A Course in Commutative Banach Algebras by Eberhard Kaniuth

By Eberhard Kaniuth

Requiring just a easy wisdom of practical research, topology, complicated research, degree conception and staff concept, this booklet presents an intensive and self-contained advent to the idea of commutative Banach algebras. The middle are chapters on Gelfand's idea, regularity and spectral synthesis. exact emphasis is put on functions in summary harmonic research and on treating many designated sessions of commutative Banach algebras, corresponding to uniform algebras, staff algebras and Beurling algebras, and tensor items. exact proofs and a number of workouts are given. The e-book goals at graduate scholars and will be used as a textual content for classes on Banach algebras, with a variety of attainable specializations, or a Gelfand conception established path in harmonic analysis.

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Let (uλ )λ and (vμ )μ be left approximate identities bounded M and N ∞ ∞ of A and B, respectively. Let x = j=1 aj ⊗bj ∈ A ⊗π B such that j=1 aj · n bj < ∞, and let > 0. Choose n ∈ N with the property that x− j=1 aj ⊗ bj ≤ and choose R ≥ 1 so that aj , bj ≤ R for 1 ≤ j ≤ n. There exist λ0 and μ0 such that aj −uλ aj ≤ /(nR) for all λ ≥ λ0 and bj −vμ bj ≤ /(nR) for all μ ≥ μ0 and all 1 ≤ j ≤ n. It follows that n n x − (uλ ⊗ vμ )x ≤ (1 + M N ) + aj ⊗ bj − (uλ ⊗ vμ ) j=1 n aj ⊗ b j j=1 aj − u λ aj · b j ≤ (1 + M N ) + j=1 n aj · bj − vμ bj + j=1 n aj − uλ aj · bj − vμ bj + j=1 ≤ (1 + M N )2 + n 2 nR ≤ (4 + M N ).

28. Let A be a Banach ∗-algebra and B a C ∗ -algebra, and let φ : A → B be a ∗-homomorphism. Show that φ is continuous and φ ≤ 1. 29. Let G be a discrete group and 0 < p < 1. Show that lp (G) with the convolution product is a commutative Banach algebra. 30. (i) Let G be nontrivial discrete group. Show that the · 1norm on the Banach ∗-algebra l1 (G) fails to be a C ∗ -norm by considering a linear combination of, say, three Dirac functions. (ii) Prove the analogous statement for L1 (Rn ). 31. Let n ∈ N, 1 ≤ p < ∞ and A = {f ∈ L1 (Rn ) : f ∈ Lp (Rn )|}.

A very good reference to the theory of multipliers is [74]. The fact that the multiplier algebra of L1 (G) is isomorphic to the measure algebra M (G), is Wendel’s theorem [135]. It is worth mentioning that Wendel’s theorem admits a generalisation to Beurling algebras. 2 Gelfand Theory Our main objective in this chapter is to develop Gelfand’s theory for commutative Banach algebras. 2). If A has an identity, Δ(A) is compact. The converse is true whenever ΓA is injective (A is semisimple), a fact that can be shown only later (Chapter 3).

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