By Eberhard Kaniuth

Requiring just a easy wisdom of sensible research, topology, complicated research, degree idea and team concept, this e-book offers an intensive and self-contained advent to the speculation of commutative Banach algebras. The center are chapters on Gelfand's thought, regularity and spectral synthesis. detailed emphasis is put on purposes in summary harmonic research and on treating many precise periods of commutative Banach algebras, resembling uniform algebras, team algebras and Beurling algebras, and tensor items. exact proofs and various workouts are given. The booklet goals at graduate scholars and will be used as a textual content for classes on Banach algebras, with numerous attainable specializations, or a Gelfand idea dependent path in harmonic analysis.

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**Extra info for A Course in Commutative Banach Algebras**

**Example text**

If (fn )n is a Cauchy sequence in L1 (G, ω), then fn ω → g for some g ∈ L1 (G) and hence g/ω ∈ L1 (G, ω) and fn → g/ω in L1 (G, ω). Thus L1 (G, ω) is complete. With convolution, L1 (G, ω) is a Banach algebra. Indeed, for f, g ∈ L1 (G, ω), |(f ∗ g)(x)|ω(x)dx ≤ G |f (xy)| · |g(y −1 )|dy dx ω(x) G G ω(xy)|f (xy)|ω(y −1 )|g(y −1 )|dydx ≤ G G |g(y −1 )|ω(y −1 )Δ(y −1 ) · = G = f |f (x)|ω(x)dx G 1,ω g 1,ω , and hence f ∗ g ∈ L1 (G, ω) and f ∗ g 1,ω ≤ f 1,ω g 1,ω . The involution on L1 (G, ω) is deﬁned in exactly the same way as for L1 (G).

1. Let A be a commutative Banach algebra and, as before, Δ(A) the set of all nonzero (hence surjective) algebra homomorphisms from A to C. We endow Δ(A) with the weakest topology with respect to which all the functions Δ(A) → C, ϕ → ϕ(x), x ∈ A, are continuous. A neighbourhood basis at ϕ0 ∈ Δ(A) is then given by the collection of sets U (ϕ0 , x1 , . . , xn , ) = {ϕ ∈ Δ(A) : |ϕ(xi ) − ϕ0 (xi )| < , 1 ≤ i ≤ n}, where > 0, n ∈ N, and x1 , . . , xn are arbitrary elements of A. This topology on Δ(A) is called the Gelfand topology.

To that end, let n ≥ 2 and consider the polynomial p(λ) = ϕ((λe − x)n ) of degree n. Denoting its roots by λ1 , . . , λn , we have for each i, 0 = p(λi ) = ϕ((λi e − x)n ) ∈ σA ((λi e − x)n ). This implies that λi ∈ σA (x) and hence |λi | ≤ rA (x). Now n (λ − λi ) = p(λ) = λn − nϕ(x)λn−1 + i=1 n 2 ϕ(x2 )λn−2 + . . + (−1)n ϕ(xn ). Comparing coeﬃcients we see that n λi = nϕ(x) and i=1 λi λj = 1≤i