# 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 conception. it's a box that has nice significance for different parts of arithmetic and physics, similar to algebraic topology, differential geometry, and quantum mechanics. It assumes a uncomplicated wisdom in sensible research yet no earlier acquaintance with operator conception is needed.

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W e shall therefore call a closed *-subalgebra o f a C*-algebra a C*-subalgebra. If a C*-algebra has a unit 1, then automatically | | 1 | | = 1, because ||1|| = ||1*1|| = | | 1 | | . Similarly, if p is a non-zero projection, then ||p|| = 1. If u is a unitary of A , then = 1, since ||u|| = ||u*u|| = | | 1 | | = 1. Hence, a(u) C T , for if A £

A „ ) = { ( T ( a i ) , . . , 6. a) n Show that ft(A) n M o r e precisely, set ft(A)}. Show that the canon- is a h o m e o m o r p h i s m . Let A be a unital Banach algebra. (a) If a is invertible in A , show that a(a~ ) = {A (b) For any element a £ A , show that r(a ) = (r(a)) . (c) If A is abelian, show that the Gelfand representation is isometric if and only if | | a | | = ||a|| for all a £ A . 1 n 2 - 1 | A£ cr{a)}. n 2 7. Let A be a Banach algebra. Show that the spectral radius function r: A —•> R is upper semi-continuous.

T h e o r e m . Let X be an infinite-dimensional Banach space, and suppose that w £ K(X), that u £ B(X), and that u is Fredholm. Then ind(u -f w) = ind(ii). Proof. 17, the function a: [0,1] - » Z , < h ind(u + tw), is continuous, and therefore a [ 0 , 1 ] is connected in the discrete space Z . Hence, a [ 0 , 1 ] is a singleton set, so i n d ( u ) = a(0) = a ( l ) = ind(u + w). • 1 . 4 . 3 . Remark. Let u be a Fredholm operator on an infinite-dimensional Banach space X. 18 i n d ( u ) = 0, since invertible operators are of course o f index zero.