Mathematics

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et. al, we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.

We investigate the pure infiniteness and stable finiteness of the Exel-Pardo C ∗ -algebras O G,E for countable self-similar graphs (G,E,φ) . In particular, we associate a specific ordinary graph E ˜ to (G,E,φ) such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph C ∗ -algebra C ∗ ( E ˜ ) imply that of O G,E . Among others, this follows a dichotomy for simple O G,E : if (G,E,φ) contains no G -circuits, then O G,E is stably finite; otherwise, O G,E is purely infinite. Furthermore, Li and Yang recently introduced self-similar k -graph C ∗ -algebras O G,Λ . We also show that when | Λ 0 |<∞ and O G,Λ is simple, then it is purely infinite.

We prove that a finite index regular inclusion of I I 1 -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of I I 1 -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner-Popa basis (respectively, a unitary orthonormal basis)

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products C(X) ??λ G , where G is a countable discrete group, and X is a compact Hausdorff space which G acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of C(X) ??λ G and to Kawabe's generalized space of amenable subgroups $\Sub_a(X,G)$. This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if C(Y)?�C(X) is an inclusion of unital commutative G -C*-algebras with X minimal and C(Y) ??λ G simple, then any intermediate C*-algebra A satisfying C(Y) ??λ G?�A?�C(X) ??λ G is simple.

A linear mapping T on a JB ∗ -triple is called triple derivable at orthogonal pairs if for every a,b,c∈E with a⊥b we have 0={T(a),b,c}+{a,T(b),c}+{a,b,T(c)}. We prove that for each bounded linear mapping T on a JB ∗ -algebra A the following assertions are equivalent: (a) T is triple derivable at zero; (b) T is triple derivable at orthogonal elements; (c) There exists a Jordan ∗ -derivation D:A→ A ∗∗ , a central element ξ∈ A ∗∗ sa , and an anti-symmetric element η in the multiplier algebra of A , such that T(a)=D(a)+ξ∘a+η∘a, for all a∈A; (d) There exist a triple derivation δ:A→ A ∗∗ and a symmetric element S in the centroid of A ∗∗ such that T=δ+S . The result is new even in the case of C ∗ -algebras. We next establish a new characterization of those linear maps on a JBW ∗ -triple which are triple derivations in terms of a good local behavior on Peirce 2-subspaces. We also prove that assuming some extra conditions on a JBW ∗ -triple M , the following statements are equivalent for each bounded linear mapping T on M : (a) T is triple derivable at orthogonal pairs; (b) There exists a triple derivation δ:M→M and an operator S in the centroid of M such that T=δ+S . \end{enumerate}

We give a local characterization for the Cuntz semigroup of AI-algebras building upon Shen's characterization of dimension groups. Using this result, we provide an abstract characterization for the Cuntz semigroup of AI-algebras.

The classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution.

We construct a pure state on the C*-algebra B( ℓ 2 ) of all bounded linear operators on ℓ 2 which is not diagonalizable, i.e., it is not of the form lim u ⟨T( e k ), e k ⟩ for any orthonormal basis ( e k ) k∈N of ℓ 2 and an ultrafilter u on N . This constitutes a counterexample to Anderson's conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison-Singer problem due to A. Marcus, D. Spielman, N. Srivastava that the restriction of our pure state to any atomic masa D(( e k ) k∈N ) of diagonal operators with respect to an orthonormal basis ( e k ) k∈N is not multiplicative on D(( e k ) k∈N ) .

We construct the first example of a C ∗ -algebra A with the properties in the title. This gives a new example of non-nuclear A for which there is a unique C ∗ -norm on A⊗ A op . This example is of particular interest in connection with the Connes-Kirchberg problem, which is equivalent to the question whether $C^*({\bb F}_2)$, which is known to have the LLP, also has the WEP. Our C ∗ -algebra A has the same collection of finite dimensional operator subspaces as $C^*({\bb F}_2)$ or $C^*({\bb F}_\infty)$. In addition our example can be made to be quasidiagonal and of similarity degree (or length) 3. In the second part of the paper we reformulate our construction in the more general framework of a C ∗ -algebra that can be described as the \emph{limit both inductive and projective} for a sequence of C ∗ -algebras ( C n ) when each C n is a \emph{subquotient} of C n+1 . We use this to show that for certain local properties of injective (non-surjective) ∗ -homomorphisms, there are C ∗ -algebras for which the identity map has the same properties as the ∗ -homomorphisms.

One of the earliest invariants introduced in the study of finite von Neumann algebras is the property Gamma of Murray and von Neumann. In this note we prove that it is not possible to classify separable II 1 factors satisfying the property Gamma up to isomorphism by a Borel measurable assignment of countable structures as invariants. We also show that the same holds true for the full II 1 factors.