An Introduction to Operator Algebras by Kehe Zhu

By Kehe Zhu

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

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Linear functional to that the above linear see between correspondence multiplicative Finally, functionals and maximal ideals is one-to-one,let

By assumption N contains no invertible elements of A. 1 \"1 - xII > 1 for all x E N. This easily implies that with norm 1. In fact, for every x E A with cp( x) i= 0 we can cp is continuous write) x=\037(x) (1- [1- \037\037x\302\273 )]) with) x 1- E N,) \037(x) so that) IIxll Pix x E N with /lx/l = = f(z) I and = > 1- 1\037(x)1 (1- \037\037x\302\273 ) 1\037(x)l. ))) of Gleason, Kahane, A Theorem Zelazko) and 25) Since) < 1

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.

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