By Kehe Zhu

An advent to Operator Algebras is a concise text/reference that specializes in the elemental ends up in operator algebras. effects mentioned comprise Gelfand's illustration of commutative C*-algebras, the GNS development, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) sensible calculus for regular operators, and kind decomposition for von Neumann algebras. routines are supplied after each one bankruptcy.

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Also aB (x) = aA (x) for all x E B. 8 Show that the spectrum of a matrix of A. then IIAII = r(A). 10 If A is a matrix r(A* A). of a Banach algebra A. 1 Functionals Linear Multiplicative In this lecture we of A is commutative, we shall with on A can be identified if cp is nontrivial tive multiplicative linear In particular, II cp II > 1. ) on A. functional shows result Banach the == 1. 1 DEFINITION that the show on a Banach algebra space. When ideal maximal the functionals linear multiplicative study the notion introduce and Functionals) Linear Multiplicative cp is a linear multiplicative a Banach on functional algebra A.

This, along with the initial condition that AI(x), where 1(0) shows that I (x) = eAX for all x in R. The boundednessof I implies that A x, so that is continuously be purely must I) imaginary. 7 THEOREM The mapping 1 of L (R, last the continuous, 1 t \037 CPt is a from R homeomorphism the onto maximal Write -I>(t) = CPt, t E R. 5, -I> ideal space of L 1 (R, dx). We next show that -I> In particular, cP is in the Banach dual Suppose cP EM. there exists a function h E Loo (R, dx) such that) PROOF maps maximal is onto.

On L I (R, dx), provided then by Fubini's theorem functional linear y)g(y) (h(x h(y)g(y)dy everywhere = e itx x) I ELI dx,) a nonzero bounded O. 8 COROLLARY h hand, I) M. Suppose for every if x < l O < R. We concludethat can show that t Q -+ t in onto -+ '1't (I) then by letting e- itx R other On the M. )cp(g) so that cp all for ! 9 Gelfand the If we transform identify in the is then f * dx g(x)h(x) the maximal itx = eitx ideal space well-known Fourier such t E R exists L f(x)e shows that h(x) 00 I) L (R).