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Register a new userStructure theory and stable rank for C*-algebras of finite higher-rank graphs
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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.
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In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $C^*$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $Lambda$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.
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 $$ |e^{it}u-exp(ih)|<epsilon. $$
We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.
We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs with no sources. We establish that strong aperiodicity - the higher-rank analogue of condition (K) - for a $k$-graph is necessary and sufficient for the associated $C^*$-algebra to have topological dimension zero. We prove that a purely infinite $2$-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the $2$-graph. We also show that a $k$-graph $C^*$-algebra with topological dimension zero is purely infinite if and only if all the vertex projections are properly infinite. We show by example that there are strongly purely infinite $2$-graphs algebras, both with and without topological dimension zero, that fail to have real-rank zero.
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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