2 Preliminaries

Throughout, G is a compact group with left Haar measure and $A,B,\ldots $ typically denote $C^*$ -algebras. We refer to [Reference Echterhoff, Kaliszewski, Quigg and RaeburnEKQR06, Appendix A] for actions and coactions. If $\delta :A\to M(A\otimes C^*(G))$ is a coaction of G on A, we say that the pair $(A,\delta )$ is a coaction.

Since G is compact, all coactions are normal (and maximal). Furthermore, the Fourier algebra $A(G)$ coincides with the Fourier–Stieltjes algebra $B(G)$ , so it is a unital Banach *-algebra (with norm from the dual space $C^*(G)^*$ , pointwise multiplication, and involution given by $f^*(x)=\overline {f(x^{-1})}$ ) that is dense in $C(G)$ . Most of the background on $A(G)$ is contained in [Reference EymardEym64]. Most importantly, [Reference EymardEym64, Théorème 3.34] says that the dual (that is, the maximal ideal space) of $A(G)$ can be identified with G via

$$ \begin{align*} \chi_x(f)=f(x)\quad\text{for }f\in A(G),x\in G. \end{align*} $$

We use the general fact that for any $C^*$ -algebra B, the dual space $B^*$ can be regarded as the dual space of the multiplier algebra $M(B)$ when the latter is given the strict topology. The zero set of an ideal I of $A(G)$ is $\{x\in G:f(x)=0\text { for all }f\in I\}$ , and a spectral set in G is a closed subset E for which $\{f\in A(G):f(x)=0\text { for all }x\in E\}$ is the only closed ideal of $A(G)$ with zero set E.

Every coaction $(A,\delta )$ gives rise to a module structure on A over the Fourier algebra $A(G)$ via slicing:

$$ \begin{align*} f\cdot a:=\langle\delta(a),\text{id}\otimes f\rangle\quad\text{for }f\in A(G),a\in A. \end{align*} $$

The same formula also makes A a module over the Fourier–Stieltjes algebra $B(G)$ .

If H is a closed subgroup of G, then any coaction of H on A can be inflated to a coaction of G on A, using the canonical homomorphism of $C^*(H)$ into $M(C^*(G))$ (see [Reference Echterhoff, Kaliszewski, Quigg and RaeburnEKQR06, Example A.29]).

We refer to [Reference Fell and DoranFD88b] for Fell bundles (called $C^*$ -algebraic bundles in [Reference Fell and DoranFD88b]) $\mathcal {A}$ over G, and to [Reference Landstad, Phillips, Raeburn and SutherlandLPRS87] for the associated coaction $\delta _{\mathcal {A}}$ —often called the dual coaction—on the full cross-sectional algebra $C^*(\mathcal {A})$ . If $(A,\delta )$ is a coaction of G, then for $x\in G$ , the spectral subspace is $A_x=\{a\in A:\delta (a)=a\otimes x\}$ and the spectrum of $\delta $ is $\operatorname {sp} \delta =\{x\in G:A_x\ne \{0\}\}$ . If $\operatorname {sp} \delta =G$ , we say the coaction is effective. Dangerous bend: the spectrum is not always a subgroup of G (see [Reference QuiggQui96, Example 5.3] for a counterexample with $G=\mathbb {Z}_4$ ; however, note that the Fell bundle described in [Reference QuiggQui96] could alternatively be regarded as coming from an effective coaction of $\mathbb {Z}_3$ , as indicated in Example 5.3). When G is finite (or, more generally, discrete, or when the fibers $A_x$ are only finitely nonzero), the family $\{A_x\}_{x\in G}$ forms a Fell bundle over G, and we say that $\delta $ is associated with $\mathcal {A}$ if $(A,\delta )\simeq (C^*(\mathcal {A}),\delta _{\mathcal {A}})$ . Every coaction of a discrete group has an associated Fell bundle, but this is not automatic for arbitrary groups (see [Reference Landstad, Phillips, Raeburn and SutherlandLPRS87, Example 2.3(6)] for a counterexample with $G=\mathbb {T}$ ). When G is finite, the isomorphism classes of Fell bundles $\mathcal {A}$ and their associated coactions $\delta _{\mathcal {A}}$ are in bijective correspondence (see [Reference QuiggQui96, Theorem 3.8 and Corollary 3.9]).

The spectral subspace $A_e$ is also denoted by $A^\delta $ and called the fixed-point subalgebra. A coaction $\delta $ is called ergodic if $A^\delta =\mathbb {C} 1$ . In this case, for every $x\in \operatorname {sp} \delta $ , the spectral subspace $A_x$ is spanned by a (nonunique) unitary $U_x$ , and the spectrum $\operatorname {sp} \delta $ is a subgroup of G.

