Polynomials of almost normal elements in C∗- algebras Sample Clauses

Polynomials of almost normal elements in C∗- algebras. The results of Chapter 5 are based on the paper [15] and are independent from the previous chapters. Let A be an arbitrary unital C∗-algebra, and suppose that A ∈ A is normal. It is well known that there exists a unique C∗-algebra homomorphism C(σ(A)) → A, f '→ f (A) from the algebra of continuous functions on the spectrum σ(A) into A such that f (z) = z is mapped into A, σ(f (A)) = f (σ(A)), and f (A) = max z∈σ(A) |f (z)| (1.5.1) (see, for example, [13]). It is called the continuous functional calculus for normal elements. The aim of Chapter 5 is to introduce an analogue of functional calculus for “almost normal” elements. More precisely, we shall always be assuming that