Let A be a unital C*-algebra. Let
$A_+$
denote the set of positive elements of A. Let
$P(A)$
denote the set of finite products of elements in
$A_+$
. Quinn showed in Reference Quinn[4] that if A is an approximately finite-dimensional C*-algebra and has no finite-dimensional representations, then
$P(A)$
is dense in A. In Reference Murphy and Phillips[3], Murphy and Phillips took up the systematic investigation of the C*-algebras with the property that
$P(A)$
is dense in A, which they termed the approximate positive factorization property (APFP). They obtained necessary conditions for the APFP to hold, confirmed the APFP for various classes of C*-algebras, and raised the following question [Reference Murphy and Phillips3, Question 4.12]: If A is an infinite-dimensional simple unital C*-algebra of real rank zero, stable rank one, and trivial
$K_1$
-group, does A have the APFP? In the theorem below, we answer this question affirmatively by establishing necessary and sufficient conditions for the APFP to hold.
Theorem 1 Let A be a unital C*-algebra. The following are equivalent:
-
(i)
$P(A)$
is dense in A, i.e., A has the APFP. -
(ii) A has stable rank one and trivial
$K_1$
-group, and the map
$\rho _A\colon K_0(A)\to \mathrm {Aff}(T(A))$
, from the
$K_0$
-group of A to the continuous affine functions on the trace space of A, has dense range.
Let us recall the meaning of the properties in (ii).
A unital C*-algebra A is said to have stable rank one if its invertible elements form a dense set.
By trivial
$K_1$
-group, we mean
$K_1(A)=0$
, where
$K_1(A)$
is the abelian group associated to A via K-theory.
Let
$T(A)$
denote the convex set of tracial states on A (endowed with the weak* topology). Let us regard traces extended to
$M_\infty (A)$
—the infinite matrices on A with finitely many nonzero entries—by setting
$\tau ((a_{ij})_{i,j})=\tau (\sum _{i=1}^\infty a_{ii})$
. Given a tracial state
$\tau $
and a
$K_0$
-group element
$[p]-[q]\in K_0(A)$
, define their pairing as
$([p]-[q],\tau )\mapsto \tau (p)-\tau (q)$
. Fixing
$[p]-[q]$
and varying
$\tau $
, we get a continuous real-valued affine function on
$T(A)$
, i.e., an element of
$\mathrm {Aff}(T(A))$
. Let us denote by
$\rho _A\colon K_0(A)\to \mathrm {Aff}(T(A))$
the mapping induced by this pairing:
$$ \begin{align*} \rho_A([p]-[q])(\tau)=\tau(p)-\tau(q). \end{align*} $$
It is known that if A is a simple C*-algebra of real rank zero, then the range of
$\rho _A$
is dense in
$\mathrm {Aff}(T(A))$
(e.g., see [Reference de la Harpe and Skandalis2, Theorem 4.1]). Thus, Theorem 1 indeed answers Murphy and Phillips’ question affirmatively.
In the second theorem of this note (which we prove first), we obtain a description of the invertible elements in
$\overline {P(A)}$
which, in a sense, extends the description as “elements of positive determinant,” valid in the case of matrices. Let
$\widetilde P(A)=\overline {P(A)}\cap GL(A)$
, where
$GL(A)$
denotes the invertible elements of A. Let
$U(A)$
denote the unitary group of A, and
$U_0(A)$
the connected component of 1 in
$U(A)$
. Let
$DU_0(A)$
denote the commutator subgroup of
$U_0(A)$
.
Theorem 2 Let A be a unital C*-algebra. Then,
$\widetilde P(A)$
is the set of invertible elements whose unitary part in the polar decomposition belongs to
$\overline {DU_0(A)}$
, i.e., the elements of the form
$ua$
, with
$u\in \overline {DU_0(A)}$
and
$a\in A_+$
positive and invertible.
1 Proofs of Theorems 1 and 2
We will use some facts around the de la Harpe–Skandalis determinant. We introduce here necessary notation, but refer the reader to [Reference Gong, Lin and Xue1, Reference Thomsen8] for background on this topic.
