The Curvature and Index of Completely Positive Maps

We study conjugacy invariants for completely positive maps thatare inspired by the concept of curvature introduced for commutingd-tuples of contractions by Arveson. 2000 Mathematics SubjectClassification 46L53, 46L55, 46L57, 46L87, 47L55, 46L07, 46L08     

Morita Equivalence for Locally C*-Algebras

The concept of Morita equivalence is generalized to the contextof locally C^*-algebras. This generalizes a well-known theoremof Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.2000 Mathematics Subject Classification 46L08, 46L05.     

On the Best Constants in Some Non-Commutative Martingale Inequalities

The optimal orders are determined for the best constants inthe non-commutative Burkholder–Gundy, Doob and Stein inequalitiesobtained recently in non-commutative martingale theory. 2000Mathematics Subject Classification 46L53, 46L51.     

Fourier Invariant Partially Approximating Subalgebras of the Irrational Rotation C*-Algebra

For a dense G-set of parameters, the irrational rotation algebrais shown to contain infinitely many C*-subalgebras satisfyingthe following properties. Each subalgebra is isomorphic to adirect sum of two matrix algebras of the same (perfect square)dimension; the Fourier transform maps each summand onto theother; the corresponding unit projection is approximately central;the compressions of the canonical generators of the irrationalrotation algebra are approximately contained in the subalgebra.2000 Mathematics Subject Classification 46L80, 46L40, 46L35.     

Analytic Rank of Graph Algebras and Quantum Spaces

We show that the analytic rank, as defined by Murphy, of a unitalgraph C*-algebra is either 1 or 0, depending on whether or notthe underlying graph possesses an initial loop. As a corollary,we show that the analytic rank of certain quantum spaces (includingsome quantum spheres, projective spaces and lens spaces) is1. 2000 Mathematics Subject Classification 46L05, 46L65, 46L87.     

Decomposition Rank of Subhomogeneous C*-Algebras

We analyze the decomposition rank (a notion of covering dimensionfor nuclear C*-algebras introduced by E. Kirchberg and the author)of subhomogeneous C*-algebras. In particular, we show that asubhomogeneous C*-algebra has decomposition rank n if and onlyif it is recursive subhomogeneous of topological dimension n,and that n is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C.Phillips to show the following. Let A be the crossed productC*-algebra coming from a compact smooth manifold and a minimaldiffeomorphism. Then the decomposition rank of A is dominatedby the covering dimension of the underlying manifold. 2000 MathematicsSubject Classification 46L85, 46L35.     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.     

Some Characterizations of Commutative Subspace Lattices

Let H be a not necessarily separable Hilbert space, and letBH denote the space of all bounded linear operators on H. Itis proved that a commutative lattice D of self-adjoint projectionsin H that contains 0 and I is spatially complete if and onlyif it is a closed subset of BH in the strong operator topology.Some related results are obtained concerning commutative lattice-orderedcones of self-adjoint operators that contain D. 2000 MathematicsSubject Classification 47D03, 47L35, 47L07, 46L10, 54F05, 54G05,46E05.     

Twisted Partial Actions: A Classification of Regular C*-Algebraic Bundles

We introduce the notion of continuous twisted partial actionsof a locally compact group on a C*-algebra. With such, we constructan associated C*-algebraic bundle called the semidirect productbundle. Our main theorem shows that, given any C*-algebraicbundle which is second countable and which satisfies a certainregularity condition (automatically satisfied if the unit fibrealgebra is stable), there is a continuous twisted partial actionof the base group on the unit fibre algebra, whose associatedsemidirect product bundle is isomorphic to the given one. 1991Mathematics Subject Classification: 46L05, 46L40, 46L55.     

Topological Entropy for the Canonical Endomorphism of Cuntz-Krieger Algebras

It is shown that Voiculescu's topological entropy for the canonicalendomorphism of a simple Cuntz–Krieger algebra O_A equalsthe logarithm of the spectral radius of A. 1991 MathematicsSubject Classification 46L55.     

C*-Algebras Characterized by Ergodic Actions of the n-Torus

Using basic harmonic analysis on the n-torus, it is shown thatthe non-commutative n-tori are the only primitive C*-algebras(up to isomorphism) on which the n-torus Tⁿ acts freely andergodically. An explicit construction recovers the generatingunitaries in a natural way from the action. 2000 MathematicsSubject Classification 46L55.     

