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.     

The Trace in Banach Jordan Pairs

The algebraic trace form (as defined by O. Loos) of an element(x, y) of a (complex) Banach Jordan pair V, where x or y isin the socle, is equal to the sum of the products of all spectralvalues and their multiplicity. The trace form is calculatedfor two examples, the Banach Jordan pair of bounded linear operatorsbetween two Banach spaces, and the Banach Jordan pair of a quadraticform. Using analytic multifunctions, it is also shown that thecomplement of the socle of a Banach Jordan pair V is eitherdense or empty. In the last case, V has finite capacity. 1991Mathematics Subject Classification 17C65, 46H70.     

On the existence of Analytic Semigroups Bounded on the Half-Disc in some Banach Algebras

Let G be a locally compact group. We show that L¹(G) does notcontain any analytic semigroups which are bounded on the half-disc,by using some facts about the analytic Radon-Nikodm propertyin Banach spaces. This solves a question raised by J. Esterle.     

Wild Spectral Behaviour of Anharmonic Oscillators

We show that the norms of the sequence of spectral projectionsof certain non-self-adjoint ordinary differential operatorscannot be polynomially bounded. The proof involves the investigationof ‘tame’ bi-orthogonal systems in Banach spaces.2000 Mathematics Subject Classification 34L10, 46B15.     

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).     

Regularity of the Fractional Maximal Function

The purpose of this work is to show that the fractional maximaloperator has somewhat unexpected regularity properties. Themain result shows that the fractional maximal operator mapsL^p-spaces boundedly into certain first-order Sobolev spaces.It is also proved that the fractional maximal operator preservesfirst-order Sobolev spaces. This extends known results for theHardy-Littlewood maximal operator. 2000 Mathematics SubjectClassification 42B25, 46E35.     

Rank Distributions on Coarse Spaces and Ideal Structure of Roe Algebras

The notions of controlled truncations for operators in the Roealgebras C^* (X) of a coarse space (X, ) with uniformly locallyfinite coarse structure, and rank distributions on (X, ) areintroduced. It is shown that the controlled propagation operatorsin an ideal I of C^* (X) are exactly the controlled truncationsof elements in I. It follows that the lattice of the idealsof C^* (X) in which controlled propagation operators are denseis isomorphic to the lattice of all rank distributions on (X,). If X is a discrete metric space with Yu's property A, thenthe ideal structure of the Roe algebra C^* (X is completely determinedby the rank distributions on X. 2000 Mathematics Subject Classification46L80, 46L89.     

Algebraic Isomorphisms and Finite Distributive Subspace Lattices

Let L₁ and L₂ be finite distributive subspace lattices on realor complex Banach spaces. It is shown that every rank-preservingalgebraic isomorphism of AlgL₁ onto AlgL₂ is quasi-spatiallyinduced. If the algebraic isomorphism in question is known onlyto preserve the rank of rank one operators, then it inducesa lattice isomorphism between L₁ and L₂.     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

Banach Spaces Whose Algebras of Operators are Unitary: A Holomorphic Approach

An element u of a norm-unital Banach algebra A is said to beunitary if u is invertible in A and satisfies ||u|| = ||u^–1||= 1. The norm-unital Banach algebra A is called unitary if theconvex hull of the set of its unitary elements is norm-densein the closed unit ball of A. If X is a complex Hilbert space,then the algebra BL(X) of all bounded linear operators on Xis unitary by the Russo–Dye theorem. The question of whetherthis property characterizes complex Hilbert spaces among complexBanach spaces seems to be open. Some partial affirmative answersto this question are proved here. In particular, a complex Banachspace X is a Hilbert space if (and only if) BL(X) is unitaryand, for Y equal to X, X* or X** there exists a biholomorphicautomorphism of the open unit ball of Y that cannot be extendedto a surjective linear isometry on Y. 2000 Mathematics SubjectClassification 46B04, 46B10, 46B20.     

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.     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

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.     

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.     

Strong Operator Modules and the Haagerup Tensor Product

Strong operator modules over a von Neumann algebra R are introducedand the so-called extended Haagerup tensor product over R ofstrong modules is studied. In the case R=C and the spaces areweak* closed this product agrees with the weak* Haagerup tensorproduct of Blecher and Smith. If C is the center of R, the extendedHaagerup tensor product R is a Banach algebra containing thecentral tensor product of Chatterjee and Smith and has very nice properties concerningslice maps and the relative commutants of subalgebras. It isshown that R is a dual Banach space, and all weak* closed two-sidedideals of the algebra R are determined. 1991 Mathematics SubjectClassification: 46L05     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

Metrizable Barrelled Countable Enlargements

In 1980, Robertson, Tweddle and Yeomans solved the metrizableBCE problem in certain special cases, for example, in the normablecase under assumption of the Continuum Hypothesis (CH). A completepositive solution was recently given by the author and L. M.Sánchez Ruiz under an assumption, weaker than CH, thatscales exist. A better partial solution for the general BCEproblem obviates scales or other extra-ZFC assumptions, andentirely lays to rest the metrizable BCE problem. The proofstimulates thought about how the least infinite-dimensionalityof normed barrelled spaces impinges on the separable quotientproblem for Banach spaces. 1991 Mathematics Subject Classification46A08.     

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.     

Hyperbolic equations, function spaces with exponential weights and Nemytskij operators

Different problems in the theory of hyperbolic equations bases on function spaces of Gevrey type are studied. Beside the original Gevrey classes, spaces defined by the behaviour of the Fourier transform were also used to prove basic results about the well-posedness of Cauchy problems for non-linear hyperbolic systems. In these approaches only the algebra property of the function spaces was used to include analytic non-linearities. Here we will generalize this dependence. First we investigate superposition operators in spaces with exponential weights. Then we show in concrete situations how a priori estimates of strictly hyperbolic type lead to the well-posedness of certain semi-linear hyperbolic Cauchy problems in suitable function spaces with exponential weights of Gevrey type. Mathematics Subject Classification (2000) 46E35, 35L80, 35L15, 47H30