Suppose that $\delta $ is an effective ergodic coaction of G on A. Interestingly, for each $x\in G$ , we get an automorphism $\operatorname {Ad} U_x$ of A and, in fact, $x\mapsto \operatorname {Ad} U_x$ is an action of G on A. When $A=M_n$ and $|G|=n^2$ , this process can be reversed, and we get a bijection between ergodic coactions and ergodic actions of G on A.

We write $w_G$ for the unitary element of

$$ \begin{align*} M(C(G)\otimes C^*(G))=C(G,M^\beta(C^*(G)))\end{align*} $$

(where the ‘ $\beta $ ’ signifies that we mean continuity with respect to the strict topology on the multiplier algebra) given by the canonical embedding $G\hookrightarrow M(C^*(G))$ . If $\mu :C(G)\to M(A)$ is a unital homomorphism, then the unitary element

$$ \begin{align*} w_\mu:=(\mu\otimes\text{id})(w_G)\in M(A\otimes C^*(G)) \end{align*} $$

satisfies

$$ \begin{align*} (\text{id}\otimes\delta_G)(w_\mu)=(w_\mu\otimes 1)(\text{id}\otimes\Sigma)(w_\mu), \end{align*} $$

where $\Sigma $ is the ‘flip automorphism’ of $C^*(G)\otimes C^*(G)$ given on elementary tensors by $\Sigma (x\otimes y)=y\otimes x$ , and where $\delta _G$ is the canonical coaction on $C^*(G)$ given on group elements by $\delta _G(x)=x\otimes x$ . Unitaries satisfying this identity are called corepresentations. In the opposite direction, [Reference Nakagami and TakesakiNT79, Theorem A.1] proves that every corepresentation $w\in M(A\otimes C^*(G))$ is of the form $w_\mu $ for a unique homomorphism $\mu $ given by slicing:

$$ \begin{align*} \mu(f)=\langle w,\text{id}\otimes f\rangle\quad\text{for }f\in A(G). \end{align*} $$

The relevance for us is that for any $\mu $ , there is an inner coaction $\delta $ of G on A given by

(2-1) $$ \begin{align} \delta(a)=\operatorname{Ad} w_\mu(a\otimes 1)\quad\text{for }a\in A. \end{align} $$

Such coactions are called unitary in [Reference Landstad, Phillips, Raeburn and SutherlandLPRS87, Examples 2.3(3)] (and [Reference QuiggQui94, Lemma 1.11] shows that conditions (b) and (c) in [Reference Landstad, Phillips, Raeburn and SutherlandLPRS87] are redundant). More generally, if $(A,\delta )$ is any coaction, we say that a unitary $w\in M(A\otimes C^*(G))$ implements $\delta $ and that $\delta $ is unitarily implemented, if (2-1) holds. Of course, not all unitaries will implement coactions and, moreover, a moment’s thought reveals that a coaction that is implemented by a unitary w might not be inner, since slicing w might not give a *-homomorphism of $A(G)$ (and hence of $C(G)$ ). For example, when G is finite abelian, we have ergodic actions that are implemented by unitaries twisted by 2-cocycles.

We need the fact that if $c\in M(C^*(G))$ , then $\delta _G(c)=c\otimes c$ implies that $c\in G$ . This is a folklore result concerning ‘group-like elements’, and is easy to verify by slicing and using

$$ \begin{align*} \langle\delta_G(c),f\otimes g\rangle=\langle c,fg\rangle\quad\text{for }f,g\in A(G) \end{align*} $$

together with the identification of G with the spectrum of $A(G)$ .

We frequently need to work with unitary multipliers of $M(C^*(G))$ and, for convenience, we write

$$ \begin{align*} UM(C^*(G))=\{u\in M(C^*(G)):u\text{ is unitary}\}. \end{align*} $$

In Section 4, we need to use the destabilization process: if A is an elementary algebra (that is, A is isomorphic to the compact operators on a Hilbert space) and $\iota :A\to M(B)$ is a nondegenerate homomorphism, we define the relative commutant of A in B as

$$ \begin{align*} C(B,\iota):=\{c\in M(B):c\iota(a)=\iota(a)c\in B\text{ for all }a\in A\}. \end{align*} $$

Then, B is a stabilization in a functorial way: there is a unique isomorphism

$$ \begin{align*} \theta:A\otimes C(B,\iota)\overset{\simeq}{\longrightarrow} B \end{align*} $$

given on elementary tensors by $\theta (a\otimes c)=\iota (a)c$ . We call this construction the destabilization process; it is probably folklore, and is described explicitly in [Reference Bédos, Kaliszewski, Quigg and TurkBKQT23, Proposition A.2], [Reference ExelExe17, Lemma 27.2], [Reference FischerFis04, Section 3], [Reference Hjelmborg and RørdamHRr98, Theorem 2.1], and [Reference Kaliszewski, Omland and QuiggKOQ16, Proposition 3.4], for example.

