By Dana P. Williams

The speculation of crossed items is very wealthy and fascinating. There are functions not just to operator algebras, yet to matters as assorted as noncommutative geometry and mathematical physics. This booklet offers a close creation to this large topic appropriate for graduate scholars and others whose study has touch with crossed product $C^*$-algebras. as well as supplying the fundamental definitions and effects, the main target of this publication is the positive excellent constitution of crossed items as published through the learn of brought on representations through the Green-Mackey-Rieffel laptop. particularly, there's an in-depth research of the imprimitivity theorems on which Rieffel's thought of prompted 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 obtainable to any graduate scholar popping out of a primary direction on operator algebras. There are appendices that care for ancillary matters, which whereas now not imperative to the topic, are however an important for an entire realizing of the fabric. a few of the appendices should be of self sustaining curiosity. To view one other booklet via this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras.

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36) would fail for large i. This establishes the claim. Now we can assume that for large i we have ǫ ϕ(xi ) − ϕ(x) ∞ < . µ(C) Since supp ϕ(xi ) ⊂ C for all i, G F (xi , s) dµ(s) − G F (x, s) dµ(s) ≤ C ϕ(xi )(s) − ϕ(x)(s) dµ(s) ≤ ǫ. This suffices. The following corollary will be very useful in the sequel. 103. Let H be a closed subgroup of G. Suppose that F ∈ C(G× H, D) is such that there is a compact set K ⊂ G such that F (s, t) = 0 if s ∈ / K. Then the function ψ(s) → F (st, t) dµH (t) H is a well-defined element of C(G) with support in KH.

Since (s, r) → s − r is continuous, there is no problem if g is continuous or even Borel. However, in general, the composition of a measurable function with a continuous function need not be measurable. But since g is the pointwise limit of simple functions in L1 (G), we can assume g is a characteristic function. Thus it suffices to see that σ(E) := { (s, r) ∈ G × G : s − r ∈ E } is µ × µ-measurable if E is µ-measurable. As above, this is automatic if E is a Borel set. If E is a null set, then there is a Borel null set F with E ⊂ F .

71. A net { ui } of self-adjoint18 elements of norm at most one in L1 (G) is called an approximate identity for L1 (G) if for all f ∈ L1 (G), f ∗ui = ui ∗f converges to f in norm. 72. L1 (G) has an approximate identity in Cc (G). Proof. 19 Now fix f ∈ Cc (G). 62 on page 19). Now if V ⊂ W we have supp uV ∗ f ⊂ supp f + W and |uV ∗ f (s) − f (s)| = G uV (r) f (s − r) − f (s) dµ(r) ≤ G uV (r) f (s − r) − f (s) dµ(r) ≤ǫ uV (r) dµ(r) G = ǫ. It follows that uV ∗ f → f uniformly and the support of uV ∗ f is eventually contained in a fixed compact set.