Type I Toeplitz algebras

The class of Toeplitz algebras associated to ordered groups is important in the analysis of Toeplitz operators on the generalised Hardy spaces defined by such groups. The conditions under which these Toeplitz algebras are Type I C^*-algebras are investigated.     

Nuclear C*-algebras and related questions

Results are given characterizing the class of nuclear C^*-algebras from various points of view, and a number of consequences of the nuclearity condition are given, and properties of C^*-algebras and W^*-algebras related to nuclearity are discussed.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 26, pp. 107–126, 1985.     

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.     

On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Tensor Products of Hilbert Modules Over Locally <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

In this paper the tensor products of Hilbert modules over locally C ^*-algebras are defined and their properties are studied. Thus we show that most of the basic properties of the tensor products of Hilbert C ^*-modules are also valid in the context of Hilbert modules over locally C ^*-algebras.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

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.     

Decomposition rank and \mathcal{Z} -stability

We show that separable, simple, nonelementary, unital C^*-algebras with finite decomposition rank absorb the Jiang–Su algebra Z\mathcal{Z} tensorially. This has a number of consequences for Elliott’s program to classify nuclear C^*-algebras by their K-theory data. In particular, it completes the classification of C^*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.     

The Proof of Inseparable Prime AW^＊—Algebras Being Primitive C^＊—Algebras

The relation between the inseparable prime C^*-algebras and primitive C^*-algebras is studied,and we prove that prime AW^*-algebras are all primitive C^*-algebras.     

Multivariable Wiener-Hopf operators II. Spectral topology and solvability

The C^*-algebras generated by Wiener-Hopf Operators on tangible closed cones are solvable: they have finite composition series of ideals with subquotients which are homogeneous C^*-algebras with continuous trace. In particular this refers to cones with piece-wise smooth boundary and to cones which are finite sets of orbits under some linear action.     

The Riccati algebraic equation in a locallyC ⋆-algebra

The stabilizability and complete stabilizability of a pair of elements and the structure of the set of selfadjoint solutions of the algebraic Riccati equation is studied inC ^*-algebras and locallyC ^*-algebras which includes certain algebras of unbounded operators. Under some mild assumptions, selfadjoint solutions are in one-to-one correspondence with a particular set of idempotents.     

Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB ^*-algebra leads to the study of properGB ^*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB ^*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC ^*-algebra and two other representation theorems forb ^*-algebras (also calledlmc ^*-algebras), one representinga b ^*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeL ^p-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andL ^w-integral of a measurable field ofC ^*-algebras are discussed briefly.     

Homotopy invariance of AF-embeddability

We prove that AF-embeddability is a homotopy invariant in the class of separable exact C ^*-algebras. This work was inspired by Spielberg's work on homotopy invariance of AF-embeddability and Dadarlat's serial works on AF-embeddability of residually finite dimensional C ^*-algebras. Submitted: February 2002.     

Certain classes of <Emphasis Type="Italic">C</Emphasis>*-algebras preserved by tracial approximation

We show that the following classes of C*-algebras in the classes Ω are inherited by simple unital C*-algebras in the classes TAΩ: (1) simple unital purely infinite C*-algebras, (2) unital isometrically rich C*-algebras, (3) unital Riesz interpolation C*-algebras.     

Quotients of partial abelian monoids and the Riesz decomposition property

Partial abelian monoids (PAMs) are structures (), where is a partially defined binary operation with domain , which is commutative and associative in a restricted sense, and 0 is the neutral element. PAMs with the Riesz decomposition properties and binary relations with special properties on PAMs are studied. Relations with abelian groups, dimension equivalence and K ₀ for AF C^*-algebras are discussed. Received September 17, 2000; accepted in final form March 13, 2002.     

Stability under Small Hilbert–Schmidt Perturbations for <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper studies the tracial stability of C^*-algebras, which is a general property of stability of relations in a Hilbert–Schmidt-type norm defined by a trace on a C^*-algebra. Precise definitions are formulated in terms of tracial ultraproducts. For nuclear C^*-algebras, a characterization of matricial tracial stability in terms of approximation of tracial states by traces of finite-dimensional representations is obtained. For the nonnuclear case, new obstructions and counterexamples are constructed in terms of free entropy theory.     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-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.