is unital. Since $v^* t_1 = t_1^* t_1 = 1_{{\mathcal {B}}({\frak {H}})}$ and, hence, $ t_1^* v = 1_{{\mathcal {B}}({\frak {H}})}$ ,

$$ \begin{align*}& \tilde{s}_1^* \tilde{s}_1 = v^* t_1^* v v^* t_1 v = v^* 1_{{\mathcal{B}}({\frak{H}})}\cdot v = v^* v = 1_{{\mathcal{B}}({\widetilde{\frak{H}}})}, \quad \text{and} \\ &\tilde{s}_j^* \tilde{s}_j = v^* t_j^* t_1^* v v^* t_1 t_j v = v^* t_j^* 1_{{\mathcal{B}}({\frak{H}})} t_j v = v^* \pi(1_{\mathcal{T}}) v = 1_{{\mathcal{B}}({\widetilde{\frak{H}}})} \quad \text{for } j\ge 2. \end{align*} $$

We also have

$$ \begin{align*} &\quad\, \tilde{s}_1 \tilde{s}_1^* = v^* t_1 v v^* t_1^* v = v^* t_1 \pi(1_{\mathcal{T}}) t_1^* v, \quad \text{and}\qquad \\ & \tilde{s}_j \tilde{s}_j^* = v^* t_1 t_j \pi(1_{\mathcal{T}}) t_j^* t_1^* v = v^* t_1 t_j t_j^* t_1^* v \quad \text{for } j\ge 2 \end{align*} $$

so that

$$ \begin{align*} \tilde{s}_1 \tilde{s}_1^* + \sum_{j=2}^m \tilde{s}_j \tilde{s}_j^* = v^* t_1 \bigg( \pi(1_{\mathcal{T}}) + \sum_{j=2}^m t_j t_j^* \bigg) t_1^* v < v^* t_1 1_{{\mathcal{B}}({\frak{H}})} t_1^* v = v^* \pi(e) v = \tilde{\pi}(f). \end{align*} $$

(ii) Since the representation $\tilde {\pi }: {\mathcal {F}} \rightarrow {\mathcal {B}}({\widetilde {\frak {H}}})$ and the isometries $\{\tilde {s}_j \}_{j \in \mathbb {N}}$ satisfy (3-3), Proposition 3.3 tells us that there exists an injective $*$ -homomorphism $\Phi : {{\mathcal {O}}_{H_{\mathcal {F}}}}\rightarrow {\mathcal {B}}({\widetilde {\frak {H}}})$ satisfying $ \Phi (S_i) = \tilde {s}_i, \, i \in \mathbb {N} $ and $ \Phi (a) = \tilde {\pi }(a), \, a \in {\mathcal {F}}, $ which gives rise to an isomorphism from $ {{\mathcal {O}}_{H_{\mathcal {F}}}} $ to $C^*(\tilde {\pi }({\mathcal {F}}), \{\tilde {s}_j\}_{j\in \mathbb {N}}). $

We next prove that $\Phi (f {{\mathcal {O}}_{H_{\mathcal {F}}}} f) = C^*(\pi (\mathcal {T}), \{ t_j \}_{j \in \mathbb {N}})$ . Put $\tilde {f}: = \tilde {\pi }(f).$ In a completely similar manner to the proofs of Lemmas 4.2 and 4.3, one knows that

(4-4) $$ \begin{align} \tilde{f}\, C^*(\tilde{\pi}({\mathcal{F}}), \{\tilde{s}_j\}_{j\in \mathbb{N}})\, \tilde{f} = C^*(\tilde{s}_1 \tilde{\pi}({\mathcal{F}}) \tilde{s}^*_1, \tilde{s}_1 \tilde{f}, \{\tilde{s}_1 \tilde{s}_j\tilde{s}_1^* \}_{j \in \mathbb{N}}). \end{align} $$

So, by using a similar manner to the proof of Lemma 4.4, it suffices to prove that $\tilde {s}_1 \tilde {\pi }({\mathcal {F}}) \tilde {s}_1^* = \pi (\mathcal {T})$ , $\tilde {s}_1 \tilde {f} = t_1$ , and $\tilde {s}_j \tilde {s}_1^* =t_j$ for $j \ge 2.$ Since $ \tilde {s}_1 v^* = v^* t_1 vv^* = t_1^* t_1 \pi (1_{\mathcal {T}}) = \pi (1_{\mathcal {T}}), $

$$ \begin{align*} \tilde{s}_1 \tilde{\pi}({\mathcal{F}}) \tilde{s}_1^* =\tilde{s}_1 v^* \pi(\mathcal{T}) v \tilde{s}_1^* = \pi(\mathcal{T}). \end{align*} $$

We also have

$$ \begin{align*} & \tilde{s}_1 \tilde{f} = v^* t_1 \pi(e) v = t_1^* t_1\cdot t_1 t_1^* v = 1_{{\mathcal{B}}({\frak{H}})}\cdot t_1 t_1^* t_1 =t_1 \quad \text{and } \\ & \tilde{s}_j \tilde{s}_1^* = v^* t_1 t_j v v^* t_1^* v = 1_{{\mathcal{B}}({\frak{H}})} t_j \pi(1_{\mathcal{T}}) 1_{{\mathcal{B}}({\frak{H}})} =t_j \quad \text{for } j\ge 2. \end{align*} $$

By (4-4),

$$ \begin{align*} \tilde{f}\, C^*(\tilde{\pi}({\mathcal{F}}), \{\tilde{s}_j\}_{j\in \mathbb{N}})\, \tilde{f} = C^*(\pi(\mathcal{T}),\{ t_j \}_{j \in \mathbb{N}}). \end{align*} $$

As $\Phi (f {{\mathcal {O}}_{H_{\mathcal {F}}}} f ) = \tilde {f}\, C^*(\tilde {\pi }({\mathcal {F}}), \{\tilde {s}_j\}_{j\in \mathbb {N}})\, \tilde {f}$ , we have $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f \cong C^*(\pi (\mathcal {T}), \{ t_j \}_{j \in \mathbb {N}}).$

Recall that the unital $C^*$ -algebra $\mathcal {T}$ is defined by $S_1 {\mathcal {F}} S_1^*$ as a subalgebra of ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ and the projection $e \in \mathcal {T}$ is defined by $e= S_1 f S_1^*\in \mathcal {K}(H)$ .

Corollary 4.8. Suppose that there exist a representation $\pi : \mathcal {T} \rightarrow {\mathcal {B}}({\frak {H}})$ and a family $t_j \in {\mathcal {B}}({\frak {H}}),\, j \in \mathbb {N}$ of partial isometries on ${\frak {H}}$ satisfying

$$ \begin{align*} \left\{\begin{array}{@{}llll} (t1) & t_1^* t_1 = 1_{{\mathcal{B}}({\frak{H}})}, & t_j ^* t_j = \pi(1_{\mathcal{T}}) & \text{ for } j\ge 2, \\ (t2) & t_1 t_1^* = \pi(e), & \pi(1_{\mathcal{T}}) + \sum_{j=2}^m t_j t_j^* < 1_{{\mathcal{B}}({\frak{H}})} & \text{ for } 2 \le m \in \mathbb{N}. \end{array}\right. \end{align*} $$

Then, there exists an isomorphism $\Phi : f {{\mathcal {O}}_{H_{\mathcal {F}}}} f \rightarrow C^*(\pi (\mathcal {T}), \{ t_j \}_{j \in \mathbb {N}})$ .

Now we are in position to give a proof of Theorem 1.1.

Proof of Theorem 1.1.

(i) Let ${\mathcal {A}}$ be a unital Kirchberg algebra with finitely generated $\mathrm {K}$ -groups. One may take a separable unital nuclear UCT $C^*$ -algebra ${\mathcal {F}}$ satisfying (2-2) as in Section 2. By (2-4) and Lemma 4.4, the reciprocal dual ${\widehat {{\mathcal {A}}}}$ is generated by the unital subalgebra $\mathcal {T}$ and partial isometries $\{ T_j \}_{j \in \mathbb {N}}$ satisfying the relations in Lemma 4.1(i). As $\mathcal {T} = S_1 {\mathcal {F}} S_1^*$ and $e = S_1 f S_1^*$ with $S_1^* S_1 = 1_{\mathcal {F}}$ ,

(4-5) $$ \begin{align} (\mathrm{K}_0({\mathcal{F}}), [f]_0, \mathrm{K}_1({\mathcal{F}})) \cong (\mathrm{K}_0(\mathcal{T}), [e]_0, \mathrm{K}_1(\mathcal{T})), \end{align} $$

showing (1-1). In the relations ( $T2$ ) of Lemma 4.1(i), $T_1^* T_1 = f$ because $T_1 = S_1 f$ , which is the unit of the algebra $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ and written as $1_{{\widehat {{\mathcal {A}}}}}$ . Hence, the relations ( $T1$ ), ( $T2$ ) in Lemma 4.1(i) are those that appeared in (1-2). This shows that the $C^*$ -algebra ${\widehat {{\mathcal {A}}}}$ is the $C^*$ -algebra satisfying the conditions (a) and (b) of Theorem 1.1(i). Its universality and uniqueness with respect to the subalgebra $\mathcal {T}$ and (1-2) follow from Corollary 4.8.

