Sheaves of C*‐algebras

Abstract

We develop the basics of a theory of sheaves of C*-algebras and, in particular, compare it to the existing theory of C*-bundles. The details of two fundamental examples, the local multiplier sheaf and the injective envelope sheaf, are discussed.

When is the second local multiplier algebra of a C *-algebra equal to the first?

“…Therefore, there is y ∈ I such that y + t = a + t and hence N (a − y)(t) = 0. The continuity of the norm function at t (see [5,Lemma 6.4…”

“…It was proved in [21] that if A is a separable unital C*-algebra, then M loc (M loc (A)) = M loc (A), provided that the primitive ideal space Prim(A) contains a dense G δ subset of closed points. One of our goals here is to see how this result can be obtained in a straightforward manner using the techniques developed in [5]. The key to our argument is the following observation.…”

“…Moreover, just as in Somerset's paper [21], Polish spaces (in the primitive spectrum) will play a decisive role. Sections 2 and 3 are fairly self-contained, while Section 4 relies on the sheaf theory developed in [5].…”

The Dauns–hofmann Theorem Revisited

“…It is noteworthy that in the early environment of the sheaf theory of Banach spaces one thought about sheaves of Banach spaces mostly in terms of those that corresponded to bundles, that is, in terms of the so called approximation sheaves ( [34], p. 423, 3.5), wheras the sheaves in the recent study [1] are sheaves in the strict categorical sense of the word. The question whether or not the sheaf M A is an approximation sheaf is not raised in [1]. On the level of logic and topos theory the article of Banaschewski and Mulvey [5] provided a new foundational caliber for the sheaf aspect at a high level of abstraction.…”

Spectral Synthesis in the Multiplier Algebra of a C0(x)-Algebra

“…As noted in [7,Section 1], every C * -algebra is a C 0 (X)-algebra, typically in many ways. C 0 (X)-algebras have been studied in [3,8,13,14,15,20,28]. In [6, Section 1], we saw that if A is a C 0 (X)-algebra then corresponding to each x ∈ X there are two natural ideals H x and J x in M(A) which one might hope would be equal but in fact need not be so.…”