Let
$\overline {[A,A]}$
denote the closure of the linear span of the commutators
$[x,y]:=xy-yx$
, with
$x,y\in A$
. Let us regard
$A/\overline {[A,A]}$
as a Banach space under the quotient norm. Let
$T\colon A\to A/\overline {[A,A]}$
denote the quotient map (or “universal trace”). Regard T extended to
$M_\infty (A)$
by setting
$T((a_{ij})_{i,j})=T(\sum _{i=1}^\infty a_{ii})$
.
Let
$GL_0^\infty (A)$
denote the inductive limit of the groups
$GL_0(M_n(A))$
,
$n\in \mathbb N$
. Given a smooth path
$\alpha \colon [t_1,t_2]\to GL_0^\infty (A)$
, its de la Harpe–Skandalis determinant is defined as:
$$ \begin{align*} \widetilde \Delta_T(\alpha)=\int_{t_1}^{t_2} T(\alpha'(t)\alpha^{-1}(t))dt\in A/\overline{[A,A]}. \end{align*} $$
If two paths
$(a_1(t))_t$
and
$(a_2(t))_t$
both connect
$1$
to an invertible element
$a\in GL_0^\infty (A)$
, then the values of
$\widetilde \Delta ((a_1(t))_t)$
and
$\widetilde \Delta ((a_2(t))_t)$
differ by at most an element of the abelian subgroup:
$$ \begin{align*} 2\pi i\{T(p)-T(q):p,q\in M_\infty(A)\text{ projections}\}. \end{align*} $$
Let us denote this subgroup by
$2\pi iT(K_0(A))$
). Given an invertible element
$a\in GL_0^\infty (A)$
, its de la Harpe–Skandalis determinant
$\Delta _T(a)$
is defined as the image of
$\widetilde \Delta _T((a(t))_t)$
in the quotient
$(A/\overline {[A,A]})/2\pi iT(K_0(A))$
, where
$t\mapsto a(t)$
is any path in
$GL_0^\infty (A)$
connecting
$1$
to a.
Proof of Theorem 2
Let Q denote the set of invertible elements whose unitary part in the polar decomposition belongs to
$\overline {DU_0(A)}$
. Thus, Q is defined by the condition
$x|x|^{-1}\in \overline {DU_0(A)}$
. By the continuity of the operations involved in this condition, it is clear that Q is a norm closed subset of
$GL(A)$
(in fact, of
$GL_0(A)$
).
Let us prove that Q is contained in
$\widetilde P(A)$
. By the form of the polar decomposition of the elements in Q, it will suffice to show that
$DU_0(A)$
is contained in
$\widetilde P(A)$
. In fact, we will show that
$DGL_0(A)$
is contained in
$\widetilde P(A)$
. It is easily checked, using polar decompositions, that
$P(A)\cap GL_0(A)$
is invariant under similarity, i.e., it is a normal subgroup of
$GL_0(A)$
. Thus,
$\widetilde P(A)$
is a closed normal subgroup of
$GL_0(A)$
. By [Reference Robert7, Theorem 1.1], to show that
$\widetilde P(A)$
contains
$DGL_0(A)$
, it suffices to show the equality of closed two-sided ideals:
$$ \begin{align*} \mathrm{Ideal}([A,A])=\mathrm{Ideal}([\widetilde P(A),A]). \end{align*} $$
Let I be the ideal on the right-hand side. It is clearly contained on the left-hand side. Observe that the image of
$\widetilde P(A)$
in the quotient
$A/I$
is central. Because the invertible positive elements of A map onto the invertible positive elements of
$A/I$
, and the latter span
$A/I$
, it follows that
$A/I$
is commutative. Thus,
$\mathrm {Ideal}([A,A])\subseteq I$
, as desired. We thus conclude by [Reference Robert7, Theorem 1.1] that
$DGL_0(A)\subseteq \widetilde P(A)$
.
