C*–Algebras and Operator Theory by Gerard J. Murphy

By Gerard J. Murphy

This e-book constitutes a primary- or second-year graduate direction in operator idea. it's a box that has nice significance for different components of arithmetic and physics, resembling algebraic topology, differential geometry, and quantum mechanics. It assumes a simple wisdom in useful research yet no previous acquaintance with operator concept is needed.

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A linear m a p p : I - ^ I o n a vector space X is idempotent if p = p. In this case X = p(X) © ker(p), since ker(p) = (1 - p)(X). In the reverse direction, if X — Y © Z , where Y and Z are vector subspaces o f X , then there is a unique idempotent p o n X such that p(X) = Y and ker(p) = Z. W e call p the projection of X on Y along Z. 2 1 . 4 . 1 2 . T h e o r e m . Let Y, Z be closed complementary vector subspaces of a Banach space X. Then the projection p of X onY along Z is bounded. Proof. Let (x ) be a sequence in X converging to 0 and suppose that (p(x )) converges t o a point y oi X.

If r G ft(#), then / ( r ( a ) ) = r ( / ( a ) ) , since the maps / H-» / ( r ( a ) ) and / ( / ( ) ) from C(cr(a)) to C are *-homomorphisms agreeing o n the generators 1 and z and hence are equal. r a 2 . 1 . 1 4 . T h e o r e m (Spectral M a p p i n g ) . Let a be a normal a unital C*-algebra A, and let f G C(

It is clear that y> is isometric and im((^>) = £ . 13, let a be a normal element o f a unital C*-algebra A, and let z be the inclusion m a p of C(a(a)) in C . W e call the unique unital *-homomorphism A such that ip(z) = a the functional calculus at a. If p is a polynomial, then (a). Note that / ( a ) is normal. Let B b e the image o f

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