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 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. workouts are supplied after each one bankruptcy.

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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. =

5 Let Bn be the open unit ball in cn and let A(Bn) be the \"ball algebra\" in Bn which can be extendedconconsisting of holomorphic functions unit to the closed ball Bn. It is clear that A( Bn) is a Banach tinuously with the and algebra sup-norm pointwise operations. Show that the maximal ideal space of A( Bn) can be identified with Bn via point-evaluations. 7 Izi = 1,) 0,) n=-oo) radius of I is generated space of x is a 1 . } and x. nonzero Show B is homeomorphic element in A. 1 The Unitization) Banach it may be necessary to considernon-unital For some applications algein the book does not lose much generality.

1 The Unitization) Banach it may be necessary to considernon-unital For some applications algein the book does not lose much generality. The bras. However, our approach Banach shows that every (non-unital) algebra can be thought following argument Banach of as a closed subalgebra of a unital algebra. without unit) equipped wit\037 a complete Let A be a complex (possibly algebra < for all x and y in A. Let A be the set norm II II satisfying Ilxyll IIxlillyll x E A and a E C. A is a vector space of all ordered pairs (x, a), where \037ith in A by) defined componentwise.