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

By Dana P. Williams

The speculation of crossed items is intensely wealthy and interesting. There are functions not just to operator algebras, yet to matters as assorted as noncommutative geometry and mathematical physics. This e-book offers a close advent to this sizeable topic compatible for graduate scholars and others whose learn has touch with crossed product $C^*$-algebras. as well as delivering the fundamental definitions and effects, the main target of this e-book is the effective perfect constitution of crossed items as printed through the research of precipitated representations through the Green-Mackey-Rieffel computer. 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 as a result of Gootman, Rosenberg, and Sauvageot. This booklet is intended to be self-contained and available to any graduate scholar popping out of a primary direction on operator algebras. There are appendices that care for ancillary matters, which whereas no longer imperative to the topic, are however the most important for an entire knowing of the cloth. a few of the appendices can be of self sufficient curiosity. To view one other booklet by way of this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras.

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86 on page 29 and [139, Appendix D]). 14 Since { ρ(r )f } is potentially a net (and not necessarily a sequence), we can’t apply the i Dominated Convergence Theorem. 62 until later. 68. Since we are not assuming that G is σ-compact, there is no reason to believe that Haar measure is σ-finite. 14) is simply f (s)∆(s) dν(s), G it certainly follows that µ and ν have the same null sets, and that we can obtain ν by integration against ∆(s−1 ) dµ(s), and we have employed the usual notation for the Radon-Nikodym derivative.

Let ǫ > 0. As usual, let Bǫ (d) be the ǫ-ball centered at d ∈ D. Since A is compact, there are d1 , . . , dn ∈ A such that n n A⊂ di + V, B 2ǫ (di ) = i=1 i=1 where V = B 2ǫ (0). Let C := { i λi di : i λi = 1 and each λi ≥ 0 }. Then C is convex, and as it is the continuous image of a compact subset of Rn , it is a compact subset of D. Therefore there are c1 , . . , cm ∈ C such that m m C⊂ ci + V. 5 Integration on Groups 31 But C + V is convex and contains A. Therefore c(A) ⊂ C + V . It follows that m m c(A) ⊂ Bǫ (ci ).

Where c = G Notice that if fi → f in the inductive limit topology on Cc (G, D), then Lfi (ϕ) → Lf (ϕ) for all ϕ ∈ D∗ . In other words, Lfi → Lf in the weak-∗ topology on in D∗∗ . Since f → Lf is certainly linear, we can construct our integral by showing that Lf ∈ ι(D) for all f ∈ Cc (G, D), and then defining I : Cc (G, D) → D by I(f ) := ι−1 (Lf ). 90. If f ∈ Cc (G, D), then Lf ∈ ι(D). Proof. Let W be a compact neighborhood of supp f in G, and let K := f (G)∪{ 0 }. Also let C be the closed convex hull of K in D.