(ii) Since

$$ \begin{align*} (\mathrm{Ext}_{\mathrm{s}}^1({\mathcal{A}}), [\iota_{\mathcal{A}}(1)]_s, \mathrm{Ext}_{\mathrm{s}}^0({\mathcal{A}})) \cong (\mathrm{K}_0({\widehat{{\mathcal{A}}}}), [1_{{\widehat{{\mathcal{A}}}}}]_0, \mathrm{K}_1({\widehat{{\mathcal{A}}}})), \end{align*} $$

together with (4-5),

$$ \begin{align*} (\mathrm{K}_0(\mathcal{T}), [e]_0, \mathrm{K}_1(\mathcal{T})) \cong (\mathrm{K}_0({\widehat{{\mathcal{A}}}}), [1_{{\widehat{{\mathcal{A}}}}}]_0, \mathrm{K}_1({\widehat{{\mathcal{A}}}})), \end{align*} $$

because of (2-2). As the $C^*$ -algebra ${\widehat {{\mathcal {A}}}}$ is a Kirchberg algebra, its isomorphism class depends only on its $K$ -theory data, together with the position of the class of the unit in $\mathrm {K}_0$ -group [Reference Kirchberg13, Reference Phillips23], so the isomorphism class of ${\widehat {{\mathcal {A}}}}$ does not depend on the choice of the unital $C^*$ -algebra $\mathcal {T}$ as long as it satisfies (1-1).

5 Ergodic automorphisms on ${\widehat {{\mathcal {A}}}}$ in terms of the generators

In [Reference Matsumoto and Sogabe19], it was proved that there exists an aperiodic ergodic automorphism of an arbitrary unital Kirchberg algebra. In this section, we concretely realize the ergodic automorphism on the Kirchberg algebra with finitely generated $\mathrm {K}$ -groups in terms of their generators in the universal representation stated in Theorem 1.1. It induces a concrete description of an ergodic automorphism of the Cuntz algebra ${\mathcal {O}}_2$ , which is stated in Section 7.

As in Theorem 3.5, let us represent ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ as the universal $C^*$ -algebra $C^*_{\text {univ}}({\mathcal {F}}, \{ S_n \}_{n \in \mathbb {Z}})$ with relations:

$$ \begin{align*} S_n^* S_n = 1 \quad \text{for } n \in \mathbb{Z} \quad \text{and} \quad \sum_{|n| < m} S_n S_n^* < f \quad \text{for any } m \in \mathbb{N}, \end{align*} $$

where $f \in \mathcal {K}(H) \subset {\mathcal {F}}$ is a fixed minimal projection. We note that in the above description, the index set of the isometries $\{S_n \}_{n \in \mathbb {Z}}$ is the integer group $\mathbb {Z}$ instead of $\mathbb {N}$ and $S_0$ plays the role of $S_1$ in the previous section. By Theorem 1.1, the reciprocal dual ${\widehat {{\mathcal {A}}}}$ of ${\mathcal {A}}$ is given by the corner $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ of ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ . Put

$$ \begin{align*} \mathcal{T}: = S_0 {\mathcal{F}} S_0^*,\quad t_0 : = S_0 f,\quad e:= S_0 f S_0^*, \quad \text{and} \quad t_n := S_n S_0^* \quad \text{for } n \in \mathbb{Z} \text{ with } n \ne 0. \end{align*} $$

We have shown that ${\widehat {{\mathcal {A}}}}$ is realized as the universal $C^*$ -algebra $C^*_{\text {univ}}(\mathcal {T}, \{t_n\}_{n \in \mathbb {Z}})$ satisfying (1-2) such that

(5-1) $$ \begin{align} \left\{\begin{array}{@{}llll} (1) & t_0^* t_0 = 1_{\widehat{{\mathcal{A}}}}, & t_j ^* t_j = 1_{\mathcal{T}} & \text{ for }\, j \in \mathbb{Z} \text{ with } j \ne 0, \\ (2) & t_0 t_0^* =e, & 1_{\mathcal{T}} + \sum_{0<|j| \le m} t_j t_j^* < 1_{\widehat{{\mathcal{A}}}} & \text{ for } m \in \mathbb{N}. \end{array}\right. \end{align} $$

Define an automorphism $\theta $ on the $C^*$ -algebra $C^*_{\text {univ}}({\mathcal {F}}, \{ S_n \}_{n \in \mathbb {Z}})$ by

(5-2) $$ \begin{align} \begin{cases} \theta(a) = a & \text{ for } a \in {\mathcal{F}},\\ \theta(S_n) = S_{n+1} & \text{ for } n \in \mathbb{Z}. \end{cases} \end{align} $$

By the universality of the $C^*$ -algebra ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ , $\theta $ gives rise to an automorphism on ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ . We know the following lemma proved in [Reference Matsumoto and Sogabe19, Lemmas 3.2 and 3.3].

Since $\theta $ satisfies $\theta (f) = f$ , it induces an automorphism written $\hat {\theta }$ on ${\widehat {{\mathcal {A}}}}$ by restricting $\theta $ to the subalgebra $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ . The automorphism $\hat {\theta }$ on ${\widehat {{\mathcal {A}}}}$ is nothing but that constructed in [Reference Matsumoto and Sogabe19, Theorem 3.1]. Lemma 5.1 together with the minimality of the projection $f$ in ${\mathcal {F}}$ directly tells us that the fixed point algebra $(f {{\mathcal {O}}_{H_{\mathcal {F}}}} f)^{\theta }$ of $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ under $f$ is a scalar multiple of the unit $f$ (that is, $f\mathcal {F}f=\mathbb {C}f$ ). The following result was proved in [Reference Matsumoto and Sogabe19] in a more general setting.

We write $\hat {\theta }$ in terms of $\mathcal {T} $ and $ t_n , n\in \mathbb {Z}$ in the following way. For $b \in \mathcal {T} = S_0 {\mathcal {F}} S_0^*$ , so that $ b = S_0 a S_0^*$ for some $ a ( = S_0^* b S_0) \in {\mathcal {F}}$ , we then have for $n \in \mathbb {Z}$ with $n \ne 0$ ,

$$ \begin{align*} \hat{\theta}^n(b) = \theta^n(S_0 a S_0^*) = S_n a S_n^* = S_nS_0^* S_0 a S_0^* S_0 S_n^* = t_n b t_n^*, \end{align*} $$

so that $\hat {\theta }^n(b) = t_n b t_n^*$ for $b \in \mathcal {T}$ and $ n \in \mathbb {Z}, n \ne 0$ . In particular, $\hat {\theta }(b) = t_1 b t_1^*$ for $b \in \mathcal {T}$ . We note that $ b = \hat {\theta }^0(b) \ne t_0 b t_0^* = S_0 b S_0^*$ . Also,

$$ \begin{align*} \hat{\theta}(t_0 ) = \theta(S_0)\theta(f)= S_1 S_0^* S_0 f = t_1 t_0, \quad \hat{\theta}(t_{-1} ) = \theta(S_{-1})\theta(S_0^*)= S_0 S_1^* = t_1^*. \end{align*} $$

Similarly, we have for $n \in \mathbb {Z}$ with $n \ne 0, -1$ ,

$$ \begin{align*} \hat{\theta}(t_n ) = \theta(S_n S_0^*) = S_{n+1} S_1^* = S_{n+1} S_0^* S_0 S_1^* = t_{n+1} t_1^*. \end{align*} $$

Therefore, we have the following proposition.

Proposition 5.3. In the reciprocal dual ${\widehat {{\mathcal {A}}}} = C^*_{\text {univ}}(\mathcal {T}, \{ t_n \}_{n \in \mathbb {Z}})$ , the correspondence $\hat {\theta }:{\widehat {{\mathcal {A}}}} \rightarrow {\widehat {{\mathcal {A}}}}$ defined by

$$ \begin{align*} \begin{cases} \hat{\theta}(b)\, = t_1 b t_1^* \quad \text{for } b \in \mathcal{T}, \\ \hat{\theta}(t_0)\, = t_1 t_0, \\ \hat{\theta}(t_{-1}) = t_1^*, \\ \hat{\theta}(t_n) \,\, = t_{n+1} t_1^* \quad \text{for } n \in \mathbb{Z}, n \ne 0, -1 \end{cases} \end{align*} $$

gives rise to an aperiodic ergodic automorphism on ${\widehat {{\mathcal {A}}}}$ .

