Mathematics

Let A be a ∗ -algebra and M be a ∗ - A -bimodule, we study the local properties of ∗ -derivations and ∗ -Jordan derivations from A into M under the following orthogonality conditions on elements in A : a b ∗ =0 , a b ∗ + b ∗ a=0 and a b ∗ = b ∗ a=0 . We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on C ∗ -algebras, group algebra, matrix algebras, algebras of locally measurable operators and von Neumann algebras.

We give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra and a discrete group. In particular, we also answer the question of when the tracial states are in canonical bijection with the invariant tracial states on the original C*-algebra. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group are all finite, and in other cases gives previously known results, for example when the original C*-algebra is commutative. We also obtain results and examples in the case of abelian groups that appear to contradict existing results in the literature.

In this paper we consider the question of what abelian groups can arise as the K -theory of C ??-algebras arising from minimal dynamical systems. We completely characterize the K -theory of the crossed product of a space X with finitely generated K -theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible K -theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their C ??-algebras are classified by their Elliott invariants. We also investigate the K -theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups G 0 and G 1 and any Choquet simplex ? with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated C ??-algebra has K -theory given by this pair of groups and tracial state space affinely homeomorphic to ? . We also improve on the second author's previous results by using our orbit-breaking construction to C ??-algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both K 0 and K 1 . These results have important applications to the Elliott classification program for C ??-algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C ??-algebras associated to étale equivalence relations.

Given a skew-symmetric real n×n matrix Θ we consider the universal enveloping C ∗ -algebra CAR Θ of the ∗ -algebra generated by a 1 ,…, a n subject to the relations a ∗ i a i + a i a ∗ i =1, a ∗ i a j = e 2πi Θ i,j a j a ∗ i , a i a j = e −2πi Θ i,j a j a i . We prove that CAR Θ has a C( K n ) -structure, where K n = [0, 1 2 ] n is the hypercube and describe the fibers. We classify irreducible representations of CAR Θ in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric Θ 1 there are only finitely many Θ 2 such that CAR Θ 1 ≃ CAR Θ 2 . Namely, CAR Θ 1 ≃ CAR Θ 2 implies ( Θ 1 ) ij =±( Θ 2 ) σ(i,j) modZ for a bijection σ of the set {(i,j):i<j, i,j=1,…,n} . For n=2 we give a full classification: CAR θ 1 ≃ CAR θ 2 iff θ 1 =± θ 2 modZ .

Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and C ∗ -algebras with additional C ∗ -algebraic structure. Our approach naturally applies to algebras arising from C ∗ -correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

We classify outer actions (or G -kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama-Takesaki type resolution group approach.

In the mid thirties Murray and von Neumann found a natural way to associate a von Neumann algebra L(Γ) to any countable discrete group Γ . Classifying L(Γ) in term of Γ is a notoriously complex problem as in general the initial data tends to be lost in the von Neumann algebraic regime. An important problem in the theory of von Neumann algebras is to completely describe all possible tensor decompositions of a given group von Neumann algebra L(Γ) . In this direction the main goal is to investigate how exactly a tensor decomposition of L(Γ) relates to the underlying group Γ . In this dissertation we introduce several new classes of groups Γ for which all tensor decompositions of L(Γ) are parametrized by the canonical direct product decompositions of Γ . Specifically, we show that whenever L(Γ)≅ M 1 ⊗ ¯ M 2 where M i are any diffuse von Neumann algebras then there exists a non-canonical direct product decomposition Γ= Γ 1 × Γ 2 such that up to amplifications we have that M 1 ≅L( Γ 1 ) and M 2 ≅L( Γ 2 ) . Our class include large classes of icc (infinite conjugacy class) amalgamated free products and wreath product groups. In addition we obtain similar classifications of tensor decompositions for the von Neumann algebras associated with the T 0 and T 1 group functors introduced by McDuff in 1969 .

We prove that Kellendonk's C ∗ -algebra of an aperiodic and repetitive tiling with finite local complexity is classifiable by the Elliott invariant. Our result follows from showing that tiling C ∗ -algebras are Z -stable, and hence have finite nuclear dimension. To prove Z -stability, we extend Matui's notion of almost finiteness to the setting of étale groupoid actions following the footsteps of Kerr. To use some of Kerr's techniques we have developed a version of the Ornstein-Weiss quasitiling theorem for general étale groupoids.

We classify ∗ -homomorphisms from nuclear C ∗ -algebras into uniform tracial sequence algebras of nuclear Z -stable C ∗ -algebras via tracial data.

We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space L 2 (Irr(G)) of functions which are square integrable with respect to the Plancherel measure. The second condition involves spectra of character-like operators associated with direct integrals of irreducible representations. As examples we study special classes of quantum groups: classical, dual to classical, compact or given by a certain bicrossed product construction.