Critères de platitude et de projectivité

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Suppose $(T,\mathcal{M})$ is a measurable space, X is a topological space, and $\emptyset \ne F(t) \subset X$ for $t \in T$. Denote ${\operatorname {Gr}}F = \{ (t,x):x \in F(t)\} $. The problem surveyed (reviewing work of others) is that of existence off: $f:T \to X$ such that $f(t) \in F(t)$ for $t \in T$ and $f^{ - 1} (U) \in \mathcal{M}$ for open $U \subset X$. The principal conditions that yield such f are (i) X is Polish, each $F(t)$ is closed, and $\{ t:F(t) \cap U \ne \emptyset \} \in \mathcal{M}$ a .tit whenever $U \subset X$ is open (Kuratowski and Ryll-Nardzewski and, under stronger assumption, Castaing), or (ii) T is a Hausdorff space, ${\operatorname {Gr}}F$ is a continuous image of a Polish space, and M is the $\sigma $-algebra of sets measurable with respect to an outer measure, among which are the open sets of T (primarily von Neumann). The latter result follows from the former by lifting F in a natural way to a map into the closed sets of a Polish space. This procedure leads to the theory ...

Survey of Measurable Selection Theorems

In topological K-theory, which can be viewed as the algebraic side of the theory of vector bundles, some of the interesting properties which one investigates are, for example, the conditions under which bundles must possess trivial direct summands, or the extent to which the cancellation property for direct (Whitney) sums of bundles holds. Such properties turn out to be controlled in part by the dimension of the base space, and results describing the nature of this control are among what are frequently called stability results [17]. During the past few years there has been a sudden flowering of a /^-theory for C*algebras. (See surveys [35, 36, 37].) Since C*-algebras are profitably thought of as 'non-commutative locally compact spaces', with the finitely generated projective modules being the appropriate generalization of vector bundles according to Swan's theorem [40], it would be natural to look for stability results for C*-algebras. But until now the theory has been focussed on the /C-groups of C*-algebras, and there has been little discussion of stability properties, presumably in part for lack of an appropriate concept of dimension for C*-algebras. One of the main objectives of this paper is to introduce a concept of dimension for C*-algebras which directly generalizes the classical concept of dimension for compact spaces [16, 20, 22], and to develop some techniques for calculating it, notably for C*algebras which are obtained as crossed-product algebras for an action of the group of integers. In algebraic K-theory, which is the generalization of topological /C-theory to rings, a substantial number of stability properties have been discovered [2, 39, 42-48]. Within that theory the concept which has played a role most analogous to that of dimension is the Bass stable rank. But despite its many successes, the Bass stable rank has often been difficult to calculate even for relatively uncomplicated rings. Another main objective of the present paper is to make accessible for application to C*algebras some of the stability results from algebraic /C-theory. This comes about from the fact that the concept of dimension which we introduce dominates the Bass stable rank. I was led to seek stability results for C*-algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation C*algebras discussed in [30] (which, as Elliott [11] has suggested, can appropriately be called 2-dimensional 'non-commutative tori'). In more concrete terms, the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent. The present paper provides the general theory which is needed for the affirmative answer which I obtained. In a subsequent paper I will derive the

Dimension and Stable Rank in the K‐Theory of C*‐Algebras

The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].)

Projective Modules over Higher-Dimensional Non-Commutative Tori

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Purity and algebraic compactness for modules

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Cancellation of modules and groups and stable range of endomorphism rings.

An Introduction to Noncommutative Noetherian Rings: Prologue

In 1959 A. F. Filippov published a paper [1] containing a lemma designed for use in the study of optimal control problems. Stated somewhat imprecisely, let k be a continuous function on a compact set Q in a finite dimensional space and with values in a finite dimensional space, and let (u'(t): a?<t <b) be a function with values in Q such that k(u'(.)) is measurable; then there exists a measurable function u from [a, b] to Q such that k(u(t)) ==k(u'(t)). For purposes of stochastic control theory it is desirable to extend this to allow arbitrary measure spaces, instead of intervals of real numbers (Kushner [4]); for purposes of the calculus of variations it is desirable to relax the requirement of compactness on Q. We do both of these (Q may be any separable metric space), and at the same time we permit the values of k to lie in any Hausdorff space; it costs nothing. In ?2 we give an application to optimal control theory. ADDED IN PROOF. In an abstract in Amer. Math. Monthly 73 (1966), p. 927, M. Q. Jacobs announces a generalization of Filippov's lemma. This generalization is a special case of our Theorem 1.

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On Filippov’s implicit functions lemma

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R. B. Warfield

Papers

Purity and algebraic compactness for modules

Cancellation of modules and groups and stable range of endomorphism rings.

An Introduction to Noncommutative Noetherian Rings: Prologue

On Filippov’s implicit functions lemma

Decompositions of injective modules