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Register a new userA tracially AF algebra which is not $mathcal Z$-absorbing
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We show that there is a simple separable unital (non-nuclear) tracially AF algebra $A$ which does not absorb the Jiang-Su algebra $mathcal Z$ tensorially, i.e., $A cong Aotimesmathcal Z$.
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We define a notion of tracial $mathcal{Z}$-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially $mathcal{Z}$-absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.
We rule out a certain $9$-dimensional algebra over an algebraically closed field to be the basic algebra of a block of a finite group, thereby completing the classification of basic algebras of dimension at most $12$ of blocks of finite group algebras.
We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.
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.
For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.
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