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.     

Vector-valued Hausdorff-Young inequality on compact groups

The main purpose of this paper is to study the validity of theHausdorff–Young inequality for vector-valued functionsdefined on a non-commutative compact group. As we explain inthe introduction, the natural context for this research is thatof operator spaces. This leads us to formulate a whole new theoryof Fourier type and cotype for the category of operator spaces.The present paper is the first step in this program, where thebasic theory is presented, the main examples are analyzed andsome important questions are posed. 2000 Mathematics SubjectClassification 43A77, 46L07.     

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.     

Jensen's Operator Inequality

Jensen's operator inequality and Jensen's trace inequality forreal functions defined on an interval are established in whatmight be called their definitive versions. This is accomplishedby the introduction of genuine non-commutative convex combinationsof operators, as opposed to the contractions considered in earlierversions of the theory by the authors, and by Brown and Kosaki.As a consequence, one no longer needs to impose conditions onthe interval of definition. It is shown how this relates tothe pinching inequality of Davis, and how Jensen's trace inequalitygeneralizes to C*-algebras. 2000 Mathematics Subject Classification47A56 (primary), 46L10, 47C15, 26A51 (secondary).     

Weak Laws of Large Numbers in Some Non-Commutative Spaces

We prove weak laws of large numbers for freely independent,uniformly bounded and non-identically distributed random variablesbelonging to non-commutative Ciach and, in particular, LorentzL^p-spaces. 2000 Mathematics Subject Classification 60B11, 60F25,46L54, 47N30.     

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     

Prime Non-Commutative JB*-Algebras

We prove that if A is a prime non-commutative JB*-algebra whichis neither quadratic nor commutative, then there exist a primeC*-algebra B and a real number with < 1 such that A =B as involutive Banach spaces, and the product of A is relatedto that of B (denoted by , say) by means of the equality xy= xy+(1 – )yx. 2000 Mathematics Subject Classification46K70, 46L70.     

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.     

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.     

The Operator Hilbert Space OH and Type III Von Neumann Algebras

A proof is given to show that the operator Hilbert space OHdoes not embed completely isomorphically into the predual ofa semi-finite von Neumann algebra. This complements Junge'srecent result, which admits such an embedding in the non-semi-finitecase. 2000 Mathematics Subject Classification 46L07, 46L54,47L25, 47L50.     

Monic non-commutative orthogonal polynomials

Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint distribution of operators on a Fock space.

     

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.     

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.     

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).     

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.     

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.     

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.     

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.     

Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.