On {sigma}-Normal C*-Algebras

A C*-algebra A is said to be monotone (respectively monotone-) complete if every increasing net (respectively increasingsequence) of elements in the ordered space A_h of all hermitianelements of A has a supremum in A_h. It is straightforward toverify that every monotone complete C*-algebra is an AW*-algebra.For type I AW*-algebras, the converse is known to be true. However,for general AW*-algebras, this question is still open, althoughan impressive attack on the problem was made by E. Christensenand G. K. Pedersen, who showed that properly infinite AW*-algebrasare monotone -complete [4]. 1991 Mathematics Subject Classification46L05, 46L06.     

Arens Regularity of the Algebra of Operators on a Banach Space

A short proof is given that if E is a super-reflexive Banachspace, then B(E), the Banach algebra of operators on E withcomposition product, is Arens regular. Some remarks are madeon necessary conditions on E for B(E) to be Arens regular. 2000Mathematics Subject Classification 47L10 (primary), 46H05 (secondary).     

Cohomology of Hopf C*-Algebras and Hopf Von Neumann Algebras

We define two canonical cohomology theories for Hopf C^*-algebrasand for Hopf von Neumann algebras (with coefficients in theircomodules). We then study the situations when these cohomologiesvanish. The cases of locally compact groups and compact quantumgroups are considered in more detail. E-mail: c.k.ng{at}qub.ac.uk2000 Mathematical Subject Classification: primary 46L05, 46L55;secondary 43A07, 22D25.     

Existence of Feller Cocycles on a C*-Algebra

The quantum stochastic differential equation is considered on a unital C^*-algebra, with separablenoise dimension space. Necessary conditions on the matrix ofbounded linear maps for the existence of a completely positivecontractive solution are shown to be sufficient. It is knownthat for completely positive contraction processes, k satisfiessuch an equation if and only if k is a regular Markovian cocycle.‘Feller’ refers to an invariance condition analogousto probabilistic terminology if the algebra is thought of asa non-commutative topological space. 2000 Mathematics SubjectClassification 81S25, 46L07, 46L53, 47D06.     

Purely Infinite Cuntz-Krieger Algebras of Directed Graphs

For arbitrary infinite directed graphs E, the characterisationof the (not necessarily simple) Cuntz–Krieger algebrasC*(E) which are purely infinite in the sense of Kirchberg–Rørdamis given. It is also shown that C*(E) has real rank zero ifand only if the graph E satisfies Condition (K). 2000 MathematicsSubject Classification 46L05.     

C*-Categories

The purpose of this paper is to give a detailed study of thebasic theory of C^*-categories. The study includes some examplesof C^*-categories that occur naturally in geometric applications,such as groupoid C^*-categories, and C^*-categories associatedto structures in coarse geometry. We conclude the paper witha brief survey of Hilbert modules over C^*-categories. 2000 MathematicalSubject Classification: 18D99, 46L05, 46L08.     

The Heat Flow of the CCR Algebra

Let Pf(x) = –if'(x) and Qf(x) = xf(x) be the canonicaloperators acting on an appropriate common dense domain in L²(R).The derivations D_P(A) = i(PA–AP) and D_Q(A) = i(QA–AQ)act on the *-algebra A of all integral operators having smoothkernels of compact support, for example, and one may considerthe noncommutative ‘Laplacian’, L = + , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linearmaps on B(L₂(R)), and this paper establishes some basic propertiesof this semigroup and its minimal dilation to an E₀-semigroup.In particular, the author shows that its minimal dilation ispure and has no normal invariant states, and he discusses thesignificance of those facts for the interaction theory introducedin a previous paper. There are similar results for the canonical commutation relationswith n degrees of freedom, where 1 n < . 2000 MathematicsSubject Classification 46L57 (primary), 46L53, 46L65 (secondary).     

Arens Regularity for Quotients Ap(E) of the Herz Algebra

If the subset T supports a synthesizable p-pseudofunction T,then the restriction A_p(E) to E of the Herz algebra on T isnot Arens regular. The proof is direct, by brute force, anddoes not show the existence of Day points for A_p(E). 2000 MathematicsSubject Classification 43A15, 43A10 (primary), 46L10 (secondary).