3 Fell bundles

We show that every coaction of a compact group on a finite-dimensional $C^*$ -algebra A is associated with a Fell bundle. In fact, the argument works just as well for amenable groups, and we state Theorem 3.1 in this context, since we feel that it might be useful elsewhere.

We need to use the fact that every finite subset of G is a spectral set for $A(G)$ ; this is implied by [Reference WarnerWar76, Theorem 4], since the empty set is a Ditkin set (a fact that is explicitly mentioned in the paragraph preceding Warner’s theorem 1).

Proof. Since G is amenable, we have $C^*(G)=C^*_r(G)$ , and it follows that the Fourier algebra $A(G)$ separates the points of $M(C^*(G))$ . We use the associated duality of the locally convex spaces $M(C^*(G))$ and $A(G)$ . Let $a\in A$ be nonzero and put

$$ \begin{align*} (a)^\circ=\{\phi\in A(G):\phi\cdot a=0\}. \end{align*} $$

Note that

$$ \begin{align*} \delta(a)=\sum_{i=1}^p e_i\otimes c_i, \end{align*} $$

where $\{e_1,\ldots ,e_p\}$ is a basis of A and $c_i\in M(C^*(G))$ . To ease the notational burden a little, put $D=\{c_1,\ldots ,c_p\}$ . Because the $e_i$ are linearly independent, it follows by slicing that $\phi \in (a)^\circ $ if and only if $\phi $ is in the annihilator $D^\perp $ in $A(G)$ .

Now, $(a)^\circ $ is an ideal of $A(G)$ of finite codimension, so is contained in only finitely many maximal ideals. By [Reference EymardEym64, Théorèm 3.34], the maximal ideals of $A(G)$ are of the form $I_x=\{\phi \in A(G):\phi (x)=0\}$ , so the zero set of $(a)^\circ $ can be identified with a finite subset $E\subseteq G$ . As we mentioned before this proof, E is spectral because it is finite (see [Reference WarnerWar76, Theorem 4]); consequently, the ideal $(a)^\circ $ coincides with the intersection

$$ \begin{align*} I_E=\bigcap_{x\in E}I_x=\{\phi\in A(G):\phi(x)=0\text{ for all }x\in E\}. \end{align*} $$

Using the duality of locally convex spaces, we can interpret the above as saying that the two subsets $D,E$ of $M(C^*(G))$ have the same annihilator in $A(G)$ , that is,

$$ \begin{align*} D^\perp=E^\perp. \end{align*} $$

It then follows by the bipolar theorem that D is contained in the double polar of E. Since E is finite, this double polar is just the linear span. Thus, we have

$$ \begin{align*} D\subseteq \operatorname{span} E. \end{align*} $$

Thus,

$$ \begin{align*} \delta(a)\in A\otimes \operatorname{span} E, \end{align*} $$

and so there are $b_x\in A$ such that

(3-1) $$ \begin{align} \delta(a)=\sum_{x\in E} b_x\otimes x. \end{align} $$

We have

$$ \begin{align*} \sum_{x\in E} \delta(b_x)\otimes x &=(\delta\otimes\text{id})\circ\delta(a) \\&=(\text{id}\otimes\delta_G)\circ\delta(a) \\&=\sum_{x\in E} b_x\otimes x\otimes x. \end{align*} $$

Since E is linearly independent in $M(C^*(G))$ , we conclude that

$$ \begin{align*} \delta(b_x)=b_x\otimes x\quad\text{for all }x\in E, \end{align*} $$

that is, $b_x$ is in the spectral subspace $A_{x}$ .

Slicing (3-1) by $1\in A(G)$ , we conclude that $a=\sum _{x\in E} b_x$ . Letting a run through a basis of A, we conclude that A is the sum of the spectral subspaces $\{A_x\}_{x\in \operatorname {sp} \delta }$ , where $\operatorname {sp} \delta $ is the spectrum of the coaction $\delta $ . Now, the family $\mathcal {A}:=\{A_{x}\}_{x\in G}$ is a Fell bundle and since A is finite-dimensional, it must coincide with the full cross-sectional algebra $C^*(\mathcal {A})$ .

