Claude Schochet

Global analysis on foliated spaces

Global analysis has as its primary focus the interplay between the local analysis and the global geometry and topology of a manifold. This is seen classicallv in the Gauss-Bonnet theorem and its generalizations. which culminate in the Ativah-Singer Index Theorem [ASI] which places constraints on the solutions of elliptic systems of partial differential equations in terms of the Fredholm index of the associated elliptic operator and characteristic differential forms which are related to global topologie al properties of the manifold. The Ativah-Singer Index Theorem has been generalized in several directions. notably by Atiyah-Singer to an index theorem for families [AS4]. The typical setting here is given by a family of elliptic operators (Pb) on the total space of a fibre bundle P = F_M_B. where is defined the Hilbert space on Pb 2 L 1p -llbl.dvollFll. In this case there is an abstract index class indlPI E ROIBI. Once the problem is properly formulated it turns out that no further deep analvtic information is needed in order to identify the class. These theorems and their equivariant counterparts have been enormously useful in topology. geometry. physics. and in representation theory.

CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

The Index Theorem

In this chapter we compute the index of a tangentially elliptic pseudodifferential operator on a compact foliated space.

Banach algebras and rational homotopy theory

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GLn(A), the group of invertible n × n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lcn(A) denote the space of “last columns” of GLn(A). We construct a natural isomorphism Ȟ(Max(A);Q) ∼= π2n−1−s(Lcn(A))⊗ Q for n > 1 2 s+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F (X,G) for G a Lie group or, more generally, a rational H-space.

Steenrod Homology and Operator Algebras

The recent work of Larry Brown, R. G. Douglas, and Peter Fillmore (referred to as BDF) [2], [3], and [4] on operator algebras has created a new bridge between functional analysis and algebraic topology. This note and a subsequent paper [5] constitute an effort to make that bridge more concrete. We first briefly describe the BDF framework. This requires the following C*-algebras: C(X), the continuous complex-valued functions on a compact metric space X; L, the bounded operators on an infinite dimensional separable Hubert space; K C L, the compact operators; and L/K, the Calkin algebra. (Let n: L —• L/K be the projection.) An extension is a short exact sequence of C*-algebras and C*-algebra morphisms of the form 0 —* K —• E —• C(X) —• 0 where E is a C*-algebra containing K and / (the identity operator) and contained in L. Unitary equivalence classes of extensions form an abelian group, denoted Ext(T). Ext(X) was invented by BDF in order to study essentially normal operators, that is, operators TG L with nT normal. Let ET denote the C*algebra generated by I, T, and K, and let X = o(jiT), the spectrum of rrT. Then the exact sequence 0 —• K —* ET —• C(X) —* 0 represents an element of Ext(X). This element is zero if and only if T is a compact perturbation of a normal operator. For I C C , BDF prove that

Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’

Suppose that A is a C∗-algebra in the bootstrap category N with KKfiltration A0 ↪→ A1 ↪→ A2 ↪→ . . . and B is a C∗-algebra with a countable approximate unit. Then the graded Kasparov group KK∗(A, B) is described both by the Universal Coefficient Theorem 0 → Ext Z (K∗(A), K∗(B)) → KK∗(A, B) → HomZ(K∗(A), K∗(B)) → 0 and by the Milnor lim ←− sequence 0 → lim ←− KK∗(Ai, B) −→ KK∗(A, B) −→ lim ←− KK∗(Ai,B) → 0. It is demonstrated that these two descriptions are closely related and that KK∗(A, B) decomposes unnaturally as the direct sum of the term

ON THE K-THEORY OF QUARTER-PLANE TOEPLITZ ALGEBRAS

Given a C*-algebra A which is filtered by a collection of closed ideals Ai, there is a spectral sequence which relates the K-theory of A to the K-theory of the various quotient algebras Ai/Ai−1. The d1 differentials in this spectral sequence are familiar index invariants, but the higher differentials are not well-understood. Considering the case of Toeplitz C*-algebras associated with certain cones in Z2, it is shown that a d2 differential in the spectral sequence is non-trivial. This differential turns out to be an obstruction to a classical lifting problem in operator theory. Analysis of this obstruction leads to necessary and sufficient conditions for the lifting problem. It is hoped that this example will illuminate the role of higher differentials in the K-theory spectral sequence.

On the structure of graded formal groups of finite characteristic

The structure of a wide range of Hopf algebras over fields is well known. Corresponding theorems for Hopf algebras over graded rings do not exist. This note is devoted to a Decomposition Theorem for certain Hopf algebras over rings (of finite characteristic) of the types which occur in topology, and a study of homological properties of the factors which therein arise.

Spanier-Whitehead K-duality for

Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier-Whitehead

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