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It is shown that, for an arbitrary free and minimal $mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $mathrm{C}(X)rtimesmathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, Gamma)$, where $Gamma$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $mathrm{C}(X)rtimesGamma$ has stable rank one. Moreover, in this case, the C*-algebra $mathrm{C}(X)rtimesGamma$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.
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Let $(X, Gamma)$ be a free and minimal topological dynamical system, where $X$ is a separable compact Hausdorff space and $Gamma$ is a countable infinite discrete amenable group. It is shown that if $(X, Gamma)$ has the Uniform Rokhlin Property and Cuntz comparison of open sets, then $mathrm{mdim}(X, Gamma)=0$ implies that $(mathrm{C}(X) rtimesGamma)otimesmathcal Z cong mathrm{C}(X) rtimesGamma$, where $mathrm{mdim}$ is the mean dimension and $mathcal Z$ is the Jiang-Su algebra. In particular, in this case, $mathrm{mdim}(X, Gamma)=0$ implies that the C*-algebra $mathrm{C}(X) rtimesGamma$ is classified by the Elliott invariant.
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $Lambda$. We begin by giving an alternative characterization of the $Lambda$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.
Let $A$ be a unital separable simple ${cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $uin U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $epsilon>0,$ there exists a self-adjoint element $hin A$ such that $$ |u-exp(ih)|<epsilon. $$ The lower bound of $|h|$ could be as large as one wants. If $uin CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $hin A$ such that $$ |u-exp(ih)| <epsilon and |h|le 2pi. $$ Examples are given that the bound $2pi$ for $|h|$ is the optimal in general. For the Jiang-Su algebra ${cal Z},$ we show that, if $uin U_0({cal Z})$ and $epsilon>0,$ there exists a real number $-pi<tle pi$ and a self-adjoint element $hin {cal Z}$ with $|h|le 2pi$ such that $$ |e^{it}u-exp(ih)|<epsilon. $$
A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and classified by the Elliott invariant. A second class of unital simple separable amenable $C^*$-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable ${cal Z}$-stable $C^*$-algebras. An isomorphism theorem for a special sub-class of those $C^*$-algebras are presented. This provides the basis for the classification of $C^*$-algebras with rational generalized tracial rank one in Part II.
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