Remark 3.3. In the proof of Theorem 3.1, we only needed the spectral subspaces $A_x$ for x in the finite subset $\operatorname {sp} \delta $ of G. This need not be a subgroup, but when it is, the rest of G is in some sense superfluous: $\delta $ restricts to a coaction $\varepsilon $ of $\operatorname {sp} \delta $ and the given coaction of G coincides with the inflation from $\operatorname {sp} \delta $ to G, as indicated in the commutative diagram.

4 Unitary implementation

We give an elementary proof of an implementation theorem (proved for arbitrary type I factors in [Reference WassermannWas88, Theorem 3], using more sophisticated machinery).

Proof. Let $\delta :A\to M(A\otimes C^*(G))$ be a coaction where $A=M_n$ . Choose a decomposition of $C^*(G)$ into a $c_0$ -direct sum of matrix algebras:

$$ \begin{align*} C^*(G)=\bigoplus_{i\in S}M_i. \end{align*} $$

For each i, we have a central projection $p_i\in C^*(G)$ such that $p_i C^*(G)\simeq M_i$ , and we fix such an isomorphism and in fact identify $p_i C^*(G)=M_i$ . Then, we get a homomorphism

$$ \begin{align*} \delta_i=(1\otimes p_i)\delta:A\to A\otimes M_i. \end{align*} $$

Since A is elementary, we can apply the destabilization process: write $C_i$ for the relative commutant of $\delta _i(A)$ in $M(A\otimes M_i)$ , so that

$$ \begin{align*} A\otimes M_i=\operatorname{span}\{\delta(A)C_i\}\simeq A\otimes C_i. \end{align*} $$

Since $A=M_n$ , we see that $C_i$ must also be a matrix algebra and, by counting dimensions, we must have $C_i\simeq M_i$ . Fixing such an isomorphism, by composition, we get an automorphism $\phi _i\in \operatorname {Aut}(A\otimes M_i)$ such that

$$ \begin{align*} \phi_i(A\otimes 1)=\delta_i(A)\quad\text{and}\quad \phi_i(1\otimes M_i)=C_i. \end{align*} $$

Since $A\otimes M_i$ is elementary, we can choose a unitary $U_i$ in $A\otimes M_i$ such that $\phi _i=\operatorname {Ad} U_i$ . It follows that

$$ \begin{align*} \delta_i(a)=\operatorname{Ad} U_i(a\otimes 1)\quad\text{for all }a\in A. \end{align*} $$

Now, we put them together: define a unitary

$$ \begin{align*} U=\sum_i U_i\in M(C^*(G)). \end{align*} $$

Then, we have

$$ \begin{align*} \delta(a)=\sum_i\delta_i(a)=\sum_i \operatorname{Ad} U_i(a\otimes 1_{M_i})=\operatorname{Ad} U(a\otimes 1_{M(C^*(G))}), \end{align*} $$

so that U implements $\delta $ .

Lemma 4.2. With the above notation, if U is a unitary implementing $\delta $ , then the set of all unitaries implementing $\delta $ is

$$ \begin{align*} \{U(1\otimes u):u\in UM(C^*(G))\}. \end{align*} $$

5 Inner

In the preceding section, we saw that all coactions of G on $M_n$ are implemented by unitaries $U\in M(M_n\otimes C^*(G))$ . The inner coactions of G on $M_n$ are precisely those for which we take $U=(\mu \otimes \text {id})(w_G)$ for a unital homomorphism $\mu :C(G)\to M_n$ .

We pause to see what $\mu $ must look like. By the spectral theorem, we choose orthogonal minimal projections $p_1,\ldots ,p_n$ and $x_1,\ldots ,x_n\in G$ such that

$$ \begin{align*} \mu(f)=\sum_i f(x_i)p_i. \end{align*} $$

Moreover, any such choice of $p_i$ and $x_i$ gives a homomorphism $\mu :C(G)\to M_n$ .

For any coaction $\delta $ of G on $M_n$ , observe that $A^\delta $ is a unital $C^*$ -subalgebra of $M_n$ ; we get some information by looking at its maximal abelian $C^*$ -subalgebras.

