Download e-book for iPad: An Introduction to Operator Algebras by Kehe Zhu

By Kehe Zhu

ISBN-10: 0849378753

ISBN-13: 9780849378751

An creation to Operator Algebras is a concise text/reference that specializes in the elemental ends up in 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) sensible calculus for regular operators, and sort decomposition for von Neumann algebras. routines are supplied after every one bankruptcy.

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Extra info for An Introduction to Operator Algebras

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In general, the Gelfand transform of 00 closed The Gelfand transform of (unital) subalgebra of L is isometric. an algebra is onto if and only if the algebra is self-adjoint; see Lecture8. Gelfand every 2 clear that It is Hoo(D). 2 Show a Banach such in a Banach if I(z) that algebra algebra A, is an A that) < r(x)r(y). 3 for Izi > r. 4 If S is a subset of a then a(x) C {O, algebra A, then the algebra, Banach I}. 5 SupposeA = {x E A: xy = yx, YES}, of A. subalgebra a Banach algebra and S is a subset of A consisting of commuting elements.

We shall show later that every C*-algebrais a C* -subalgebra finite dimensional of B(H). It is also true that every C*-algebra is a C*subalgebra of some Mr& (C). See [111. 7 DEFINITION a in C*-Algebra Suppose A is a (a) We say that x is self-adjoint (b) We that x is unitary say (c) We say that (d) We say that (e) We say that C*-algebra and x is = x. = xx* = 1, or equivalently in A. if x* if x* x x is normal if x*x = xx*. ) in = X-I. x* A. are normal. It is also elements It is clear that both self-adjoint and unitary and are positive.

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). This L1(R,dx). 7, there dx = f(x)h(x) L - * g),) cp(f = dx f(x)h(x) By Theorem is multiplicative. cp(f) L Banach Algebras) that dx for almost M of L1 (R, all x dx) on R.

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An Introduction to Operator Algebras by Kehe Zhu

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