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 value for different components of arithmetic and physics, comparable to algebraic topology, differential geometry, and quantum mechanics. It assumes a uncomplicated wisdom in useful research yet no past acquaintance with operator thought is needed.
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Additional info for C*–Algebras and Operator Theory
1 0 . Let A = C ^ O . 6. Let a: [0,1] C be the inclusion. Show that x generates A as a Banach algebra. If t G [0,1], show that r belongs to where r is denned by r (f) = / ( * ) , and show that the m a p [0,1] —» Q,(A), t H-* r , is a homeomorphism. Deduce that r(f) = | | / | | o o ( / € A). Show that the Gelfand representation is not surjective for this example. t t t t 1 1 . Let A be a unital Banach algebra and set C(a)= V inf ||6||=1 J ||a6|| 11 11 (a G A). v 7 W e say that an element a of A is a left topological zero divisor if there is a sequence of unit vectors (a ) of A such that l i m _ , o o aa = 0.
2 Proof. 1. Suppose that c is another element o f A + such that c = a. A s c commutes with a it must c o m m u t e with 6, since b is the limit o f a sequence of polynomials in a. Let B be the (necessarily abelian) C*-subalgebra o f A generated b y 6 and c, and let ip: B —> Co(ft) b e the Gelfand representation of J5. (c), and therefore b = c. • 2 2 If A is a C*-algebra and a is a positive element, we denote by a / unique positive element 6 such that b = a. 1 2 the 2 If c is a hermitian element, then c is positive, and w e set | c | = ( c ) / , c = | ( | c | 4- c ) , and c ~ = | ( | c | — c ) .
C*-Algebras and Hilbert Space Operators 36 A n element a in A is self-adjoint or hermitian if a = a*. For each a £ A there exist unique hermitian elements 6, c £ A such that a = 6 + ic (6 = | ( a - f a*) and c = ^ ( a — a*)). T h e elements a* a and aa* are hermitian. T h e set o f hermitian elements of A is denoted by A . W e say a is normal ii a*a = aa*. In this case the *-algebra it generates is abelian and is in fact the linear span of all a a * , where ra,n £ N and n + m > 0. A n element p is a projection if p = p* = p .