Read e-book online Crossed Products of C^* Algebras PDF

By Dana P. Williams

ISBN-10: 0821842420

ISBN-13: 9780821842423

ISBN-10: 1219628859

ISBN-13: 9781219628858

ISBN-10: 6619862632

ISBN-13: 9786619862635

The idea of crossed items is very wealthy and fascinating. There are purposes not just to operator algebras, yet to topics as different as noncommutative geometry and mathematical physics. This booklet presents a close advent to this big 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 focus of this booklet is the advantageous perfect constitution of crossed items as printed through the research of caused representations through the Green-Mackey-Rieffel computer. 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 therapy of the generalized Effros-Hahn conjecture and its evidence because of Gootman, Rosenberg, and Sauvageot. This ebook is intended to be self-contained and obtainable to any graduate pupil popping out of a primary direction on operator algebras. There are appendices that take care of ancillary matters, which whereas now not relevant to the topic, are however an important for a whole realizing of the fabric. a number of the appendices could be of self sustaining curiosity. To view one other ebook by means of this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras

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Download PDF by Dana P. Williams: Crossed Products of C^* Algebras

The idea of crossed items is very wealthy and interesting. There are functions not just to operator algebras, yet to matters as diverse as noncommutative geometry and mathematical physics. This publication offers an in depth advent to this gigantic topic compatible for graduate scholars and others whose examine has touch with crossed product $C^*$-algebras.

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Now α and β are restrictions to D of the canonical projections P × Q → P and P × Q → Q, which are immediately seen to be Z-maps. 3, α and β are Z-maps. Direct verification shows that γ α = δβ. 6 Let f = ( f 1 , . . , f k ) be a k-tuple of McNaughton functions of M([0, 1]n ), P ⊆ [0, 1]n a rational polyhedron, and Q = f (P). Then the subalgebra M o f M(P) generated by f 1 P, . . , fk P is isomorphic to M(Q). Proof For every g ∈ M(Q) the composite function h = g f P is piecewise linear and every linear piece of h has integer coefficients.

Vk ) and T = conv(ι(v1 ), . . , ι(vk )). By (i), ι extends to a unique linear Z-homeomorphism of T onto T . 1) it follows that ιT1 ∩ ιT2 = ιT1 ∩T2 . Now let ι¯ = T ∈ ιT . (iii) The set η( ) = {η(T ) | T ∈ } is a regular triangulation of the rational polyhedron Q = η(P). Now apply (ii). 15 Given integers m, n > 0, let P ⊆ [0, 1]n and Q ⊆ [0, 1]m be rational polyhedra, and η a one–one Z-map of P onto Q. Then the following conditions are equivalent: (i) η is a Z-homeomorphism. (ii) For some regular η-triangulation of P, den(η(v)) = den(v) for all vertices of , and the simplex η(T ) is regular for each T ∈ .

Let the rational point y ∈ T be defined by y˜ = q. 7, den(y) < den(y1 ) +· · ·+ den(yt ). The inverse image x = η−1 (y) is a rational point satisfying den(x) < den(x 1 ) +· · ·+ den(x t ), because η preserves denominators. The regularity of S ensures that the homogeneous correspondent x˜ is a positive integer combination of x˜1 , . . , x˜t . Since x lies in the relative interior of S, all the coefficients in this combination are ≥ 1. It follows that den(x) ≥ den(x1 ) + · · · + den(xt ), a contradiction.

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Crossed Products of C^* Algebras by Dana P. Williams


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