6 Invariant states

In this section, we show that a $\hat {\theta }$ -invariant state on ${\widehat {{\mathcal {A}}}}$ is unique, so that the automorphism $\hat {\theta }$ on ${\widehat {{\mathcal {A}}}}$ is uniquely ergodic. In [Reference Matsumoto and Sogabe19, Theorem 1.5], it is shown that there exists a $\hat {\theta }$ -invariant state on ${\widehat {{\mathcal {A}}}}$ in a more general setting. The construction of the $\hat {\theta }$ -invariant state in our setting is as follows. Let ${\mathcal {F}}$ be a separable unital nuclear UCT $C^*$ -algebra containing $\mathcal {K}(H)$ as an essential ideal and $f \in \mathcal {K}(H)$ as a minimal projection satisfying (2-2). Then, we have a $C^*$ -algebra $\mathcal {T}_{H_{\mathcal {F}}}$ as in the proof of Lemma 3.1. Let $E_{\mathcal {F}}: \mathcal {T}_{H_{\mathcal {F}}}\rightarrow {\mathcal {F}}$ be the conditional expectation arising from a projection on the Fock space ${\mathcal {E}}_{\mathcal {F}} ={\mathcal {F}} \oplus \bigoplus _{n=1}^\infty H_{\mathcal {F}}^{\otimes n}$ onto the subspace ${\mathcal {F}}$ . As $f$ is a minimal projection in ${\mathcal {F}}$ , there exists a scalar $\varphi _{{\widehat {{\mathcal {A}}}}}(f x f) \in \mathbb {C}$ for $x \in {\mathcal {O}}_{H_{\mathcal {F}}}$ satisfying

$$ \begin{align*} f E_{\mathcal{F}}(f x f) f = \varphi_{{\widehat{{\mathcal{A}}}}}(f x f) f \quad \text{for } x \in \mathcal{T}_{H_{\mathcal{F}}}. \end{align*} $$

Under the identification between $\mathcal {T}_{H_{\mathcal {F}}}$ and ${\mathcal {O}}_{H_{\mathcal {F}}}$ , we have a state $\varphi _{{\widehat {{\mathcal {A}}}}}$ on $f {\mathcal {O}}_{H_{\mathcal {F}}} f = {\widehat {{\mathcal {A}}}}$ , which was proved in [Reference Matsumoto and Sogabe19] to be invariant under the automorphism $\hat {\theta }$ .

Let us represent the reciprocal algebra ${\widehat {{\mathcal {A}}}}$ as $C^*_{\text {univ}}(\mathcal {T}, \{ t_n \}_{n \in \mathbb {Z}})$ as in the previous section. We show the following proposition.

To prove the proposition, we provide several lemmas. The first one is obtained from (5-1).

The next one is straightforward from Proposition 5.3.

Lemma 6.3.

$$ \begin{align*} \hat{\theta}(t_n t_n^*) = \begin{cases} t_1 e t_1^* & \text{for } n=0, \\ 1_{\mathcal{T}} & \text{for } n= -1, \\ t_{n+1} t_{n+1}^* & \text{for } n\ne 0, -1. \end{cases} \end{align*} $$

By using the lemma above, we have the following lemma.

Proof. We first see that $\varphi (t_1 t_1^*) =0.$ Put $\epsilon =\varphi (t_1 t_1^*).$ By Lemma 6.3, we have for $n \in \mathbb {N}$ ,

(6-1) $$ \begin{align} \varphi(t_{n+1} t_{n+1}^*) =\varphi\circ \hat{\theta}^n(t_1 t_1^*) =\varphi(t_1 t_1^*) =\epsilon. \end{align} $$

The inequality $1_{\mathcal {T}} + \sum _{n=1}^m t_j t_j^* < 1_{{\widehat {{\mathcal {A}}}}}$ tells us that $\varphi (1_{\mathcal {T}}) + m \epsilon <1$ for all $m \in \mathbb {N}$ , so that we have $\epsilon =0$ , and (6-1) shows $\varphi (t_{n} t_{n}^*) =0$ for all $n \in \mathbb {N}$ . Since

$$ \begin{align*} 0 \le \varphi(t_0 t_0^*) =\varphi\circ \hat{\theta}(t_0 t_0^*) =\varphi(t_1e t_1^*) \le \varphi(t_1 t_1^*), \end{align*} $$

we have $\varphi (t_0 t_0^*) =0.$ We also have

(6-2) $$ \begin{align} 0 \le \varphi(t_{-1} t_{-1}^*) =\varphi\circ \hat{\theta}^2(t_{-1} t_{-1}^*) =\varphi\circ \hat{\theta}(1_{\mathcal{T}}) =\varphi(t_1 1_{\mathcal{T}} t_1^*) \le \varphi(t_1 t_1^*) \end{align} $$

and, similarly for $n \ge 2$ ,

$$ \begin{align*} \varphi(t_{-n} t_{-n}^*) =\varphi\circ \hat{\theta}^{n-1}(t_{-n} t_{-n}^*) =\varphi(t_{-1} t_{-1}^*) =0. \end{align*} $$

Therefore, $\varphi (t_j t_j^*) =0$ for all $j \in \mathbb {Z}.$ As in (6-2), we have $\varphi (1_{\mathcal {T}}) = \varphi \circ \hat {\theta }(1_{\mathcal {T}}) =0$ , so that

$$ \begin{align*} |\varphi(b)| = |\varphi( 1_{\mathcal{T}} b )| \le \varphi(1_{\mathcal{T}})^{\frac{1}{2}} \varphi(b^* b)^{\frac{1}{2}} =0 \quad \text{for } b \in \mathcal{T}. \\[-36pt] \end{align*} $$

Recall that $e\in \mathcal {T}$ is a minimal projection (that is, $e\mathcal {T} e=\mathbb {C}e$ ). Let $a\in C^*_{\text {univ}}(\mathcal {T}, \{t_j\}_{j \in \mathbb {Z}})$ be of the form

(6-3) $$ \begin{align} a = a_1 a_2 \cdots a_m, \quad \text{where } a_i \in \mathcal{T} \cup \{ t_j \}_{j \in \mathbb{Z}} \cup \{ t_j^* \}_{j \in \mathbb{Z}}. \end{align} $$

Assume that $\varphi (a) \ne 0$ . If $a_1 \in \mathcal {T}\cup \{ t_j \}_{j \in \mathbb {Z}}\cup \{ t_j^* \}_{j \in \mathbb {Z}, j\ne 0}$ , the inequality

$$ \begin{align*} |\varphi(a) | = |\varphi(a_1 a_2 \cdots a_m ) | \le \varphi(a_1 a_1^*)^{\frac{1}{2}} \varphi((a_2 \cdots a_m)^* (a_2 \cdots a_m) )^{\frac{1}{2}} \end{align*} $$

together with Lemma 6.4 show us $\varphi (a_1 a_1^*) =0$ so that $\varphi (a)=0$ , which is a contradiction to the hypothesis that $\varphi (a) \ne 0.$ If $a_m \in \mathcal {T}\cup \{ t_j^* \}_{j \in \mathbb {Z}}\cup \{ t_j \}_{j \in \mathbb {Z}, j \ne 0}$ , the inequality

$$ \begin{align*} |\varphi(a) | = |\varphi(a_1 \cdots a_{m-1} a_m ) | \le \varphi((a_1 \cdots a_{m-1}) (a_1 \cdots a_{m-1})^*) ^{\frac{1}{2}} \varphi(a_m^* a_m)^{\frac{1}{2}} \end{align*} $$

similarly shows us that $\varphi (a_m^* a_m)=0$ and, hence, $\varphi (a) =0$ . Therefore, we have $a_m =t_0$ as well as $a_1 = t_0^*$ , so that the element $a$ must be of the form

(6-4) $$ \begin{align} a = t_0^* a_2 \cdots a_{m-1}t_0 \quad \text{where}\ a_i \in \mathcal{T} \cup \{ t_j \}_{j \in \mathbb{Z}} \cup \{ t_j^* \}_{j \in \mathbb{Z}}. \end{align} $$

If $a_2, \ldots ,a_{m-1} \in \mathcal {T}$ , then by putting $b:= a_2 \cdots a_{m-1} \in \mathcal {T}$ , we have $ a = t_0^* b t_0$ , so that by Lemma 6.2, we see that $a = c 1_{{\widehat {{\mathcal {A}}}}}$ for some $c \in \mathbb {C}$ , showing the desired conclusion of Proposition 6.1. Henceforth, we assume that there exists $k$ with $ 2 \le k < m-1$ such that

$$ \begin{align*} a_2, \ldots,a_k \in \mathcal{T} \quad \text{and} \quad a_{k+1} \not\in \mathcal{T}. \end{align*} $$

Put $b_1 := a_2 \cdots a_k \in \mathcal {T}$ . In case $a_2 \not \in \mathcal {T}$ , we take the number $k$ as $k=1$ , and then $t_0 b_1$ is written as $t_0$ . Hence, (6-4) goes to

(6-5) $$ \begin{align} a = t_0^* b_1 a_{k+1} a_{k+2}\cdots a_{m-1}t_0, \end{align} $$

where $b_1 \in \mathcal {T}, \, a_{k+1} \in \{ t_j \}_{j \in \mathbb {Z}} \cup \{ t_j^* \}_{j \in \mathbb {Z}} $ and $ a_{k+2}, \ldots , a_{m-1} \in \mathcal {T} \cup \{ t_j \}_{j \in \mathbb {Z}} \cup \{ t_j^* \}_{j \in \mathbb {Z}}. $ We provide one more lemma to prove Proposition 6.1.

