Inverse semigroups and combinatorial C*-algebras

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case. *Partially supported by CNPq.     

Integrability of dual coactions on Fell bundle C*-algebras

We study integrability for coactions of locally compact groups. For abelian groups, this corresponds to integrability of the associated action of the Pontrjagin dual group. The theory of integrable group actions has been previously studied by Ruy Exel, Ralf Meyer and Marc Rieffel. Our goal is to study the close relationship between integrable group coactions and Fell bundles. As a main result, we prove that dual coactions on C*-algebras of Fell bundles are integrable, generalizing results by Ruy Exel for abelian groups.     

Fell bundles over groupoids

We study the C*-algebras associated to Fell bundles over groupoids and give a notion of equivalence for Fell bundles which guarantees that the associated C*-algebras are strongly Morita equivalent. As a corollary we show that any saturated Fell bundle is equivalent to a semi-direct product arising from the action of the groupoid on a C*-bundle.

     

Infinite simple C*-algebras and Reduced cross products of abelian C*-algebras and free groups

Summary Let Γ=〈g ₁〉*〈g ₂〉*...*〈g _n 〉*... be a free product of cyclic groups with generators {g _i }, andC _r ^* (Γ,℘ _Λ) be the C*-algebra generated by the reduced group C*-algebraC _r ^* Γ and a set of projectionsP _gL associated with a subset Λ of {g _i }. We prove the following: (1)C _r ^* (Γ,℘ _Λ) is *-isomorphic to the reduced cross product for certain Hausdorff compact spaceX _Λ constructed from Γ and its boundary ∂Γ. (2)C _r ^* (Γ,℘ _Λ) is either a purely infinite, simple C*-algebra or an extension of a purely infinite, simple C*-altebra, depending on the pair (Γ, Λ). (3)C _r ^* (Г,℘ _Λ) is nuclear if and only if the subgroup Γ_Λ generated by {g _i }/Λ is amenable. Partially supported by RMC grant 45/290/603 from the University of Newcastle Partially supported by NSF grant DMS-9225076 and a Taft travel grant from the University of Cincinnati     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

Decomposition of operator semigroups on W*-algebras

We consider semigroups of operators on a W^∗-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a “stable” and “reversible” part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W^∗-algebras.     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

Continuous fields of C*-algebras over finite dimensional spaces

Let X be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of C(X)-algebras by C(X)-subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over X with fibers isomorphic to a fixed Cuntz algebra On, n∈{2,3,…,∞}, are locally trivial. They are trivial if n=2 or n=∞. For n?3 finite, such a field is trivial if and only if (n−1)[A₁]=0 in K₀(A), where A is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras D satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over X with fibers isomorphic to D is automatically locally trivial or trivial. In a more general context, we show that a separable unital continuous field over X with fibers isomorphic to a KK-semiprojective Kirchberg C*-algebra is trivial if and only if it satisfies a K-theoretical Fell type condition.     

Projectionless C*-algebras

We give a sufficient condition for a unital C*-algebra to have no nontrivial projections, and we apply this result to known examples and to free products. We also show how questions of existence of projections relate to the norm-connectedness of certain sets of operators.

     

Long-time asymptotic properties of dynamical semigroups onW *-algebras

Mathematische Zeitschrift -     

Ring C*-algebras

We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples. Originally, our motivation comes from algebraic number theory.     

AW*-algebras Which are Enveloping C*-algebras of JC-algebras

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$ , $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I₂, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$ , i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

Noncommutative Jordan C*-algebras

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.     

Separably injective C*-algebras

We show that a C*-algebra is a 1-separably injective Banach space if and only if it is linearly isometric to the Banach space \({C_0(\Omega)}\) of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space \({\Omega}\).     

Topologically injective C*-algebras

A criterion for the topological injectivity of an AW*-algebra as a right Banach module over itself is given. A necessary condition for a C* -algebra to be topologically injective is obtained.     

Non-Hausdorff symmetries of C*-algebras

Symmetry groups or groupoids of C*-algebras associated to non-Hausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C*-algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.