Proof. First suppose that $M^\delta _n$ has an abelian $C^*$ -subalgebra of dimension n. Then, we can find rank-one projections $p_1,\ldots p_n\in A^\delta $ such that $\sum _ip_i=1$ . Choose matrix units $e_{ij}$ such that $e_{ii}=p_i$ for $i=1,\ldots ,n$ . By Theorem 4.1, we can choose a unitary U implementing $\delta $ . We can write

$$ \begin{align*} U=\sum_{i,j} e_{ij}\otimes y_{ij}\quad\text{with }y_{ij}\in M(C^*(G)). \end{align*} $$

For each k,

$$ \begin{align*} p_k\otimes 1 &=\delta(p_k) \\&=\sum_{i,r,j,s}e_{ir}p_k e_{sj}\otimes y_{ir}y_{js}^* \\&=\sum_{i,j}e_{ij}\otimes y_{ik}y_{jk}^*. \end{align*} $$

Because the $e_{ij}$ are linearly independent, all terms except when $i=j=k$ are zero, and $y_{kk}y_{kk}^*=1$ , so $y_{kk}$ is a unitary. However, with $i\ne k$ and $j=k$ ,

$$ \begin{align*} 0=y_{ik}y_{kk}^*, \end{align*} $$

so $y_{ik}=0$ since $y_{kk}\in UM(C^*(G))$ . Thus,

$$ \begin{align*} U=\sum_i p_i\otimes y_i, \end{align*} $$

where we have written $y_i=y_{ii}$ to simplify the notation. For all $i,j$ ,

$$ \begin{align*} \delta(e_{ij})=\sum_{r,s} p_re_{ij}p_s\otimes y_ry_s^*=e_{ij}\otimes y_iy_j^*. \end{align*} $$

The coaction identity implies that $\delta _G(y_iy_j^*)=y_iy_j^*\otimes y_iy_j^*$ , so $y_iy_j^*\in G$ and $e_{ij}\in A_{y_iy_j^*}$ .

Now, we adjust U so that $y_i$ itself is in G: recall that if $U'$ is any unitary implementing $\delta $ , then $U'=U(1\otimes u)$ for some $u\in UM(C^*(G))$ and, moreover, every choice of $u\in UM(C^*(G))$ gives a suitable $U'$ . If we take $u=y_n^*$ , then $x_i:=y_iy_n^*\in G$ for $i=1,\ldots ,n$ (and $x_n=e$ ) and, therefore, we now can replace the original U by

$$ \begin{align*} U=\sum_i p_i\otimes x_i \end{align*} $$

with $x_i\in G$ .

We define a homomorphism $\mu :C(G)\to M_n$ by

$$ \begin{align*} \mu(f)=\sum_i p_if(x_i) \end{align*} $$

and then

$$ \begin{align*} U=(\mu\otimes\text{id})(w_G), \end{align*} $$

so $\delta $ is inner.

Conversely, suppose that $\delta $ is inner, determined by a unital homomorphism $\mu :C(G)\to A$ . Choose mutually orthogonal minimal projections $p_1,\ldots ,p_n\in M_n$ and $x_1,\ldots ,x_n\in G$ such that

$$ \begin{align*} \mu(f)=\sum_i f(x_i)p_i. \end{align*} $$

Thus, $\delta $ is implemented by

$$ \begin{align*} (\mu\otimes\text{id})(w_G)=\sum_i p_i\otimes x_i, \end{align*} $$

and a quick computation shows that

$$ \begin{align*} \delta(a)=\sum_{i,j} p_iap_j\otimes x_ix_j^{-1}. \end{align*} $$

Choosing matrix units $e_{ij}$ such that $e_{ii}=p_i$ , we have $e_{ij}\in A_{x_ix_j^{-1}}$ . In particular, $p_i=e_{ii}\in A_e=A^\delta $ for all i, so the fixed-point algebra $A^\delta $ has an abelian $C^*$ -subalgebra of dimension n.

Paraphrasing the above theorem and its proof, we see that an inner coaction $\delta $ on $A=M_n$ is implemented by

$$ \begin{align*} U=\sum_1^np_i\otimes x_i, \end{align*} $$

where $p_i=e_{ii}$ and $x_i\in G$ . For the matrix units,

$$ \begin{align*} e_{ij}\in A_{x_ix_j^{-1}}. \end{align*} $$

Moreover, the $x_i$ terms are only unique up to multiplying (on the right) by any fixed $x\in G$ . Here are some easy consequences.

Example 5.4. For inner coactions $\delta $ of G on $A=M_3$ , if, for example,

$$ \begin{align*} A^\delta=\operatorname{span}\{e_{ij}:1\le i,j\le 2\text{ or }i=j=3\}, \end{align*} $$

then $\delta $ is implemented by a corepresentation of the form

$$ \begin{align*} U=e_{11}\otimes 1+e_{22}\otimes 1+e_{33}\otimes x \end{align*} $$

for some $x\in G$ . This is a routine application of Proposition 5.2 and the discussion immediately preceding it.

