By Dana P. Williams

ISBN-10: 0821842420

ISBN-13: 9780821842423

The idea of crossed items is intensely wealthy and interesting. There are purposes not just to operator algebras, yet to topics as assorted as noncommutative geometry and mathematical physics. This publication presents an in depth advent to this monstrous topic appropriate for graduate scholars and others whose learn has touch with crossed product $C^*$-algebras. as well as delivering the elemental definitions and effects, the focus of this e-book is the superb excellent constitution of crossed items as printed through the learn of prompted representations through the Green-Mackey-Rieffel computing device. specifically, there's an in-depth research of the imprimitivity theorems on which Rieffel's thought of precipitated representations and Morita equivalence of $C^*$-algebras are dependent. there's additionally an in depth remedy of the generalized Effros-Hahn conjecture and its evidence as a result of Gootman, Rosenberg, and Sauvageot. This booklet is intended to be self-contained and obtainable to any graduate scholar popping out of a primary path on operator algebras. There are appendices that care for ancillary topics, which whereas now not critical to the topic, are however the most important for a whole figuring out of the fabric. the various appendices could be of autonomous curiosity. To view one other booklet via this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras.

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**Example text**

21 The idea is to assign to a function f ∈ Cc (G, D) an element I(f ) of D which is meant to be the integral of f : I(f ) := f (s) dµ(s). 22) G Naturally, we want I to be linear and be bounded in some sense. Note that if f ∈ Cc (G, D), then s → f (s) is in Cc (G) and f 1 := G f (s) dµ(s) ≤ f ∞ · µ(supp f ) < ∞. If z ∈ Cc (G) and a ∈ D, then the function s → z(s)a is called an elementary tensor and will be denoted by z ⊗ a. We also want our integral to have the following properties: G f (s) dµ(s) ≤ f 1 and G (z ⊗ a)(s) dµ(s) = a z(s) dµ(s).

12] implies, among other things, that every irreducible representation of G is finite dimensional. However, there are locally compact groups with infinite dimensional irreducible representations. However, even for nice groups — for example abelian groups — we’ll still want to work with infinite-dimensional representations in the sequel. 85 on the preceding page is that in the nonabelian case, the natural analogue for characters is going to be unitary representations and irreducible representations in particular.

Suppose that D is a Banach space, that f ∈ Cc (G, D) and that ǫ > 0. Then there is a neighborhood V of e in G such that either sr−1 ∈ V or s−1 r ∈ V implies f (s) − f (r) < ǫ. 30 Locally Compact Groups Proof. 62 on page 19). 87. Fix f ∈ Cc (G, D) with K := supp f . Let W be a compact neighborhood of e in G. 88 on the preceding page, choose a symmetric neighborhood V ⊂ W of e such that sr−1 ∈ V implies that ǫ f (s) − f (r) < . 2 n Then there are ri ∈ K such that K ⊂ i=1 V ri . Then K c ∪ { V ri }ni=1 is an open cover of G.

### Crossed products of C*-algebras by Dana P. Williams

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