Sheaves of \(C^*\)-algebras. (English) Zbl 1213.46044
The authors provide a systematic account of sheaves of \(C^*\)-algebras, developing the basics of the theory and comparing it to the existing theory of \(C^*\)-bundles. Both theories of sheaves and (upper semicontinuous) \(C^*\)-bundles are shown to be closely related to each other, and in some special situations there is a perfect correspondence between them.
One of the basic examples of sheaves studied in the paper is the so-called multiplier sheaf of a \(C^*\)-algebra \(A\). This is a sheaf over the primitive ideal space \(\mathrm{Prim}(A)\) which is built from the multiplier algebras \(\mathcal{M}(A(U))\) of the ideals \(A(U)\) in \(A\) associated to open subsets \(U\) in \(\mathrm{Prim}(A)\), together with the restriction maps \(\mathcal{M}(A(U))\to \mathcal{M}(A(V))\) for \(U\subseteq V\). Another basic example appearing in the article is the injective envelope sheaf built from the injective envelope \(\mathcal{I}(A)\) of a \(C^*\)-algebra \(A\).
One of the basic examples of sheaves studied in the paper is the so-called multiplier sheaf of a \(C^*\)-algebra \(A\). This is a sheaf over the primitive ideal space \(\mathrm{Prim}(A)\) which is built from the multiplier algebras \(\mathcal{M}(A(U))\) of the ideals \(A(U)\) in \(A\) associated to open subsets \(U\) in \(\mathrm{Prim}(A)\), together with the restriction maps \(\mathcal{M}(A(U))\to \mathcal{M}(A(V))\) for \(U\subseteq V\). Another basic example appearing in the article is the injective envelope sheaf built from the injective envelope \(\mathcal{I}(A)\) of a \(C^*\)-algebra \(A\).
Reviewer: Alcides Buss (Florianopolis)
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
| 46M15 | Categories, functors in functional analysis |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
Keywords:
sheaves; bundles of \(C^*\)-algebras; local multiplier algebra; injective envelope; Hausdorff primitive ideal spaceReferences:
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