6 Ergodic

Focusing on coactions of our compact group G on $M_n$ , there are some immediate, easy facts.

Lemma 6.1 is almost trivial. The (seemingly) slightly stronger Proposition 6.2 below is easy but relies on somewhat more sophisticated technology.

In our study of effective ergodic coactions on $M_n$ , we need to make contact with finite groups of central type (we thank Erik Bédos for informing us of this terminology and the references [Reference Ginosar and SchnabelGS19, Reference SchnabelSch16]), defined in the literature as groups having a simple twisted group algebra; the motivation for the terminology has to do with central group extensions. Groups of central type have been of increasing interest in the past few decades. The abelian groups of central type are quite well understood. The smallest nonabelian examples have order 16, where there are two nonisomorphic groups of central type. In Example 6.7, we discuss a couple of examples that we found in [Reference SchnabelSch16].

We need a bit of the theory of 2-cocycles and we adopt the conventions of [Reference KleppnerKle62]. Keep in mind that for any ergodic coaction $\delta $ of a finite group G on a unital $C^*$ -algebra A, each spectral subspace $A_x$ is spanned by a unitary, and any choice of a spanning set of unitaries $\{U_x\}_{x\in G}$ , where we always choose $U_e=1$ , gives a projective representation of G in A that by definition has an associated 2-cocycle $\omega \in Z^2(G,\mathbb {T})$ determined by

$$ \begin{align*} U_xU_y=\omega(x,y)U_{xy}\quad\text{for }x,y\in G, \end{align*} $$

and that satisfies:

1. (1) $\omega (x,e)=\omega (e,x)=1$ for all $x\in G$ and

2. (2) $\omega (x,y)\omega (xy,z)=\omega (x,yz)\omega (y,z)$ for all $x,y,z\in G$ .

Here, U as above is called an $\omega $ -representation.

As explained in [Reference Fell and DoranFD88b, Propositions VIII.10.10 and VIII.16.2], $\omega $ -representations of G, representations of a cocycle bundle $\mathcal {A}$ with cocycle $\omega $ , and central extensions of $\mathbb {T}$ by G correspond up to isomorphism. Choosing the spanning unitaries $U_x$ is almost the same as choosing the associated cocycle $\omega $ —the unitaries determine the cocycle uniquely, but a different choice of unitaries could lead to the same cocycle. Two cocycles $\omega $ and $\omega '$ are cohomologous if there is a map $m:G\to \mathbb {T}$ such that

$$ \begin{align*} \omega'(x,y)=m(x)m(y)\overline{m(xy)}\omega(x,y)\quad\text{for }x,y\in G, \end{align*} $$

and $\omega '$ is trivial if it is cohomologous to $\omega =1$ . Every cocycle is cohomologous to a normalized one, meaning that

$$ \begin{align*} \omega(x,x^{-1})=1\quad\text{for all }x\in G, \end{align*} $$

and we always assume that our cocycle $\omega $ is normalized. Choosing $\omega $ to be normalized is the same as choosing the spanning unitaries so that

$$ \begin{align*} U_x^*=U_{x^{-1}}\quad\text{for all }x\in G. \end{align*} $$

A routine computation shows that for $\omega $ normalized,

$$ \begin{align*} \omega(x,y)^{-1}=\omega(y^{-1},x^{-1})\quad\text{for all }x,y\in G. \end{align*} $$

There, in general, are many $\omega $ -representations for any given cocycle $\omega $ . When G is finite, there are $\omega $ -representations U that are universal in the sense that for any $\omega $ -representation V, there is a unique homomorphism $\pi _V$ making the diagram

commute. As usual, the pair $(C^*(U(G)),U)$ is unique up to isomorphism and we write $C^*_\omega (G)$ for any such algebra $C^*(U(G))$ , and call it the twisted group $C^*$ -algebra for $\omega $ (although we do not, in general, give a special notation to the associated U). One choice of such a universal $\omega $ -representation is the right regular $\omega $ -representation [Reference KleppnerKle62, page 556] R of G on $L^2(G)$ , defined by

$$ \begin{align*} (R_x\xi)(y)=\omega(y,x)\xi(yx). \end{align*} $$

We have

$$ \begin{align*} C^*_\omega(G)=\operatorname{span} U(G) \end{align*} $$

for any universal U, so

$$ \begin{align*} \dim C^*_\omega(G)=|G|. \end{align*} $$

Also, any $\omega $ -representation V is universal if and only if the associated homomorphism $\pi _V:C^*_\omega (G)\to C^*(V(G))$ is faithful, if and only if the unitaries $\{V_x:x\in G\}$ are linearly independent.

