By Eberhard Kaniuth

ISBN-10: 0387208437

ISBN-13: 9780387208435

Requiring just a simple wisdom of useful research, topology, advanced research, degree conception and workforce thought, this e-book offers a radical and self-contained advent to the speculation of commutative Banach algebras. The center are chapters on Gelfand's conception, regularity and spectral synthesis. designated emphasis is put on functions in summary harmonic research and on treating many unique periods of commutative Banach algebras, corresponding to uniform algebras, crew algebras and Beurling algebras, and tensor items. distinctive proofs and a number of routines are given. The e-book goals at graduate scholars and will be used as a textual content for classes on Banach algebras, with a number of attainable specializations, or a Gelfand conception dependent direction in harmonic analysis.

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

**Sample text**

Thus x cannot be contained in I. As an application of Urysohn’s lemma, we now determine all the closed ideals of C0 (X). 6. Let X be a locally compact Hausdorﬀ space, and for each subset E of X let I(E) = {f ∈ C0 (X) : f (x) = 0 for all x ∈ E}. Then the map E → I(E) is a bijection between the collection of nonempty closed subsets of X and the proper closed ideals of C0 (X). Moreover, I(E) is a modular ideal if and only if E is compact, and I(E) ∈ Max(C0 (X)) if and only if E is a singleton. Proof.

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 )|}.

2. The projective tensor norm on A ⊗ B is an algebra norm. Proof. Let x = n i=1 ai ⊗ bi and y = n ⊗ dj . 5 Tensor products of Banach algebras n n m m ai cj · bi dj ≤ i=1 j=1 31 ai · b i cj · dj i=1 . j=1 Taking the inﬁma over all such representations of x and y, we conclude that π(xy) ≤ π(x)π(y). Now let A and B be Banach algebras. 2 we can extend the product on A ⊗ B to A ⊗π B. Then A ⊗π B becomes a Banach algebra, which is commutative if and only if both A and B are commutative. Moreover, if A and B are ∗-algebras, then A ⊗π B is a Banach ∗-algebra for the involution deﬁned by (a ⊗ b)∗ = a∗ ⊗ b∗ for a ∈ A and b ∈ B.