We prove that two dual operator spaces $X$ and $Y$ are stably isomorphic if and only if there exist completely isometric normal representations $ϕ$ and $ψ$ of $X$ and $Y$, respectively, and ternary rings of operators $M_1, M_2$ such that $ϕ(X)= [M_2^*ψ(Y)M_1]^{-w^*}$ and $ψ(Y)=[M_2ϕ(X)M_1^*].$ We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. We provide examples motivated by CSL algebra theory.
We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic.
Rigged modules over an operator algebra are a generalization of Hilbert modules over a $C^{\star}$-algebra. We characterize the rigged modules over an operator algebra $\mathcal A$ which are orthogonally complemented in $C_\infty(\mathcal A),$ the space of infinite columns with entries in $\mathcal A.$ We show that every such rigged module `restricts' to a bimodule of Morita equivalence between appropriate stably isomorphic operator algebras.
It is shown that the coloured isomorphism class of a unital, simple, $\mathcal{Z}$-stable, separable amenable C$^*$-algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex.
The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of the n by n matrices to its dual basis is a complete order isomorphism and complete co-order isomorphism. In the case of the standard matrix units this map is a complete order isomorphism and this is a restatement of the correspondence between completely positive maps and the Choi matrix. However, we exhibit natural orthonormal bases for the matrices such that this map is an order…
We prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz-Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behavior of the stable conjugacy between the associated stable Cuntz-Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Marko…
In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a *-isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.
For a Banach D-bimodule M over an abelian unital C*-algebra D, we define E(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual of M. Equip E(M) with the weak-* topology. We develop general properties of E(M). It is properly viewed as a coordinate system for M when M is a subset of C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E(C) coincides with Kumjian's twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal which contains D. For triangular subalg…
We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Δ$-equivalent if they have completely isometric representations $α$ and $β$ respectively and there exists a ternary ring of operators $M$ such that $α(A)$ (resp. $β(B)$) is equal to the norm closure of the linear span of the set $M^*β(B)M, $ (resp. $Mα(A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Δ$-equivalent, then the operator algebras $A\otimes \cl K$ and $B\otimes \cl K$ are iso…
Let $A$ be a separable $C^*$-algebra and $B$ a stable $C^*$-algebra containing a strictly positive element. We show that the group $\Ext(SA,B)$ of unitary equivalence classes of extensions of $SA$ by $B$, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between $\Ext(SA,B)$ and the $E$-theory group $E(A,B)$ of homotopy classes of asymptotic homomorphisms from $S^2A$ to $B$.
It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious "missing T-duals.'' Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative to…
We give an elementary proof for Lewis Bowen's theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and every free product Γof n infinite amenable groups, the factor K^Γ/K of the Bernoulli action of Γon K^Γ by the diagonal action of K, is isomorphic with a Bernoulli action of Γ.
We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part…
It is shown that, for a C*-algebra of stable rank one (i.e., in which the invertible elements are dense), two well-known isomorphism invariants, the Cuntz semigroup and the Thomsen semigroup, contain the same information. More precisely, these two invariants, viewed appropiately, determine each other in a natural way.
Rigidity for Isomorphisms between Operator Algebras with Commutative Diagonals
East Carolina University（东卡罗来纳大学）；Shandong Technology and Business University（山东工商学院）；Chongqing Normal University（重庆师范大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）；University of Chinese Academy of Sciences（中国科学院大学）
We establish axiomatic characterizations of $K$-theory and $KK$-theory for real C*-algebras. In particular, let $F$ be an abelian group-valued functor on separable real C*-algebras. We prove that if $F$ is homotopy invariant, stable, and split exact, then $F$ factors through the category $KK$. Also, if $F$ is homotopy invariant, stable, half exact, continuous, and satisfies an appropriate dimension axiom, then there is a natural isomorphism $K(A) \to F(A)$ for a large class of separable real C*-algebras $A$. Furthermore, we prove that a natural transformation $F(A) \to G(A)$ of homotopy invari…
We consider isomorphisms between quotient algebras of $\prod_{n=0}^{\infty} \mathbb{M}_{k(n)}(\mathbb{C})$ associated with Borel ideals on $\mathbb{N}$ and prove that it is relatively consistent with \textbf{ZFC} that all of these isomorphisms are trivial, in the sense that they lift to a *-homomorphism from $\prod_{n=0}^{\infty} \mathbb{M}_{k(n)}(\mathbb{C})$ into itself. This generalizes a result of Farah-Shelah who proved this result for centers of these algebras (in its dual form). We also use a simpler forcing notion and completely remove the large cardinal assumption used by Farah-Shelah…
We first study situations where the stable AF-algebras defined by two square primitive nonsingular incidence matrices with nonnegative integer matrix elements are isomorphic even though no powers of the associated automorphisms of the corresponding dimension groups are isomorphic. More generally we consider neccessary and sufficient conditions for two such matrices to determine isomorphic dimension groups. We give several examples.
We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).
We investigate to what extent a nilpotent Lie group is determined by its $C^*$-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension $\le 5$ are uniquely determined by the Morita equivalence class of their $C^*$-algebras. We also find that this last property is shared by the filiform Lie groups and the $6$-dimensional free two-step nilpotent Lie group.
We consider two categories of C*-algebras; in the first, the isomorphisms are ordinary isomorphisms, and in the second, the isomorphisms are Morita equivalences. We show how these two categories, and categories of dynamical systems based on them, crop up in a variety of C*-algebraic contexts. We show that Rieffel's construction of a fixed-point algebra for a proper action can be made into functors defined on these categories, and that his Morita equivalence then gives a natural isomorphism between these functors and crossed-product functors. These results have interesting applications to non-a…
The notion of isomorphism of stable AF-C*-algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C*-isomorphism induces an equivalence relation on these matrices, called C*-equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e., there is an algorithm that can be used to check in a finite number of steps whether two given primitive nonsingular matrices are C*-equivalent or not.
