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Register a new userThe isomorphism relation for separable C*-algebras
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We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.
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We study the range of a classifiable class ${cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${cal A}$ contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle $C^*$-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable $C^*$-algebras. We also show that there are many other $C^*$-algebras in the class. We prove that, for any partially ordered, simple weakly unperforated rationally Riesz group $G_0$ with order unit $u,$ any countable abelian group $G_1,$ any metrizable Choquet simplex $S,$ and any surjective affine continuous map $r: Sto S_u(G_0)$ (where $S_u(G_0)$ is the state space of $G_0$) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable $C^*$-algebra $A$ in the classifiable class ${cal A}$ such that $$ ((K_0(A), K_0(A)_+, [1_A]), K_1(A), T(A), lambda_A)=((G_0, (G_0)_+, u), G_1,S, r).
The class of simple separable KK-contractible (KK-equivalent to ${0}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $Aotimes mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.
Using non-selfadjoint techniques, we establish the Hao-Ng isomorphism for the reduced crossed product and all discrete groups. For the full crossed product an analogous result holds for all discrete groups but the C*-correspondences involved have to be hyperrigid. These results are obtained by calculating the C*-envelope of the reduced crossed product of an operator algebra by a discrete group.
Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $Aotimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $textrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.
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