Lemma 6.5. For the expression (6-5) of the element $a$ , there exist $b_1\in \mathcal {T}$ and $k \ge 1$ such that

$$ \begin{align*} & t_0^* b_1 a_{k+1} a_{k+2}\cdots a_{m-1}t_0 \\ &\quad= \begin{cases} (1) & t_0^*a_{k+3} a_{k+4}\cdots a_{m-1}t_0 \, \, (=a_{k+2} a_{k+3}\cdots a_{m-1}t_0 ) \quad \text{if } a_{k+1} = t_0, \\ (2) & 0 \quad\quad\quad\quad\quad\quad\qquad\qquad\, \text{if } a_{k+1} = t_i, \, i\ne 0, \\ (3) & t_0^* b_1 t_0^* a_{k+2} a_{k+3}\cdots a_{m-1}t_0 \quad \text{if } a_{k+1} = t_0^*, \\ (4) & 0 \qquad\qquad\qquad\qquad\qquad \text{if } a_{k+1} = t_{j_1}^*, \, j_1\ne 0, \, \, a_{k+2} \in \mathcal{T}, \\ (5) & t_0^* b_1 a_{k+3}\cdots a_{m-1}t_0 \qquad\quad\ \text{if } a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_i \text{ with } i =j_1, \\ (6) & 0 \qquad\qquad\qquad\qquad\qquad \text{if } a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_i \text{ with } i \ne j_1, \\ (7) & t_0^* b_1 t_{j_1}^* t_0^* a_{k+3}\cdots a_{m-1}t_0 \qquad\! \text{if } a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_0^*, \\ (8) & 0 \qquad\qquad\qquad\qquad\qquad \text{if } a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_i^*, \, i \ne 0, \\ \end{cases} \end{align*} $$

up to scalar multiples.

Proof. (1), (2) Assume that $a_{k+1} = t_i$ for some $i \in \mathbb {Z}$ . By Lemma 6.2(iv),

$$ \begin{align*} t_0^* b_1 t_i^* = \begin{cases} c 1_{{\widehat{{\mathcal{A}}}}} \text{ for some } c \in {\mathbb{C}} & \text{ if } i=0, \\ 0 & \text{ otherwise. } \end{cases} \end{align*} $$

For $i=0$ ,

(6-6) $$ \begin{align} t_0^* b_1 a_{k+1} a_{k+2}\cdots a_{m-1}t_0 = c a_{k+2}\cdots a_{m-1}t_0. \end{align} $$

By applying the preceding argument to reduce (6-4) from (6-3) to (6-6), we have $a_{k+2} = t_0^*$ .

Condition (3) is obvious. (4) Assume that $a_{k+1} = t_{j_1}^*, \, j_1\ne 0, \, \, a_{k+2} \in \mathcal {T}.$ By Lemma 6.2(i), we have $a_{k+1}a_{k+2} =0.$

(5), (6) Assume that $a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_i$ for some $i \in \mathbb {Z}.$ By Lemma 6.2(iii),

$$ \begin{align*} a_{k+1}a_{k+2} = t_{j_1}^* t_i = \begin{cases} 1_{\mathcal{T}} & \text{ if } i= j_1, \\ 0 & \text{ otherwise}, \end{cases} \end{align*} $$

so that Conditions (5) and (6) hold.

(7), (8) Assume that $a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_i^*$ for some $i \in \mathbb {Z}$ . As $j_1 \ne 0$ , by Lemma 6.2(ii),

$$ \begin{align*} a_{k+1}a_{k+2} = t_{j_1}^* t_i^* = \begin{cases} t_{j_1}^* t_0^* & \text{ if } i= 0, \\ 0 & \text{ if } i \ne 0, \end{cases} \end{align*} $$

so that Conditions (7) and (8) hold.

Proof of Proposition 6.1.

Now, assume that an element $a\in C^*_{\text {univ}}(\mathcal {T}, \{t_j\}_{j \in \mathbb {Z}})$ is of the form (6-3) such that $\varphi (a) \ne 0$ . Since $a \ne 0$ , Lemma 6.5 implies that the cases (2), (4), (6), and (8) are excluded. Hence, for the word $a$ having the form (6-5), we have the following four cases:

(6-7) $$ \begin{align} \begin{cases} (1) & t_0^*a_{k+3} a_{k+4}\cdots a_{m-1}t_0 \qquad\ \ \, \text{if }\, a_{k+1} = t_0, \\ (3) & t_0^* b_1 t_0^* a_{k+2} a_{k+3}\cdots a_{m-1}t_0 \quad \text{if } a_{k+1} = t_0^*, \\ (5) & t_0^* b_1 a_{k+3}\cdots a_{m-1}t_0 \qquad\quad\hspace{5pt} \text{if } a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, a_{k+2}= t_i \text{ with } i =j_1, \\ (7) & t_0^* b_1 t_{j_1}^* t_0^* a_{k+3}\cdots a_{m-1}t_0 \qquad \text{if } a_{k+1} = t_{j_1}^*,\, j_1\ne 0, \, \, a_{k+2}= t_0^*. \end{cases} \end{align} $$

In (6-4), the number of elements in $\mathcal {T} \cup \{t_i\}\cup \{t_i^*\}$ between the leftmost $t_0^*$ and the rightmost $t_0$ is $m-2.$ However, we have the numbers of elements in $\mathcal {T}\cup \{t_i\}\cup \{t_i^*\}$ as in (6-7):

$\bullet $ between the leftmost $t_0^*$ and the rightmost $t_0$ in case (1) is $m-k-3;$

$\bullet $ between the second leftmost $t_0^*$ and the rightmost $t_0$ in case (3) is $m-k-2;$

$\bullet $ between the leftmost $t_0^*$ and the rightmost $t_0$ in case (5) is $m-k-2$ ;

$\bullet $ between the second leftmost $t_0^*$ and the rightmost $t_0$ in case (7) is $m-k-3.$

In each case in the above four, the numbers are less than $m-2$ because $1\le k < m-1$ .

Although the reduction of $a$ as in Lemma 6.5 is a procedure from the leftmost of $A$ , we may symmetrically apply the discussion of Lemma 6.5 to $a$ from the rightmost, or apply it to $a^*$ instead of $a$ . By alternatively applying the reductions of Lemma 6.5 from the leftmost and the rightmost of $a$ toward the middle of the word, two-sided inductions from left and right on the numbers of elements in $\mathcal {T} \cup \{t_i\}\cup \{t_i^*\}$ between $t_0^*$ and $t_0$ work, and we then obtain the following expression for $a$ up to a scalar multiple:

(6-8) $$ \begin{align} a = t_0^* b_1 {\tilde{t}}_{j_1}^* b_2 {\tilde{t}}_{j_2}^* \cdots b_r {\tilde{t}}_{j_r}^* t_0^* t_0 {\tilde{t}}_{j_{-s}} b_{-s} \cdots {\tilde{t}}_{j_{-2}} b_{-2} {\tilde{t}}_{j_{-1}} b_{-1} t_0 \end{align} $$

for some $j_1, \ldots , j_r, \, j_{-1}, \ldots , j_{-s} \in \mathbb {Z}$ with $r,s \in \mathbb {N},$ where

$$ \begin{align*} {\tilde{t}}_{j_{\ell}}^* = t_0^* \text{ or } t_{j_{\ell}}^*t_0^*, \quad {\tilde{t}}_{j_{-{\ell}}} = t_0 \text{ or } t_0 t_{j_{-{\ell}}} \end{align*} $$

and $b_1,\ldots , b_r, \, b_{-1}, \ldots , b_{-s} \in \mathcal {T}$ . As $t_0^* t_0= 1_{{\widehat {{\mathcal {A}}}}}$ , which is in the middle of (6-8), by using Lemma 6.2(i), (iii), (iv) suitably many times, we finally obtain that the element expressed as (6-8) is reduced to the following form:

$$ \begin{align*} a = c \overbrace{t_0^*\cdots t_0^*}^{p}\overbrace{t_0\cdots t_0}^{q} \quad \text{for some } c \in {\mathbb{C}} \text{ and } p, q \in \mathbb{N}. \end{align*} $$

As $| \varphi (t_0^n)| \le \varphi (t_0 t_0^*)^{\frac 12}\varphi {(t_0^{n-1} t_0^{n-1*})}^{\frac 12}, $ we have $\varphi (t_0^{n}) = 0$ and similarly $ \varphi (t_0^{*n}) = 0$ for $n \in \mathbb {N}$ by Lemma 6.4. We thus conclude that $p=q$ and $a=1_{{\widehat {{\mathcal {A}}}}}$ up to a scalar multiple.

