An Introduction to Operator Algebras by Kehe Zhu

By Kehe Zhu

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

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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.

1'(/)h(y) equation all = '1'(ly ), we obtain) + y) = '1'(/)h(x so that h(x t E R such '1'(/)h(x)h(y), = h(x)h(y) for all x and y in R. 6, there exists = e itx for almost every x E R. Therefore, '1' = '1't and q. + y) that h( x) is onto. (t) in dx), (R, r lim a for fi in every L 1 (R, dx), f(x) - e itX dx = ) (eito

General Let K lies 'P(fo) (I)) 'Pzo the maximal ideal space of we identify If Zo topology is D and transform point \037 (I) 'Pzo (I) for every f in A(D). By the in the maximal ideal space of A (D), we have a one-to-one continuous mapping from D onto MA(D)' weak-star Thus 'Pzo' = I(zo) for every polynomial I. This implies dense in A(D). So is onto. Suppose Za \037 Zo in D . Then 'PZOl Since both the linear show that the case, continuity, nonvanishing. ker'P = there exists then for Xo each {I E there the mapping consider) C(K) : 'P(/) in x in K has an the By compactness of each = I in C (K) Xl i= X2 such that is one-to-one.

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