March 7, 2017

C*-Algebras by Jacques Dixmier

By Jacques Dixmier

Nearly four-fifths of this publication bargains with the examine of C*-algebras, and the most effects due, between others, to Fell, Glimm, Kadison, Kaplansky, Mackey and Segal are expounded. as a result volume of fabric gathered on unitary representations of teams, the latter pages of the publication are dedicated to a quick account of a few elements of this topic, rather because the concept of teams offers probably the most attention-grabbing examples of C*-algebras. the speculation of C*-algebras continues to be increasing speedily, yet this paintings is still a transparent and obtainable advent to the basics of the topic.

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An element of S. which we denote by 1J’(X); if x E A. then 1J’(X)(x) = X(1J(x)) is a continuous function of x; 1J’ is thus a continuous map of T into Sand 1J’(T) is a compact subset of S. If 1J’(T) # S, there exists a continuous complex-valued function f on S such that f# 0 and f 11J’(T) = O. 1. f is the Gelfand transform of an x E A. We have x# O. X(1J(x)) = 0 for every X E T, hence 1J(x) = 0, and this is absurd. 2. PROPOSITION. Let A be a C*-algebra and I a closed two-sided ideal of A. Then I is self-adjoint and AII.

2. 2. Let A be an involutive algebra and f a positive form on A. For x, yEA put (x I y) = f(y*x). This scalar product is linear in x, anti-linear in y and (x Ix) ~ 0 for each x. A is thus endowed with a pre-Hilbert space structure. In particular, we have f(y*x) = f(x*y) (1) (x E A, yEA), (x E A, yEA). If(y*xW~f(x*x)f(y*y) (2) If A is unital, we deduce, putting y = 1 in (1) and (2), that = f(x), (3) f(x*) (4) If(xW ~ f(1)f(x*x). Let H be the Hausdorff pre-Hilbert space constructed canonicaIly from the pre-Hilbert space A, so that H = AIN where N is the set of those x E A for which f(x*x) = O.

F(x*x)"’;;;SUPgEPg(X*X). 7, 111T(X)II",;;; Ilxll for every 1T E R, so that IIxll’",;;;[r]. Moreover, each function x ~ 1I1T(x)1I is a seminorm on A, hence x ~ IIxll’ is a seminorm on A. We have for each 1T E R, and so IIx*II’ = IIxll and IIx*xll’ = IIxll,2. 2. Let I be the set of x E A such that Ilxll’ = 0, which is a closed self-adjoint two-sided ideal of A. The map x ~ IIxll’ defines a norm on the quotient AlI. Endowed with this norm, All satisfies all the C*-algebra axioms except that AI I is not complete in general.

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