Proof of Theorem 1.2.

Since the set of linear combinations of finite words of elements of $\mathcal {T} \cup \{ t_j\}_{j \in \mathbb {Z}} \cup \{ t_j^* \}_{j \in \mathbb {Z}}$ is dense in ${\widehat {{\mathcal {A}}}}$ , Proposition 6.1 shows that a $\hat {\theta }$ -invariant state of $\widehat {{\mathcal {A}}}$ is unique and it has to be $\varphi _{\widehat {{\mathcal {A}}}}$ .

We next show that the state $\varphi _{{\widehat {{\mathcal {A}}}}}$ on ${\widehat {{\mathcal {A}}}}$ is pure. Let ${\mathcal {F}}$ be a separable unital $C^*$ -algebra containing $\mathcal {K}(H)$ as an essential ideal and $f \in \mathcal {K}(H)$ a minimal projection satisfying (2-2). Take a unit vector $\xi _0 \in H$ such that $ f = \xi _0 \otimes \xi _0^*$ . Let $H_{\mathcal {F}}$ be the Hilbert $C^*$ -bimodule $H \otimes _{\mathbb {C}}\ell ^2(\mathbb {Z}) \otimes _{\mathbb {C}} {\mathcal {F}}$ over ${\mathcal {F}}$ . One may regard $f$ as an element of ${\mathcal {O}}_{H_{\mathcal {F}}}= \mathcal {T}_{H_{\mathcal {F}}}$ as in (2-3). Let $L^2({\widehat {{\mathcal {A}}}}, \varphi _{{\widehat {{\mathcal {A}}}}})$ be the Hilbert space of the Gelfand--Naimark--Segal (GNS) representation of ${\widehat {{\mathcal {A}}}}$ by the state $\varphi _{{\widehat {{\mathcal {A}}}}}$ . Under the identification ${\widehat {{\mathcal {A}}}} = f {\mathcal {O}}_{H_F} f$ as in (2-4), one may naturally identify $L^2({\widehat {{\mathcal {A}}}}, \varphi _{{\widehat {{\mathcal {A}}}}})$ with the Hilbert space spanned by

$$ \begin{align*} fF(H_{\mathcal{F}})f &:= f{\mathcal{F}} f\Omega \oplus (\xi_0 \otimes_{\mathbb{C}} \ell^2(\mathbb{Z}) \otimes_{\mathbb{C}} {\mathcal{F}} f) \\ & \qquad\oplus \bigoplus_{k=2}^\infty (\xi_0 \otimes_{\mathbb{C}}\ell^2(\mathbb{Z}) \otimes_{\mathbb{C}} {\mathcal{F}}) \otimes (H \otimes_{\mathbb{C}}\ell^2(\mathbb{Z}) \otimes_{\mathbb{C}} {\mathcal{F}})^{\otimes k-2}\\ & \qquad\otimes (H \otimes_{\mathbb{C}}\ell^2(\mathbb{Z}) \otimes_{\mathbb{C}} {\mathcal{F}} f) \end{align*} $$

through the correspondence

$$ \begin{align*} f T_\xi f \in f \mathcal{T}_{H_{\mathcal{F}}} f = f {\mathcal{O}}_{H_{\mathcal{F}}} f \longrightarrow f \xi f \in f F(H_{\mathcal{F}}) f \quad \text{for } \xi \in H_{\mathcal{F}}, \end{align*} $$

where $\Omega $ is a vacuum vector. Note that $fF(H_{\mathcal {F}})f$ is a right Hilbert $f{\mathcal {F}} f$ -module and $f{\mathcal {F}} f=\mathbb {C}f$ . For any $X \in {\mathcal {B}}(L^2({\widehat {{\mathcal {A}}}}, \varphi _{{\widehat {{\mathcal {A}}}}}))\cap ({\widehat {{\mathcal {A}}}})^\prime $ and $\xi \in (\xi _0 \otimes _{\mathbb {C}}\ell ^2(\mathbb {Z}) \otimes _{\mathbb {C}} {\mathcal {F}} f) \cup (\xi _0 \otimes _{\mathbb {C}}\ell ^2(\mathbb {Z}) \otimes _{\mathbb {C}} {\mathcal {F}})\otimes (H \otimes _{\mathbb {C}}\ell ^2(\mathbb {Z}) \otimes _{\mathbb {C}} {\mathcal {F}})^{\otimes k-2} \otimes (H \otimes _{\mathbb {C}}\ell ^2(\mathbb {Z}) \otimes _{\mathbb {C}} {\mathcal {F}} f) $ with $k\ge 2$ , we have $T_\xi ^* \Omega =0$ so that

$$ \begin{align*} \langle X \Omega \mid \xi \rangle = \langle X \Omega \mid T_\xi \Omega \rangle = \langle X T_\xi^* \Omega \mid \Omega \rangle = 0 \end{align*} $$

so that $ X \Omega = \lambda \Omega $ for some $\lambda \in \mathbb {C}.$ Since $X\xi = X T_\xi \Omega = T_\xi X\Omega = \lambda \xi $ , we have $X = \lambda \in \mathbb {C}$ , showing that ${\mathcal {B}}(L^2({\widehat {{\mathcal {A}}}}, \varphi _{{\widehat {{\mathcal {A}}}}}))\cap {\widehat {{\mathcal {A}}}}^\prime = \mathbb {C}$ . Hence, the state $\varphi _{{\widehat {{\mathcal {A}}}}}$ is pure.

Since any unital Kirchberg algebra ${\mathcal {A}}$ with finitely generated $\mathrm {K}$ -groups is the reciprocal dual of the reciprocal dual of itself (that is, ${\mathcal {A}} \cong \widehat {{\widehat {{\mathcal {A}}}}}$ ), we reach the conclusion of Theorem 1.2.

Remark 6.6. (1) As in [Reference Abadie and Dykema1], an automorphism $\alpha $ of a unital $C^*$ -algebra ${\mathcal {A}}$ is uniquely ergodic if and only if for every $a \in {\mathcal {A}}$ , the ergodic sums $({1}/{n})\sum _{i=0}^{n-1}\alpha ^i(a)$ converge in norm to a scalar multiple of the unit $1_{{\mathcal {A}}}$ of ${\mathcal {A}}$ as $n$ goes to infinity. By Theorem 3.5, one may represent ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ as the universal $C^*$ -algebra $C^*_{\text {univ}}({\mathcal {F}}, \{ S_n \}_{n \in \mathbb {Z}})$ with (3-5). Consider the automorphism $\theta $ on ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ defined by (5-2). The restriction of $\theta $ to $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ is our ergodic automorphism $\hat {\theta }$ on ${\widehat {{\mathcal {A}}}}$ . By using the fact that the family

(6-9) $$ \begin{align} S_n {\mathcal{F}} S_n^*, \quad S_n f, \quad S_n S_i S_n^*, \quad n, i\in \mathbb{N} \end{align} $$

generates $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ as in Lemma 4.2, one may prove that the ergodic sums $({1}/{n})\sum _{i=0}^{n-1}\hat {\theta }^i(a)$ converge in norm to a scalar multiple of the unit $1_{f{{\mathcal {O}}_{H_{\mathcal {F}}}} f}$ on the dense subalgebra of ${{\mathcal {O}}_{H_{\mathcal {F}}}}$ algebraically generated by the elements in (6-9) by overlapping discussions with our proof of Proposition 6.1. Consequently, it is possible to prove that the ergodic sums $({1}/{n})\sum _{i=0}^{n-1}\hat {\theta }^i(a)$ converge in norm to a scalar multiple of the unit $1_{f{{\mathcal {O}}_{H_{\mathcal {F}}}} f}$ of $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ for all elements of $f {{\mathcal {O}}_{H_{\mathcal {F}}}} f$ , showing that $\hat {\theta }$ is uniquely ergodic on ${\widehat {{\mathcal {A}}}}$ .

(2) Recall that there is another characterization of a state $\varphi $ on a $C^*$ -algebra ${\mathcal {A}}$ being pure, that is, every positive linear functional majorized by $\varphi $ is a positive scalar multiple of $\varphi $ (see [Reference Takesaki28, Definition 9.21]). By using this characterization of a pure state, one may show that $\varphi _{{\widehat {{\mathcal {A}}}}}$ is pure in the following way. Let $\phi $ be a positive linear functional on ${\widehat {{\mathcal {A}}}}$ such that $\phi \le \varphi _{{\widehat {{\mathcal {A}}}}}$ . Since the state $\varphi _{{\widehat {{\mathcal {A}}}}}$ satisfies the conclusion of Lemma 6.4, so does the positive linear functional $\phi $ . Hence, the proof of Proposition 6.1 works for $\phi $ so that one may conclude that ${\phi }/({\phi (1)})$ coincides with $\varphi _{{\widehat {{\mathcal {A}}}}}$ .

