By Jacques Dixmier

ISBN-10: 0444557474

ISBN-13: 9780444557476

ISBN-10: 0720407621

ISBN-13: 9780720407624

Virtually four-fifths of this booklet bargains with the learn of C*-algebras, and the most effects due, between others, to Fell, Glimm, Kadison, Kaplansky, Mackey and Segal are expounded. a result of volume of fabric accrued on unitary representations of teams, the latter pages of the e-book are dedicated to a quick account of a few points of this topic, really because the idea of teams offers the most attention-grabbing examples of C*-algebras. the idea of C*-algebras continues to be increasing speedily, yet this paintings is still a transparent and available advent to the basics of the topic.

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**Example text**

For each x E A, let 7T’(X) be the operator CR. 2, §4] 39 POSITIVE FORMS AND REPRESENTATIONS in AI N obtained from left multiplication by x in A by passage to the quotient. Let g be the canonical image of 1 in A f. (i) Each 7T’(X) has a unique extension to a continuous linear operator 7T(X) in At• (ii) The map x ~ 7T(X)(X E A) is a representation of A in At. (iii) g is a cyclic vector for 7T(A). (iv) f(x) = (7T(X)g g) for each x E A. 5 (ii), we have, for x, yEA, (7T’(X )7T’(y)g I 7T’(X )7T’(y )g) =/(y*x*xy) ~ Ilx* xll/{Y*y) = Ilx* xl/( 7T’(y)g 17T’(y )g), from which (i) follows.

Let A be a C*-algebra. 1. The set of positive elements of A is denoted by A ". 10 that the statement x ~ 0 has the same meaning whether it is interpreted in A or in B. Since A + is a convex cone and A + n (- A +) ={O}, the relation x - y ~ 0 is a partial ordering in A which is compatible with the real vector space structure of A; we write this relation x ~ y or y ... x. 7, formulae (I); z" (resp. z ") is called the positive (resp. negative) part of z. 6. Let A be the C*-algebra of continuous complex-valued functions vanishing at infinity on a locally compact space T and suppose f E A.

3, and identify A h with a subspace of Cf5(B). By the Hahn-Banach theorem, g can be extended to a linear form p. 11 = IIgli (d. 6). The measure p. on B can be written p. = p. - where p. -II• Let f = p. A h , I’ = p. 4 is unique. References: [618], [619], [681], [848], [1097], [1101], [1323], [1455], [1589]. 7. 1. PROPOSITION. Let A be an involutive Banach algebra with an approximate identity. Let R be the set of representations of A, R’ the set of topologically irreducible representations of A, B the set of continuous positive forms of norm ~ 1 on A, and P the set of pure states of A.

### C*-Algebras by Jacques Dixmier

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