Simple, Z-stable, projectionless C*-algebras

Do simple, $\mathcal{Z}$ -stable, stably projectionless C*-algebras have stable rank one?

Definitions: A C*-algebra $A$ is said to be $\mathcal{Z}$ -stable if it tensorially absorbs the Jiang-Su algebra $\mathcal{Z}$ , that is, $A\cong\mathcal{Z}\otimes A$ . Further, a simple C*-algebra is projectionless if it contains no nonzero projections, and it is stably projectionless if $A\otimes\mathcal{K}$ is projectionless. (In the nonsimple case, one should require that no quotient of $A$ contains a nonzero projection – see [1].) A unital C*-algebra $A$ is said to have stable rank one if the invertible elements in $A$ are dense. A nonunital C*-algebra $A$ has stable rank one if its minimal unitization $\widetilde{A}$ does.

Background/Motivation: Rørdam showed in Theorem 6.7 in [2] that every unital, simple, finite $\mathcal{Z}$ -stable C*-algebra has stable rank one. Using this, one can show that every simple, finite, $\mathcal{Z}$ -stable C*-algebra $A$ that is not stably projectionless has stable rank one. Indeed, one may first reduce to the separable case (to show that a given element $x_0+\lambda 1\in\widetilde{A}$ with $x_0\in A$ is approximated by invertible elements, consider any separable, simple, $\mathcal{Z}$ -stable sub-C*-algebra of $A$ that is not stably projectionless and that contains $x$ .) Let $p\in A\otimes\mathcal{K}$ be a nonzero projection. By Brown’s stabilization theorem, we have $A\otimes\mathcal{K}\cong (p(A\otimes\mathcal{K})p)\otimes\mathcal{K}$ . Then the hereditary sub-C*-algebra $p(A\otimes\mathcal{K})p$ is separable, unital, simple, finite and $\mathcal{Z}$ -stable (by Corollary 3.2 in [3]) and therefore has stable rank one. Since stable rank one is invariant under stable isomorphism, it follows that $A$ has stable rank one.

Thus, a simple, $\mathcal{Z}$ -stable C*-algebra that is not stably projectionless has stable rank one if and only if it is (stably) finite. A simple, stably projectionless C*-algebra is automatically stably finite, and it is therefore natural to expect that every simple, $\mathcal{Z}$ -stable, stably projectionless C*-algebra has stable rank one.

Let $A$ be a simple, $\mathcal{Z}$ -stable, stably projectionless C*-algebra. By Corollary 3.2 in [1], $A$ almost has stable rank one, that is, every hereditary sub-C*-algebra $B\subseteq A$ satisfies $B\subseteq\overline{\mathrm{Gl}(\widetilde{B})}$ . In particular, every element in $A$ can be approximated by invertible elements in $\widetilde{A}$ . To show that $A$ has stable rank one, one would need to show that every element in $\widetilde{A}$ is approximated by invertibles. In Theorem 6.13 in [4] it is shown that $A$ has stable rank at most two, that is, the tuples $(x,y)$ in $(\widetilde{A})^2$ such that $x^*x+y^*y$ is invertible are dense in $(\widetilde{A})^2$ . One can also show:

Simple, $\mathcal{Z}$ -stable, stably projectionless C*-algebras have general stable rank one.

Here, the general stable rank of a unital C*-algebra $A$ , denoted $\mathrm{gsr}(A)$ , is the least integer $n\geq 1$ such that $\mathrm{Gl}_m(A)$ acts transitively on $\mathrm{Lg}_m(A)$ for all $m\geq n$ . (We use $\mathrm{Gl}_m(A)$ to denote the set of invertible elements in the matrix algebra $M_m(A)$ . Further, $\mathrm{Lg}_m(A)$ denotes the set of $m$ -tuples $(a_1,\ldots,a_m)\in A^m$ that generate $A$ as a left ideal, that is, such that $a_1^*a_1+\ldots+a_m^*a_m$ is invertible.) For a nonunital C*-algebra $\mathrm{gsr}(A):=\mathrm{gsr}(\widetilde{A})$ . In general, one has $\mathrm{gsr}(A)\leq\mathrm{sr}(A)+1$ , as noted in Theorem 3.3. of the overview article [5]. If $A$ is unital, then the action of $\mathrm{Gl}_m^0(A)$ , the connected component of the unit in $\mathrm{Gl}_m(A)$ , on $\mathrm{Lg}_m(A)$ has open orbits. It follows that the orbits of the action of $\mathrm{Gl}_m(A)$ on $\mathrm{Lg}_m(A)$ are also open (and hence also closed).

