By Eberhard Kaniuth

ISBN-10: 0387208437

ISBN-13: 9780387208435

Requiring just a uncomplicated wisdom of sensible research, topology, complicated research, degree concept and crew idea, this publication presents a radical and self-contained creation to the speculation of commutative Banach algebras. The middle are chapters on Gelfand's thought, regularity and spectral synthesis. particular emphasis is put on purposes in summary harmonic research and on treating many specific sessions of commutative Banach algebras, equivalent to uniform algebras, crew algebras and Beurling algebras, and tensor items. special proofs and quite a few routines are given. The ebook 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 thought dependent path in harmonic analysis.

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**Additional resources for A Course in Commutative Banach Algebras**

**Sample text**

I=1 This being true for all n, we conclude that ϕ(x2 ) = ϕ(x)2 for all x ∈ A. 1 that ϕ is multiplicative. 46 2 Gelfand Theory Throughout the book, for any Banach algebra A, Δ(A) denotes the set of all nonzero multiplicative linear functionals on A. It is very important to know how Δ(A) and Δ(Ae ) are related. 3. Because ψ(e) = 1 for every ψ ∈ Δ(Ae ), each ϕ ∈ Δ(A) has a unique extension ϕ ∈ Δ(Ae ) given by ϕ(x + λe) = ϕ(x) + λ, x ∈ A, λ ∈ C. Let Δ(A) = {ϕ : ϕ ∈ Δ(A)}. Moreover, let ϕ∞ denote the homomorphism from Ae to C with kernel A, that is, ϕ∞ (x + λe) = λ.

Let T ∈ M (A) and suppose that T is bijective. Show that T −1 ∈ M (A). 46. Let A be a faithful Banach algebra. Show that the multiplier algebra M (A) is complete in the strong operator topology on B(A) in which a net (Tα )α converges to T if and only if Tα x − T x → 0 for all x ∈ A. 47. Let X and Y be locally compact Hausdorﬀ spaces. For f ∈ C0 (X, C0 (Y )), deﬁne φ(f ) on X × Y by φ(f )(x, y) = f (x)(y). Show that the mapping φ : f → φ(f ) is an isometric isomorphism from C0 (X, C0 (Y )) onto C0 (X × Y ).

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 )|}. Show that A becomes a Banach algebra under the norm f = f 1 + f p. 32. Let L1 (R+ ) denote the subalgebra of L1 (R) consisting of all functions f such that f (t) = 0 for all t < 0.

### A Course in Commutative Banach Algebras by Eberhard Kaniuth

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