C∗–Algebras Christian B¨ar1 and Christian Becker2 1 Institut f¨urMathematik, Universit¨atPotsdam,
[email protected] 2 Institut f¨urMathematik, Universit¨atPotsdam,
[email protected] In this chapter we will collect those basic concepts and facts related to C∗– algebras that will be needed later on. We give complete proofs. In Sections 1, 2, 3, and 6 we follow closely the presentation in [1]. For more information on C∗–algebras see e. g. [2, 4, 5, 7, 8]. 1 Basic definitions Definition 1. Let A be an associative C-algebra, let k · k be a norm on the C-vector space A, and let ∗ : A → A, a 7→ a∗, be a C-antilinear map. Then (A, k · k, ∗) is called a C∗–algebra, if (A, k · k) is complete and we have for all a, b ∈ A: 1. a∗∗ = a (∗ is an involution) 2. (ab)∗ = b∗a∗ 3. kabk ≤ kak kbk (submultiplicativity) 4. ka∗k = kak (∗ is an isometry) 5. ka∗ak = kak2 (C∗–property). A (not necessarily complete) norm on A satisfying conditions (1) to (5) is called a C∗–norm. Remark 1. Note that the Axioms 1–5 are not independent. For instance, Ax- iom 4 can easily be deduced from Axioms 1,3 and 5. Example 1. Let (H, (·, ·)) be a complex Hilbert space, let A = L(H) be the algebra of bounded operators on H. Let k · k be the operator norm, i. e., kak := sup kaxk. x∈H kxk=1 Let a∗ be the operator adjoint to a, i. e., 2 Christian B¨arand Christian Becker (ax, y) = (x, a∗y) for all x, y ∈ H.