7 Examples

(1) The reciprocal dual $\widehat {\mathcal {O}}_A$ of a simple Cuntz–Krieger algebra ${{\mathcal {O}}_A}$ . Let ${\mathcal {A}}$ be a simple Cuntz–Krieger algebra ${{\mathcal {O}}_A}$ for an irreducible nonpermutation matrix $A =[A(i,j)]_{i,j=1}^N$ with entries in $\{0,1 \}$ . For $m \in \mathbb {N}$ , let us denote by $ B_m(X_A) $ the set of words $(\mu _1,\ldots ,\mu _m), \, \mu _i =1,\ldots , N $ of length $m$ such that $A(\mu _i, \mu _{i+1}) =1, i=1,\ldots , m-1. $ We put $B_*(X_A) = \bigcup _{m=0}^\infty B_m(X_A)$ , where $B_0(X_A)$ denotes the empty word. The Toeplitz algebra ${{\mathcal {T}}_A}$ for the matrix $A$ introduced in [Reference Enomoto, Fujii and Watatani6] (compare with [Reference Evans7]) is the universal unital $C^*$ -algebra generated by $N$ -partial isometries $T_1,\ldots , T_N$ and a nonzero projection $e$ subject to the following relations:

(7-1) $$ \begin{align} \sum_{j=1}^N T_j T_j^* + e = 1, \quad T_i^* T_i = \sum_{j=1}^N A(i,j) T_j T_j^* + e, \quad i=1,\ldots,N. \end{align} $$

For $\mu =(\mu _1,\ldots ,\mu _m)\in B_m(X_A),$ we write $T_{\mu _1}\cdots T_{\mu _m}$ as $T_\mu $ . As the identities

$$ \begin{align*} T_\mu e T_\nu^* \cdot T_\xi e T_\eta^* = \begin{cases} T_\mu e T_\eta^* & \text{ if } \nu = \xi, \\ 0 & \text{ otherwise} \end{cases} \end{align*} $$

for $\mu , \nu , \xi , \eta \in B_*(X_A)$ hold, ${{\mathcal {T}}_A}$ is the closed linear span of elements of the form

$$ \begin{align*} T_\mu e T_\nu^*, \quad T_\xi T_\eta^*, \quad \mu, \nu, \xi, \eta \in B_*(X_A) \end{align*} $$

and the $C^*$ -subalgebra of ${{\mathcal {T}}_A}$ generated by $ T_\mu e T_\nu ^*, \mu , \nu \in B_*(X_A)$ is isomorphic to the $C^*$ -algebra $\mathcal {K}(H)$ of compact operators on a separable infinite dimensional Hilbert space $H$ , such that there is a natural short exact sequence

$$ \begin{align*} 0 \longrightarrow \mathcal{K}(H) \longrightarrow \mathcal{T}_A \longrightarrow {{\mathcal{O}}_A} \longrightarrow 0. \end{align*} $$

We consider the Toeplitz algebra ${{\mathcal {T}}_{A^t}}$ for the transposed matrix $A^t =[A^t(i,j)]_{i,j=1}^N$ of $A$ . It was proved in [Reference Matsumoto15, Reference Matsumoto16] (compare with [Reference Matsumoto and Sogabe17, Reference Matsumoto and Sogabe20]) that

$$ \begin{align*} (\mathrm{K}_0({{\mathcal{T}}_{A^t}}), [e]_0, \mathrm{K}_1({{\mathcal{T}}_{A^t}})) \cong (\mathrm{Ext}_{\mathrm{s}}^1({{\mathcal{O}}_A}), [\iota_{{{\mathcal{O}}_A}}(1)]_s, \mathrm{Ext}_{\mathrm{s}}^0({{\mathcal{O}}_A})). \end{align*} $$

Hence, the reciprocal dual $\widehat {\mathcal {O}}_A$ is realized as a universal $C^*$ -algebra $C^*_{\text {univ}}(\mathcal {T}_{A^t}, \{t_j\}_{j \in \mathbb {N}})$ generated by ${{\mathcal {T}}_{A^t}}$ and a family $\{t_j\}_{j \in \mathbb {N}}$ of partial isometries satisfying (1-2). The algebra ${{\mathcal {T}}_{A^t}}$ itself is a universal $C^*$ -algebra generated by partial isometries $T_j, j=1,\ldots N$ and a projection $e$ satisfying (7-1) for the transposed matrix $A^t$ . By unifying the generators $T_j, j=1,\ldots , N$ and $\{t_j\}_{j \in \mathbb {N}}$ with their relations (7-1) and (1-2), we have the following realization of the reciprocal dual $\widehat {\mathcal {O}}_A$ as a simple Exel–Laca algebra [Reference Exel and Laca8], which has been already seen in [Reference Matsumoto and Sogabe18, Theorem 1.3].

Proposition 7.1 [Reference Matsumoto and Sogabe18, Theorem 1.3].

The reciprocal dual $\widehat {\mathcal {O}}_A$ is the simple Exel–Laca algebra ${\mathcal {O}}_{\widehat {A}}$ defined by the infinite matrix $\widehat {A}$ over $\mathbb {N}$ with entries in $\{0,1\}$ such that

(7-2) $$ \begin{align} \widehat{A}(i,j) := \begin{cases} A(j,i) & \text{ if } 1 \le i , j \le N, \\ 1 & \text{ if } 1 \le i \le N,\, \, j= N+1, \\ 0 & \text{ if } 1 \le i \le N,\, \, j\ge N+2, \\ 1 & \text{ if } i \, =N+1, \\ 1 & \text{ if } i \, \ge N+2,\, \, j \le N+1, \\ 0 & \text{ if } i \, \ge N+2,\, \, j \ge N+2 \end{cases} \end{align} $$

for $i,j \in \mathbb {N}$ . The matrix $\widehat {A}$ is written as

$$ \begin{align*} \widehat{A}= \left[ \begin{array}{@{}ccccccc} & &1 &0 &0 &\cdots \\ &\Huge{\text{A}}^t&\vdots&\vdots&\vdots&\vdots \\ & &1 &0 &0 &\cdots \\ 1&\dots &1 &1 &1 &\cdots \\ 1&\dots &1 &0 &0 &\cdots \\ 1&\dots &1 &0 &0 &\cdots \\ \vdots &\dots &\vdots &\vdots&\vdots&\vdots \\ \end{array} \right]. \end{align*} $$

Proof. Theorem 1.1(i) tells us that the $C^*$ -algebra $\widehat {\mathcal {O}}_A$ is the universal $C^*$ -algebra $C^*_{\text {univ}}({{\mathcal {T}}_{A^t}}, \{t_j\}_{j \in \mathbb {N}})$ generated by ${{\mathcal {T}}_{A^t}}$ and partial isometries $\{t_j\}_{j \in \mathbb {N}}$ satisfying (1-2), which are

$$ \begin{align*} t_1^* t_1 = 1_{\widehat{\mathcal{O}}_A}, \quad t_j^* t_j = 1_{{{\mathcal{T}}_{A^t}}} \text{ for } j \ge 2, \quad t_1 t_1^* = e, \quad 1_{{{\mathcal{T}}_{A^t}}} + \sum_{j=2}^m t_j t_j^* < 1_{\widehat{\mathcal{O}}_A} \end{align*} $$

for $m \ge 2$ . Let us define a family of partial isometries $\{ s_j \}_{j \in \mathbb {N}}$ by setting

$$ \begin{align*} s_j = \begin{cases} T_j & \text{for } 1\le j \le N, \\ t_{j-N} & \text{for } j>N. \end{cases} \end{align*} $$

It is direct to see that the $C^*$ -algebra $C^*(\{s_j \}_{j \in \mathbb {N}})$ generated by the partial isometries $\{ s_j \}_{j\in \mathbb {N}}$ is the Exel–Laca algebra ${\mathcal {O}}_{\widehat {A}}$ defined by the matrix $\widehat {A}$ . As $C^*(\{s_j \}_{j \in \mathbb {N}})$ coincides with $C^*_{\text {univ}}({{\mathcal {T}}_{A^t}}, \{t_j\}_{j \in \mathbb {N}})$ , we have $\widehat {\mathcal {O}}_A$ is the Exel–Laca algebra ${\mathcal {O}}_{\widehat {A}}$ for the matrix $\widehat {A}$ .

