By Dana P. Williams

ISBN-10: 0821842420

ISBN-13: 9780821842423

The speculation of crossed items is very wealthy and interesting. There are purposes not just to operator algebras, yet to topics as diverse as noncommutative geometry and mathematical physics. This publication presents an in depth advent to this massive topic compatible for graduate scholars and others whose examine has touch with crossed product $C^*$-algebras. as well as offering the elemental definitions and effects, the focus of this booklet is the wonderful excellent constitution of crossed items as published by way of the learn of prompted representations through the Green-Mackey-Rieffel desktop. specifically, there's an in-depth research of the imprimitivity theorems on which Rieffel's thought of caused representations and Morita equivalence of $C^*$-algebras are dependent. there's additionally a close therapy of the generalized Effros-Hahn conjecture and its evidence because of Gootman, Rosenberg, and Sauvageot. This e-book is intended to be self-contained and obtainable to any graduate pupil popping out of a primary path on operator algebras. There are appendices that care for ancillary matters, which whereas now not critical to the topic, are however the most important for a whole knowing of the fabric. many of the appendices may be of autonomous curiosity. To view one other publication by means of this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras.

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**Sample text**

Two measures that are mutually absolutely continuous are said to be equivalent, and equivalence is an equivalence relation on the set of Radon measures on G. An equivalence class is called a measure class. 67 on the preceding page implies that Haar measure and right-Haar measure are in the same measure class C. Since all measures µ′ ∈ C have the same null sets, it follows that if N is a µ′ -null set, then rE and Er are µ′ -null for all r ∈ G. A Radon measure on G is called quasi-invariant if N null implies that rN is null for all r ∈ G.

Fix ǫ > 0. 88 on page 29, there is neighborhood V of e in G such that f (s) − f (r) < ǫ provided s−1 r ∈ V . We can shrink V if necessary so that supp(f )V ⊂ W . Let s1 , . . , sn ∈ supp f be such that n supp f ⊂ si V. 43 on page 12, there are ϕi ∈ Cc+ (G) such that supp ϕi ⊂ si V and such that i ϕi (s) is bounded by 1 for all s, and equal to 1 when s ∈ supp f . Let g(s) = i f (si )ϕi (s). Then supp g ⊂ W and g(s) − f (s) < ǫ for all s ∈ G. Furthermore, n ϕi (s) dµ(s) ι f (si ) . Lg = i=1 G 32 Locally Compact Groups Since n i=1 G ϕi (s) dµ(s) := d ≤ µ(W ), it follows that Lg ∈ dι(C) which is in µ(W )ι(C) since 0 ∈ C.

Every locally compact group G has a Haar measure which is unique up to a strictly positive scalar. 57 need not concern us here. 1. For the moment, be aware that the precise formulation of the regularity conditions (given by the supremum and infimum conditions) varies a bit from reference to reference (if the group is not second countable). 3 Haar Measure 17 where λ(r)f (s) := f (r−1 s). 5) is called a Haar functional on G. 19] implies that a Haar measure assigns strictly positive measure to each nonempty open set.

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

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