Primitive partial permutation representations of the polycyclic monoids and branching function systems

We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of Kawamura on certain kinds of representations of the Cuntz C*-algebras and to the branching function systems of Bratteli and Jorgensen.      

Geometric structure of dimension functions of certain continuous fields

For a continuous field of C^?-algebras A, we give a criterion to ensure that the stable rank of A is one. In the particular case of a trivial field this leads to a characterization of stable rank one, completing accomplishments by Nagisa, Osaka and Phillips. Further, for certain continuous fields of C^?-algebras, we study when the Cuntz semigroup satisfies the Riesz interpolation property, and we also analyze the structure of its functionals. As an application, we obtain a positive answer to a conjecture posed by Blackadar and Handelman in a variety of situations.     

A purely infinite AH-algebra and an application to AF-embeddability

We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit ofC*-algebras of the formC ₀([0, 1),M _k ) and it absorbs the Cuntz algebra ctive limit of the finite and elementaryC*-algebrasC ₀([0, 1),M _k ). As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exactC*-algebra is AF-embeddable, and we exhibit a concrete AF-algebra into which this class ofC*-algebras can be embedded.     

Strict Topologies as Topological Algebras

Let X be a completely regular Hausdorff space, C_b(X) the space of all scalar-valued bounded continuous functions on X with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally m-convex.     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.     

Semigroup C⁎-algebras and amenability of semigroups

We construct reduced and full semigroup C^?-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz. Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C^?-algebras in analogy to the group case.     

Noncommutative differential geometry,quantization, and smooth symmetries of the C*-algebras associated to topological dynamics

Noncommutative differential geometric structures are considered for a class of simple C^*-algebras. This structure is defined in terms of smooth Lie group actions on the C^*-algebra in question together with a certain quantization mapping motivated directly by the known cohomological obstructions for the quantum mechanical quantization correspondence. We show that such a quantization mapping may be constructed for the C^*-algebras associated to antisymmetric bi-characters and for the Cuntz/Cuntz-Krieger C^*-algebras associated to topological dynamics. A certain curvature obstruction is defined in terms of the quantization mapping. It is shown that existence of smooth Lie group actions is determined by the curvature obstruction.     

MAURER-CARTAN EQUATION OF A∞-ALGEBRAS

The axioms of A∞-algebras can be written as Maurer-Cartan equation. Infinitesimal multiplication of a graded associative algebra is defined and the integrability of infinitesimal multiplication is discussed through the Massey F-product.     

Extension algebras of Cuntz algebra

In this paper, we construct representatives for all equivalence classes of the unital essential extension algebras of Cuntz algebra by the C^*-algebras of compact operators on a separable infinite-dimensional Hilbert space. We also compute their K-groups and semigroups and classify these extension algebras up to isomorphism by their semigroups.     

Harmonic Analysis of Fractal Processes via C* - Algebras

We construct a harmonic analysis of iteration systems which include those which arise from wavelet algorithms based on multiresolutions. While traditional discretizations lead to asymptotic formulas, we argue here for a direct Fourier duality; but it is based on a non - commutative harmonic analysis, specifically on representations of the Cuntz C* -algebras. With this approach the waling from the wavelet takes the form of an endomorphism of B(H), H a Hilbert space derived from the lattice of translations. We use this to describe, and to calculate, new invariants for the wavelets. those iteration systems which arise from wavelets and from Julia sets, we show that the associated endomorphisms are in fact Powers shifts.     

Topological algebras with maximal regular ideals closed

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.     

Representations and positive functionals

We discuss the representation theory of both the locally convex and non-locally convex topological^*-algebras. First we discuss the^*-representation of topological^*-algebras by operators on a Hilbert space. Then we study those topological^*-algebras so that every^*-representation of which on a Hilbert space is necessarily continuous. It is well-known that each^*-representation of aB ^*-algebra on a Hilbert space is continuous. We show that this is true for a large class of^*-algebras more general thanB ^*-algebras, including certain non-locally convex^*-algebras. Finally, we investigate the conditions under which a positive functional on a topological^*-algebra is representable.The research of the first-named author was partially supported by an NSERC grant. This work was done by the second-named author when he was a post-doctoral fellow at McMaster University.     

C*-Bundles for Certain Liminal C*-Algebras

It is shown that certain liminal C*-algebras whose limit sets in their primitive ideal space are discrete can be described as algebras of continuous sections of a C*-bundle associated with them. Their multiplier algebras are also described in a similar manner. The class of C*-algebras under discussion includes all the liminal C*-algebras with Hausdorff primitive ideal spaces but also many other liminal algebras. A large sub-class of examples is examined in detail.      

K-theory of Continuous Deformations of C＊-algebras

We study K-theory of continuous deformations of C＊-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.     

Universal continuous calculus for Su*-algebras

Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*-algebra (an ordered *-algebra that is symmetric, i.e., “strictly” positive elements are invertible and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for $C^{\ast }$-algebras to a class of generally unbounded ordered *-algebras. On the way, some results about *-algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any representation theory.     

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.     

Joint continuity of multiplication on the dual of the left uniformly continuous functions

Let G be a locally compact group and LUC(G) the C*-algebra of the bounded left uniformly continuous functions on G. The spectrum G ^LUC of LUC(G) is the universal semigroup compactification of G with respect to the joint continuity property: the multiplication on G×G ^LUC is jointly continuous. The paper studies the joint weak* continuity of multiplication on LUC(G)^* and, in particular, the question how the joint continuity property of G ^LUC can be related to a property concerning the whole algebra LUC(G)^*. The group G is naturally replaced by the measure algebra M(G), and LUC(G)^* can be identified with M(G ^LUC), the space of regular Borel measures on G ^LUC. It is shown that the joint weak* continuity can fail even on bounded sets of M(G)×M(G ^LUC), but, on the other hand, the multiplication on M(G)×M(G ^LUC) is positive continuous in the sense of Jewett.