The class of the reciprocal duals $\widehat {\mathcal {O}}_A$ of simple Cuntz–Krieger algebras is exactly the class of unital Kirchberg algebras ${\mathcal {A}}$ with finitely generated $\mathrm {K}$ -groups having torsion free $\mathrm {K}_1$ -groups such that $ \mathrm {rank}(\mathrm {K}_0({\mathcal {A}})) - \mathrm {rank}(\mathrm {K}_1({\mathcal {A}})) =1$ [Reference Matsumoto and Sogabe18, Proposition 4.4]. Hence, Proposition 7.1 shows that the Kirchberg algebras belonging to the class are explicitly represented as Exel–Laca algebras by the above matrices.

(2) The reciprocal dual $\widehat {\mathcal {P}}_\infty $ of ${{\mathcal {P}}_\infty }$ . Let ${{\mathcal {P}}_\infty }$ be the Kirchberg algebra such that $\mathrm {K}_0({{\mathcal {P}}_\infty }) = 0, \, \mathrm {K}_1({{\mathcal {P}}_\infty }) = \mathbb {Z}$ . It is the unique Kirchberg algebra realized as the unital Exel–Laca algebra ${\mathcal {O}}_{P_\infty }$ by the infinite matrix  $P_\infty $ :

(7-3) $$ \begin{align} P_\infty = \begin{bmatrix} 1 & 0 & 1 & 1 & 1 & 1 &\cdots & & \\ 0 & 1 & 1 & 1 & 1 & 1 &\cdots & & \\ 1 & 0 & 1 & 0 & 0 & 0 &\cdots & & \\ 0 & 1 & 0 & 1 & 0 & 0 &\cdots & & \\ 0 & 0 & 1 & 0 & 1 & 0 &\cdots & & \\ &\ddots &\ddots &\ddots &\ddots &\ddots&\ddots & & \\ & & & & & & & & \end{bmatrix} \quad (\text{see } [29, \text{Example 4.2}]). \end{align} $$

We concretely represent the reciprocal dual $\widehat {\mathcal {P}}_\infty $ of $\mathcal {P}_\infty $ by generators and their relations as a unital simple Exel–Laca algebra. We know that $\mathrm {Ext}_{\mathrm {w}}^1({{\mathcal {P}}_\infty }) =\mathbb {Z}, \mathrm {Ext}_{\mathrm {w}}^0({{\mathcal {P}}_\infty }) = 0 $ by the UCT (see [Reference Blackadar2, Theorem 23.1.1]) so that the cyclic six-term exact sequence (2-1) for ${\mathcal {A}} = {{\mathcal {P}}_\infty }$ goes to

Since there is an isomorphism between $\mathrm {Ext}_{\mathrm {s}}^1({{\mathcal {P}}_\infty })$ and $\mathbb {Z}\oplus \mathbb {Z}$ such that $\iota _{{{\mathcal {P}}_\infty }}(1) $ is mapped to $0\oplus 1\in \mathbb {Z}\oplus \mathbb {Z},$

$$ \begin{align*} (\mathrm{Ext}_{\mathrm{s}}^1({{\mathcal{P}}_\infty}), [\iota_{{{\mathcal{P}}_\infty}}(1)]_s, \mathrm{Ext}_{\mathrm{s}}^0({{\mathcal{P}}_\infty})) \cong (\mathbb{Z}\oplus \mathbb{Z},0\oplus 1, 0 ). \end{align*} $$

Let $\mathcal {T}_\infty $ be the universal unital $C^*$ -algebra generated by a countable family $T_i, i \in \mathbb {N}$ of isometries and one nonzero projection $e$ such that

(7-4) $$ \begin{align} \sum_{j=1}^m T_j T_j^* + e <1 \quad \text{for } m \in \mathbb{N}. \end{align} $$

Since

$$ \begin{align*} T_\mu e T_\nu^* T_\xi e T_\eta^* = \begin{cases} T_\mu e T_\eta^* & \text{ if } \nu = \xi, \\ 0 & \text{ otherwise,} \end{cases} \end{align*} $$

the $C^*$ -subalgebra $C^*(T_\mu e T_\nu ^* \mid \mu , \nu \in \mathbb {N}^*)$ generated by partial isometries $ T_\mu e T_\nu ^*, \mu , \nu \in \mathbb {N}^* $ is isomorphic to the $C^*$ -algebra $\mathcal {K}(H)$ of compact operators. It forms an essential ideal of $\mathcal {T}_\infty $ such that the sequence

(7-5) $$ \begin{align} 0 \longrightarrow \mathcal{K}(H) \longrightarrow \mathcal{T}_\infty \longrightarrow {{\mathcal{O}}_\infty} \longrightarrow 0 \end{align} $$

is exact. That is, the exact sequence (7-5) is a trivial extension of ${{\mathcal {O}}_\infty }$ . The $\mathrm {K}$ -theory standard six-term exact sequence for (7-5), together with $\mathrm {K}_0({{\mathcal {O}}_\infty }) = \mathbb {Z}, \mathrm {K}_1({{\mathcal {O}}_\infty }) = 0$ yields the cyclic six-term exact sequence

(7-6)

As the downward arrow in the diagram (7-6) is induced by the map $ \mathbb {Z} \ni 1 \rightarrow [e]_0 \in \mathrm {K}_0({{\mathcal {T}}_\infty })$ , there is an isomorphism between $\mathrm {K}_0({{\mathcal {T}}_\infty })$ and $\mathbb {Z}\oplus \mathbb {Z}$ such that $[e]_0 $ is mapped to $0\oplus 1\in \mathbb {Z}\oplus \mathbb {Z},$ so that

$$ \begin{align*} (\mathrm{K}_0({{\mathcal{T}}_\infty}), [e]_0, \mathrm{K}_1({{\mathcal{T}}_\infty})) \cong (\mathbb{Z}\oplus \mathbb{Z},0\oplus 1, 0), \end{align*} $$

showing that

$$ \begin{align*} (\mathrm{K}_0({{\mathcal{T}}_\infty}), [e]_s, \mathrm{K}_1({{\mathcal{T}}_\infty})) \cong (\mathrm{Ext}_{\mathrm{s}}^1({{\mathcal{P}}_\infty}), [\iota_{{{\mathcal{P}}_\infty}}(1)]_s, \mathrm{Ext}_{\mathrm{s}}^0({{\mathcal{P}}_\infty})). \end{align*} $$

By Theorem 1.1, the reciprocal dual $\widehat {\mathcal {P}}_\infty $ is realized as a universal $C^*$ -algebra $C^*_{\text {univ}}({{\mathcal {T}}_\infty }, \{t_j\}_{j \in \mathbb {N}})$ generated by ${{\mathcal {T}}_\infty }$ and a family $\{t_j\}_{j \in \mathbb {N}}$ of partial isometries satisfying (1-2) for $\mathcal {T} = {{\mathcal {T}}_\infty }$ . The trivial extension ${{\mathcal {T}}_\infty }$ of ${{\mathcal {O}}_\infty }$ itself is a universal $C^*$ -algebra generated by isometries $T_j, j \in \mathbb {N}$ and a nonzero projection $e$ satisfying (7-4). By unifying them, we have the following relations:

$$ \begin{align*} &\qquad\ \ \qquad T_i^* T_i = 1_{{{\mathcal{T}}_\infty}} \text{ for } i \in \mathbb{N}, \quad \sum_{i=1}^m T_i T_i^* + e <1_{{{\mathcal{T}}_\infty}}, \\ & t_1^* t_1 = 1_{\widehat{\mathcal{P}}_\infty}, \quad t_j^* t_j = 1_{{{\mathcal{T}}_\infty}} \text{ for } j \ge 2, \quad t_1 t_1^* = e, \quad 1_{{{\mathcal{T}}_\infty}} + \sum_{j=2}^m t_j t_j^* < 1_{\widehat{\mathcal{P}}_\infty} \end{align*} $$

for $m \ge 2$ . They go to the relations:

$$ \begin{align*} & \sum_{i=1}^m T_i T_i^* + t_1 t_1^* <1_{{{\mathcal{T}}_\infty}}, \quad 1_{{{\mathcal{T}}_\infty}} + \sum_{j=2}^n t_j t_j^* < 1_{\widehat{\mathcal{P}}_\infty}\quad \text{for } m,n \in \mathbb{N} \text{ with } n\ge 2, \\ &\qquad\ \,\quad t_1^* t_1 = 1_{\widehat{\mathcal{P}}_\infty}, \quad T_i^* T_i = t_j^* t_j = 1_{{{\mathcal{T}}_\infty}}\quad \text{for } i, j \in \mathbb{N} \text{ with } j \ne 1. \end{align*} $$

Therefore, we have the following proposition.