Later, we are interested in the possibility that $C^*_\omega (G)$ is simple, that is, is isomorphic to $M_n$ for some n. This obviously puts restrictions on G—for example, the order $|G|$ must be $n^2$ . However, this alone is insufficient. It seems natural to say that G has a simple twisted $C^*$ -algebra if there exists a cocycle $\omega $ for which $C^*_\omega (G)$ is simple; equivalently, there exists an irreducible projective representation U of G such that the unitaries $\{U_x\}_{x\in G}$ are linearly independent. In the literature, such a group is said to be of central type. There is a subtlety: the literature also contains numerous studies of faithful irreducible projective representations U, meaning that the composition of U with the quotient map $\operatorname {U}(n)\to \operatorname {PU}(n)$ is faithful; equivalently, the only $x\in G$ for which $U_x$ is a scalar operator is $x=e$ . This is weaker than the unitaries $\{U_x\}_{x\in G}$ being linearly independent. For example, it follows from [Reference NgNg76, Theorem 4.4] that there are groups of order 16 with faithful irreducible projective representations U on $\mathbb {C}^2$ and, trivially, $\{U_x\}_{x\in G}$ must be linearly dependent. Interestingly, these groups are nonabelian. This behavior cannot occur for abelian groups, because Frucht [Reference FruchtFru32] proved that a finite abelian group has a faithful irreducible projective representation if and only if it is isomorphic to $H\times H$ for some (abelian) group H, and by [Reference KleppnerKle62, paragraph following Corollary 1], such a group does have a simple twisted $C^*$ -algebra. The smallest nonabelian finite groups of central type have order 16.

Proof. Suppose that $\delta $ is an ergodic coaction on $A=M_n$ . Then, the spectrum $\operatorname {sp} \delta $ is a subgroup of order $n^2$ , and without loss of generality, we suppose that $G=\operatorname {sp} \delta $ . Choose spanning unitaries $U_x$ for the spectral subspaces $A_x$ and let $\omega $ be the associated 2-cocycle. Then, U is an irreducible $\omega $ -representation of G. In [Reference KleppnerKle62, page 559], Kleppner defines $x\in G$ to be $\omega $ -regular if

$$ \begin{align*} \omega(x,y)=\omega(y,x)\quad\text{whenever }xy=yx, \end{align*} $$

and it follows from [Reference KleppnerKle62, Corollary 1] that $G\setminus \{e\}$ contains no $\omega $ -regular elements. Thus, picking any $x\ne e$ , we find $y\ne e$ such that $xy=yx$ and $\omega (x,y)\ne \omega (y,x)$ . Let $H=\langle x,y\rangle $ be the abelian subgroup generated by x and y, so $\omega $ restricted to $H\times H$ is a nontrivial cocycle. As we mentioned in the preceding proof, H must be noncyclic.

Proposition 6.4. If G has an irreducible projective representation U on $\mathbb {C}^p$ with p prime and $\{U_x:x\in G\}$ linearly independent, then $G=\mathbb {Z}_p\times \mathbb {Z}_p$ . With $\gamma =\exp (2\pi i/p)$ , then up to equivalence,

$$ \begin{align*} U(1,0)=\operatorname{diag}(\gamma^j : j=1,\ldots, p) \quad\text{and}\quad U(0,1)=\{\delta(j,j+1): j\in \mathbb{Z}_p\}. \end{align*} $$

Proof. From the assumptions, there are $x,y\in G$ with $yx=xy$ and

$$ \begin{align*} \omega(x,y)\ne \omega(y,x). \end{align*} $$

Since $|G|=p^2$ , the order of x and y must be $1$ , p, or $p^2$ (remember that we assume G is finite). One quickly rules out $1$ and $p^2$ . Then, one realizes that one can pick $x,y$ such that

$$ \begin{align*} \omega(x,y)\overline{\omega(y,x)}=\exp(2\pi i/p)=:\gamma. \end{align*} $$

We must have $H\cap K=\{e\}$ , so $|HK|= |H| \, |K|=p^2= |G|$ and, hence, $G=\mathbb {Z}_p\times \mathbb {Z}_p$ . We leave as an exercise to show that U as described is (up to equivalence) the only irreducible projective representation of $\mathbb {Z}_p\times \mathbb {Z}_p$ satisfying the theorem.