For the inclusion
$\widetilde P(A)\subseteq Q$
, it will suffice to show that Q is a group, as it is a closed set (in
$GL_0(A)$
) and contains all invertible positive elements. Let
$u_1a_1$
and
$u_2a_2$
be in Q, with
$u_1,u_2\in \overline {DU_0(A)}$
and
$a_1,a_2\in A_+$
. Then,
$$ \begin{align*} u_1a_1u_2a_2=u_1u_2 (u_2^*a_1u_2)a_2. \end{align*} $$
Set
$a_1'=u_2^*a_1u_2$
. Let
$a_1'a_2=w|a_1a_2|$
be the polar decomposition of
$a_1'a_2$
. Then,
$$ \begin{align*} u_1a_1u_2a_2=u_1u_2w |a_1a_2|. \end{align*} $$
We will be done once we have shown that
$w\in \overline {DU_0(A)}$
. That is, we must show that given two invertible positive elements
$a,b$
, the unitary
$u=ab|ab|^{-1/2}$
(in the polar decomposition of
$ab$
) belongs to
$\overline {DU_0(A)}$
. Write
$a=e^{c}$
and
$b=e^{d}$
, with
$c,d\in A$
self-adjoint. Define
$u_t=e^{tc}e^{td}|e^{td}e^{tc}|^{-1/2}$
, for
$t\in [0,1]$
. This is a path of unitaries connecting 1 to u. Let us calculate its determinant:
$$ \begin{align*} \widetilde{\Delta}_T((u_t)_t) &=\widetilde\Delta_T((e^{tc})_t)+\widetilde\Delta_T((e^{td})_t)-\widetilde\Delta_T((|e^{tc}e^{td}|)_t)\\ &=T(c)+T(d)-\widetilde\Delta_T((|e^{tc}e^{td}|)_t). \end{align*} $$
Because the path
$$ \begin{align*} t\mapsto |e^{tc}e^{td}|^2=e^{td}e^{2tc}e^{td}\end{align*} $$
has determinant
$2(T(c)+T(d))$
, the path
$t\mapsto |e^{tc}e^{td}|$
has determinant
$T(c)+T(d)$
. Hence, from the above calculation, we get
$\widetilde \Delta _T((u_t)_t)=0$
. By (the proof of) [Reference Gong, Lin and Xue1, Lemma 3],
$$ \begin{align*} u=\prod_{k=1}^N e^{ih_k},\end{align*} $$
where
$h_k$
is self-adjoint for all k and
$\sum _{k=1}^N h_k\in \overline {[A,A]}$
. (From the proof of [Reference Gong, Lin and Xue1, Lemma 3],
$e^{ih_k}=u_{t_{k-1}}^{-1}u_{t_k}$
for all k, where
$0=t_0<t_1<\cdots <t_N=1$
is a sufficiently fine partition of the interval
$[0,1]$
.) Now,
$\prod _{k=1}^N e^{ih_k}$
belongs to
$\overline {DU_0(A)}$
, by [Reference Robert7, Theorem 6.2(b)] (alternatively, by the proof of [Reference Gong, Lin and Xue1, Proposition 4]).▪
In the proof of Theorem 1, we rely on some results by Thomsen from Reference Thomsen[8]. We isolate these facts in a lemma:
Lemma 3 Let A be a unital C*-algebra. If
$U_0(A)=\overline {DU_0(A)}$
, then
$\rho _A\colon K_0(A)\to \mathrm {Aff}(T(A))$
has dense range. Conversely, if
$\rho _A$
has dense range and A has stable rank one, then
$U_0(A)=\overline {DU_0(A)}$
.