Proposition 7.3. The reciprocal dual $\widehat {\mathcal {P}}_{\infty }$ of ${{\mathcal {P}}_\infty }$ is the universal Kirchberg algebra generated by two families of partial isometries $\{T_i\}_{i \in \mathbb {N}}$ and $\{ t_j \}_{j \in \mathbb {N}}$ with orthogonal ranges, which is realized as a simple Exel–Laca algebra ${\mathcal {O}}_{\widehat {P}_\infty }$ defined by a matrix $\widehat {P}_\infty $ such as

$$ \begin{align*} \widehat{P}_\infty = \begin{bmatrix} 1 & 1 & 1 & \cdots & 1 & 0 & 0 & 0 & \cdots \\ 1 & 1 & 1 & \cdots & 1 & 0 & 0 & 0 & \cdots \\ 1 & 1 & 1 & \cdots & 1 & 0 & 0 & 0 & \cdots \\ \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & \\ 1 & 1 & 1 & \cdots & 1 & 1 & 1 & 1 & \cdots \\ 1 & 1 & 1 & \cdots & 1 & 0 & 0 & 0 & \cdots \\ 1 & 1 & 1 & \cdots & 1 & 0 & 0 & 0 & \cdots \\ 1 & 1 & 1 & \cdots & 1 & 0 & 0 & 0 & \cdots \\ \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & \end{bmatrix}, \end{align*} $$

where the entries of the above matrix are ordered as follows:

$$ \begin{align*} T_1T_1^*, \,\, T_2T_2^*, \,\, T_3T_3^*, \,\, \ldots, \, \, & t_1t_1^*, \,\, t_2t_2^*, \,\, t_3t_3^*, \,\, t_4t_4^*, \, \, \ldots, \quad \text{in rows,}\\ T_1^*T_1, \,\, T_2^*T_2, \,\, T_3^*T_3, \, \, \ldots, \, \, & t_1^*t_1, \,\, t_2^*t_2, \,\, t_3^*t_3, \,\, t_4^*t_4, \, \, \ldots, \quad \text{in columns.} \end{align*} $$

For example, the first and second rows in the matrix $\widehat {P}_\infty $ show us that

$$ \begin{align*} T_1T_1^* + T_2T_2^* + T_3T_3^* + & \cdots + t_1t_1^* < T_1^* T_1 =1_{\mathcal{T}_\infty}, \\ T_1T_1^* + T_2T_2^* + T_3T_3^* + & \cdots + t_1t_1^* < T_2^* T_2 =1_{\mathcal{T}_\infty} \end{align*} $$

and the middle row shows us that

$$ \begin{align*} T_1T_1^* + T_2T_2^* + T_3T_3^* + \cdots + t_1t_1^* + t_2 t_2^* + t_3t_3^* + t_4 t_4^*+ \cdots < 1_{\widehat{\mathcal{P}}_\infty}, \end{align*} $$

and so on.

(3) The reciprocal dual $\widehat {{\mathcal {O}}}_\infty $ of ${{\mathcal {O}}_\infty }$ . The $C^*$ -algebra ${{\mathcal {P}}_\infty }$ is the Exel–Laca algebra ${\mathcal {O}}_{P_\infty }$ for the matrix $P_\infty $ defined by (7-3). It is the universal $C^*$ -algebra generated by a family $\{S_i\}_{i\in \mathbb {N}}$ of partial isometries such that

$$ \begin{align*} \begin{cases} S_1^* S_1 + S_2 S_2^* = S_2^* S_2 + S_1 S_1^* =1, \\ S_3^* S_3 = S_1 S_1^* + S_3 S_3^*, \\ S_4^* S_4 = S_2 S_2^* + S_4 S_4^*, \\ S_5^* S_5 = S_3 S_3^* + S_5 S_5^*, \\ \cdots \end{cases} \text{and} \quad \sum_{j=1}^m S_j S_j^* < 1 \text{ for } m \in \mathbb{N}. \end{align*} $$

Let us consider the following universal $C^*$ -algebra, written $\mathcal {T}_{P_\infty }$ , and generated by partial isometries $s_i, i \in \mathbb {N}$ and a nonzero projection $e$ :

$$ \begin{align*} \begin{cases} s_1^* s_1 + s_2 s_2^* +e = s_2^* s_2 + s_1 s_1^* =1, \\ s_3^* s_3 = s_1 s_1^* + s_3 s_3^*, \\ s_4^* s_4 = s_2 s_2^* + s_4 s_4^*, \\ s_5^* s_5 = s_3 s_3^* + s_5 s_5^*, \\ \cdots \end{cases} \text{and} \quad e + \sum_{j=1}^m s_j s_j^* < 1 \text{ for } m \in \mathbb{N}. \end{align*} $$

Since the $C^*$ -subalgebra $C^*(s_\mu s_2 e s_2^* s_\nu ^* \mid \mu , \nu \in \{1,2,3,\ldots \}^*)$ of $\mathcal {T}_{P_\infty }$ forms an essential ideal of $\mathcal {T}_{P_\infty }$ isomorphic to $\mathcal {K}(H)$ , we have a short exact sequence

(7-7) $$ \begin{align} 0 \longrightarrow \mathcal{K}(H) \longrightarrow \mathcal{T}_{P_\infty} \longrightarrow {\mathcal{O}}_{P_\infty} \longrightarrow 0. \end{align} $$

As $ s_1^* s_1 + s_2 s_2^* +e = s_2^* s_2 + s_1 s_1^* =1$ and $[s_is_i^*]_0 = [s_i^* s_i]_0$ in $\mathrm {K}_0(\mathcal {T}_{P_\infty })$ , we have $[e]_0 = 0$ in $\mathrm {K}_0(\mathcal {T}_{P_\infty }).$ By the $\mathrm {K}$ -theory standard cyclic six-term exact sequence for (7-7) together with $\mathrm {K}_1({{\mathcal {P}}_\infty }) = \mathbb {Z}, \, \mathrm {K}_0({{\mathcal {P}}_\infty }) = \{ 0 \}$ , we have the cyclic six-term exact sequence

Since the downward arrow $\iota _*: \mathbb {Z} \rightarrow \mathrm {K}_0(\mathcal {T}_{P_\infty })$ satisfies $\iota _*(1) = [e]_0 =0 $ in $\mathrm {K}_0(\mathcal {T}_{P_\infty })$ , so that $\iota _*: \mathbb {Z} \rightarrow \mathrm {K}_0(\mathcal {T}_{P_\infty })$ is the zero map, hence, we have $\mathrm {K}_0(\mathcal {T}_{P_\infty }) = 0,$ and the upper horizontal arrow $\partial : \mathbb {Z} \rightarrow \mathbb {Z}$ is surjective and hence isomorphic. We then have $\mathrm {K}_1(\mathcal {T}_{P_\infty }) = 0$ and, hence,

$$ \begin{align*} (\mathrm{K}_0(\mathcal{T}_{P_\infty}), [e]_0, \mathrm{K}_1(\mathcal{T}_{P_\infty})) \cong ( 0, \, 0,\, 0). \end{align*} $$

As $\mathrm {Ext}_{\mathrm {s}}^i({{\mathcal {O}}_\infty }) = 0$ for $i=0,1$ ,

$$ \begin{align*} (\mathrm{K}_0(\mathcal{T}_{P_\infty}), [e]_0, \mathrm{K}_1(\mathcal{T}_{P_\infty})) \cong (\mathrm{Ext}_{\mathrm{s}}^1({{\mathcal{O}}_\infty}), [\iota_{{{\mathcal{O}}_\infty}}]_s, \mathrm{Ext}_{\mathrm{s}}^0({{\mathcal{O}}_\infty})). \end{align*} $$

Hence, the $C^*$ -algebra $\widehat {{\mathcal {O}}}_\infty $ is the universal $C^*$ -algebra $C^*_{\text {univ}}(\mathcal {T}_{P_\infty }, \{t_i\}_{i \in \mathbb {N}}) $ generated by $\mathcal {T}_{P_\infty }$ and partial isometries $\{t_j \}_{j \in \mathbb {N}}$ satisfying (1-2) for $\mathcal {T} = \mathcal {T}_{P_\infty }$ . By [Reference Matsumoto and Sogabe18, Section 8], $\widehat {{{\mathcal {O}}}}_\infty $ is isomorphic to the Cuntz algebra ${\mathcal {O}}_2$ (see [Reference Cuntz5]). One may construct an aperiodic ergodic automorphism on ${\mathcal {O}}_2$ as in Section 5. We then choose the index set of $\{ t_j\}_{j \in \mathbb {N}}$ as the integer group $\mathbb {Z}$ such that $\{ t_j\}_{j \in \mathbb {Z}}$ satisfies (5-1). By Theorem 1.1(i) and Proposition 5.3, we may describe our ergodic automorphism on ${\mathcal {O}}_2$ as an automorphism on the universal $C^*$ -algebra $C^*_{\text {univ}}(\{s_i\}_{i \in \mathbb {N}}, f, \{t_j\}_{j \in \mathbb {Z}})$ in the following way.