Download Crossed products of C-star algebras by Dana P. Williams PDF

By Dana P. Williams

The speculation of crossed items is intensely wealthy and fascinating. There are purposes not just to operator algebras, yet to topics as various as noncommutative geometry and mathematical physics. This e-book presents a close creation to this enormous topic compatible 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 ebook is the high-quality perfect constitution of crossed items as printed through the research of brought about representations through the Green-Mackey-Rieffel desktop. specifically, there's an in-depth research of the imprimitivity theorems on which Rieffel's idea of caused 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 ebook is intended to be self-contained and available 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 primary to the topic, are however the most important for a whole knowing of the cloth. the various appendices should be of self sufficient curiosity. To view one other publication by way of this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras.

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Let U : G → U (H) be a unitary representation. A subspace V ⊂ H is invariant for U if Us V ⊂ V for all s ∈ G. If the only closed invariant subspaces for U are the trivial ones — { 0 } and H — then U is called an irreducible representation. 85. 1), if G is abelian, then the irreducible representations of G are one-dimensional and correspond to the characters of G in the obvious way. 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.

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 Lg = ϕi (s) dµ(s) ι f (si ) . 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.

Let U : G → U (H) be a unitary representation. A subspace V ⊂ H is invariant for U if Us V ⊂ V for all s ∈ G. If the only closed invariant subspaces for U are the trivial ones — { 0 } and H — then U is called an irreducible representation. 85. 1), if G is abelian, then the irreducible representations of G are one-dimensional and correspond to the characters of G in the obvious way. 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.

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