By Kehe Zhu

An creation to Operator Algebras is a concise text/reference that makes a speciality of the basic leads to 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) useful calculus for regular operators, and kind decomposition for von Neumann algebras. routines are supplied after every one bankruptcy.

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Banach algebra A. Then n = r(x)n for n. 7 If B is a maximal subalgebra the unit 1. 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.

H(x) dx f. L h( for on R a/most x R. all x. ) 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).

1 PROPOSITION If A is a in A, then on Let A If 7\037. o : A \037 Banach non-unital functional PROOF 43) Transform) cP C is and algebra ( x) I < = {(x, a) :x II x II E A, is a : A \037 C cP all x for a E C} linear a multiplicative in A. ) A. 1) 10(x,a) I Letting a = 0 yields < I cp( II (x, x) I a) < = II II x II Ilxlj + all x E for x lal, A. E A, a E C. 1, the maximal the closed unit ball of A *. It is easy to see that MA U {O} is closed, and hence compact by Alaoglu's theorem, in the weak-star topology of A*.