Example 6.5. It follows from Proposition 6.2 that $\operatorname {SU}(2)$ has no ergodic coactions on any $M_n$ with $n>1$ , because every finite abelian subgroup is cyclic (folklore). Interestingly, the quotient group $\operatorname {SO}(3)$ has an ergodic coaction on $M_2$ , because it contains the Klein-4 group $\mathbb {Z}_2\times \mathbb {Z}_2$ as a subgroup; for example, the elements

$$ \begin{align*} \begin{pmatrix} 1&0&0\\0&-1&0\\0&0&-1 \end{pmatrix} \quad\text{and}\quad \begin{pmatrix} -1&0&0\\0&-1&0\\0&0&1 \end{pmatrix} \end{align*} $$

are suitable generators.

Remark 6.6. Reference [Reference KleppnerKle62, Discussion following Corollary 1] shows that if H is any abelian group of order n, then there is a 2-cocycle $\omega $ on $G=H\times \widehat H$ given by

$$ \begin{align*} \omega\bigl((x,\chi),(y,\sigma)\bigr)=\chi(y), \end{align*} $$

and no element of $G\setminus \{e\}$ is $\omega $ -regular, so G is of central type and hence has an effective ergodic coaction on $M_n$ . Note that the set-up in Proposition 6.4 is a special case. In fact, Frucht proved in [Reference FruchtFru32] that a finite abelian group has a faithful irreducible projective representation if and only if it is isomorphic to the square of an abelian group, in which case, it is of central type, as we explained in the preliminaries. This was rediscovered somewhat later by Olesen et al. [Reference Olesen, Pedersen and TakesakiOPT80, Theorem 5.9].

Ng [Reference NgNg76] generalized this to nonabelian groups; his setting was primarily metacyclic groups (extensions of cyclic by cyclic). Reference [Reference NgNg76, Proposition 4.1] shows that if G is such a group, with generators a and b, if there is a faithful irreducible projective representation, then by [Reference NgNg76, Theorem 5.2], G is the semidirect product of $\langle a\rangle $ and $\langle b\rangle $ . Moreover, if $|G|$ is a power of an odd prime, then G has a faithful irreducible projective representation if and only if the subgroups $\langle a\rangle $ and $\langle b\rangle $ are isomorphic. So, for G to be nonabelian, the smallest such example would have order $81=3^4$ .

7 $G=S_3$

It is interesting to examine coactions $\delta $ of the smallest nonabelian group $S_3$ (the permutations of a set of three elements) on $A=M_n$ . We fix a presentation

$$ \begin{align*} S_3=\langle\alpha,\beta\mid\alpha^3=\beta^2=e,\alpha\beta=\beta\alpha^2\rangle. \end{align*} $$

Since all proper subgroups of $S_3$ are cyclic, to avoid trivialities, we look for effective coactions. By Lemma 6.1, this requires $n\ge 3$ , and we cannot expect $\delta $ to be ergodic.

Here is where we can make some progress: look for effective inner coactions. As we have observed, we need $n\ge 3$ . However, even $n=3$ will not work: in the notation of the proof of Theorem 5.1, we have $\operatorname {sp} \delta =\{x_ix_j^{-1}\}$ with $x_i\in S_3$ , and we can assume $x_3=1$ . So, we can choose $x_1, x_2$ , and then we have $e_{12}\in A_{x_1x_2^{-1}}$ . Since $S_3$ has three self-inverse elements, trivial computations show that we can have at most three non-identity elements in $\operatorname {sp} \delta $ , which is absurd. Therefore, we cannot have an effective inner coaction of $S_3$ on $M_3$ .

Therefore, an effective inner coaction of $S_3$ on $M_n$ requires $n\ge 4$ . Then it is easy to generate examples: as we mentioned above, we might as well choose $x_4=1$ . Then, for example, we can choose

$$ \begin{align*} x_1=\alpha, \quad x_2=\beta \quad\text{and}\quad x_3=\alpha\beta. \end{align*} $$

Then, $\{x_ix_j^{-1}:i,j=1,\ldots ,3\}$ contains

$$ \begin{align*} x_1x_2^{-1}=\alpha\beta\quad\text{and}\quad x_2x_3^{-1}=\beta\alpha=\alpha^2\beta, \end{align*} $$

and we have accounted for all six elements of $S_3$ . Trivially, we could have made other choices and we could do something similar for any $n\ge 5$ as well. We can conclude as follows.