Proof. We maintain the notation used in [Reference Thomsen8, Section 3]. By [Reference Thomsen8, Theorem 3.2],
$$ \begin{align} U_0(A)/\overline{DU_0(A)}\cong \mathrm{Aff}(T(A))/\overline{\Delta_1^0(\pi_1(U_0(A)))}, \end{align} $$
where
$\Delta _1^0(\pi _1(U_0(A))$
is a subgroup of
$\rho _A(K_0(A))$
. Thus, from
$U_0(A)=\overline {DU_0(A)}$
, we get that
$\Delta _1^0(\pi _1(U_0(A))$
is dense in
$\mathrm {Aff}(T(A))$
, and a fortiori,
$\rho _A$
has dense range. (The map
$\Delta _1^0$
is defined in terms of the de la Harpe–Skandalis determinant for paths recalled above. More specifically, given a loop
$\alpha $
in
$U_0(A)$
starting and ending in 1, one computes
$h=\frac {1}{2\pi i}\widetilde \Delta _T(\alpha )\in A/\overline {[A,A]}$
, and takes the affine function
$\widehat h\in \mathrm {Aff}(T(A))$
induced by this element. Then,
$\Delta _1^0([\alpha ]):=\widehat h$
. By the homotopy invariance of the determinant, and the isomorphism
$\pi _1(U_0^\infty (A))\cong K_0(A)$
from Bott periodicity,
$\Delta _1^0([\alpha ])\in \rho _A(K_0(A))$
.)
Suppose now that
$\rho _A$
has dense range and A has stable rank one. By the isomorphism (1.1), to show that
$U_0(A)=\overline {DU_0(A)}$
, it suffices to show that
$\Delta _1^0(\pi _1(U_0(A)))=\rho _A(K_0(A))$
. For C*-algebras of stable rank one, the inclusion of
$U_0(A)$
in
$U_0^\infty (A)$
induces an isomorphism of homotopy groups
$\pi _1(U_0(A))\cong \pi _1(U_0^\infty (A))$
. This follows from combining [Reference Rieffel5, Corollary 8.6], which says if A has stable rank one, then
$C(\mathbb {T},A)$
has connected stable rank at most two, and [Reference Rieffel6, Proposition 2.6] applied to
$C(\mathbb {T},A)$
. Hence,
$\pi _1(U_0(A))\cong \pi _1(U_0^\infty (A))\cong K_0(A)$
. The subgroup
$\Delta _1^0(\pi _1(U_0(A)))$
is then precisely
$\rho _A(K_0(A))$
.▪
Proof of Theorem 1
(i)
$\Rightarrow $
(ii). In fact, this implication is already present in Reference Murphy and Phillips[3]. Suppose that A has the APFP. By [Reference Murphy and Phillips3, Theorem 2.4], A has stable rank one and trivial
$K_1$
-group. (These are also fairly straightforward consequences of the APFP.) The density of the range of
$\rho _A$
is essentially proved in [Reference Murphy and Phillips3, Lemma 2.8], but because that result is stated differently, and in less generality, we include an argument here. Let
$u\in U_0(A)$
. Then, u is the limit of a sequence of elements in
$\widetilde P(A)$
. By Theorem 2, they have the form
$w_na_n$
with
$w_n\in \overline {DU_0(A)}$
and
$a_n\in A_+$
. Comparing unitary parts, we get that
$w_n\to u$
. It follows that
$U_0(A)=\overline {DU_0(A)}$
. Hence, by Lemma 3,
$\rho _A$
has dense range.
(ii)
$\Rightarrow $
(i). Because A has stable rank one, it suffices to approximate invertible elements of A by elements of
$P(A)$
. By polar decomposition, every invertible element has the form
$ua$
, where u is a unitary and
$a\in A_+$
. By the description of
$\widetilde P(A)$
from Theorem 2, we will be done once we have shown that
$U(A)=\overline {DU_0(A)}$
. Now, by the
$K_1$
-injectivity/surjectivity of stable rank one C*-algebras [Reference Rieffel6, Theorem 2.10],
$U(A)/U_0(A)\cong K_1(A)$
. Because we have assumed that
$K_1(A)=0$
, we get that
$U(A)=U_0(A)$
. Thus, it remains to show that
$U_0(A)=\overline {DU_0(A)}$
. This follows from Lemma 3.▪
For a nonunital C*-algebra A, Murphy and Phillips define the APFP by asking that
$1+P(A)$
be dense in
$1+A$
, in the unitization of A. A different, weaker, property is obtained by simply asking that
$P(A)$
be dense in A. We have not pursued the investigation of these APFPs in the nonunital setting.