Let $A$ be a simple, $\mathcal{Z}$ -stable, stably projectionless C*-algebra. Let us prove that $\mathrm{gsr}(A)=1$ . By Theorem 6.13 in [4], we have $\mathrm{sr}(A)\leq 2$ , and so $\mathrm{gsr}(A)\leq 3$ . To verify that $\mathrm{gsr}(A)\leq 2$ , we need to show that $\mathrm{Gl}_2(\widetilde{A})$ acts transitively on $\mathrm{Lg}_2(\widetilde{A})$ . So let $x=(\lambda_1+a_1,\lambda_2+a_2)\in\mathrm{Lg}_2(\widetilde{A})$ with $\lambda_1,\lambda_2\in\mathbb{C}$ and $a_1,a_2\in A$ . First, we find $u\in\mathrm{Gl}_2(\mathbb{C})\subseteq\mathrm{Gl}_2(\widetilde{A})$ such that $ux=(b_1,\kappa_2+b_2)$ with $\kappa_2\in\mathbb{C}$ and $b_1,b_2\in A$ . (If $\lambda_1=0$ , we use the identity matrix; if $\lambda_2=0$ , we use the flip matrix; and if $\lambda_1\neq 0$ and $\lambda_2\neq 0$ we use the upper-triangular matrix with diagonal entries $1$ and upper right entry $-\lambda_1/\lambda_2$ .) Given $\varepsilon>0$ , we use that $A\subseteq\overline{\mathrm{Gl}(\widetilde{A})}$ to approximate $b_1$ by some invertible $\kappa_1+c\in\widetilde{A}$ such that $\| x - u^{-1}(\kappa_1+c,\kappa_2+b_2) \|<\varepsilon$ . We note that $(\kappa_1+c,\kappa_2+b_2)$ is in the orbit of $(1,0)$ , that is, there exists $v\in\mathrm{Gl}_2(\widetilde{A})$ such that $(\kappa_1+c,\kappa_2+b_2)=v(1,0)$ . Then $\| x - u^{-1}v(1,0) \|<\varepsilon$ . Since the orbits of the action of $\mathrm{Gl}_2(\widetilde{A})$ on $\mathrm{Lg}_2(\widetilde{A})$ are closed, it follows that $x=w(1,0)$ for some $w\in\mathrm{Gl}_2(\widetilde{A})$ . Finally, since $\widetilde{A}$ is finite, it follows that $\mathrm{gsr}(A)=1$ .

We remark that $\mathrm{sr}(A)=1$ implies $\mathrm{gsr}(A)=1$ , but not conversely in general.

[1]
L. Robert, Remarks on Z-stable projectionless C*-algebras, Glasgow Math. J. (2015) 273–277. https://doi.org/10.1017/s0017089515000117.
[2]
M. Rørdam, The stable and the real rank of ${\mathcal Z}$ -absorbing C*-algebras, Int. J. Math. (2004) 1065–1084. https://doi.org/10.1142/s0129167x04002661.
[3]
A.S. Toms, W. Winter, Strongly self-absorbing $C^{*}$ -algebras, Trans. Amer. Math. Soc. (2007) 3999–4029. https://doi.org/10.1090/s0002-9947-07-04173-6.
[4]
L. Robert, L. Santiago, A revised augmented Cuntz semigroup, ArXiv:1904.03690. (2019). https://arxiv.org/abs/1904.03690v1.
[5]
B. Nica, Homotopical stable ranks for Banach algebras, Journal of Functional Analysis. (2011) 803–830. https://doi.org/10.1016/j.jfa.2011.03.001.

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