A Course in Commutative Banach Algebras by Eberhard Kaniuth

By Eberhard Kaniuth

Requiring just a simple wisdom of useful research, topology, advanced research, degree concept and staff idea, this booklet offers a radical and self-contained advent to the idea of commutative Banach algebras. The middle are chapters on Gelfand's idea, regularity and spectral synthesis. certain emphasis is put on purposes in summary harmonic research and on treating many unique sessions of commutative Banach algebras, akin to uniform algebras, crew algebras and Beurling algebras, and tensor items. distinctive 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 conception dependent direction in harmonic analysis.

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Example text

12. Every involution on a semisimple commutative Banach algebra A is continuous. Proof. Let · be the given norm an A. We define a new norm | · | on A by |x| = x∗ . It is clear that | · | is submultiplicative. If xn ∈ A, n ∈ N, form a Cauchy sequence for | · |, then (x∗n )n is a Cauchy sequence for · . Consequently, x∗n − x → 0 for some x ∈ A, and hence |xn − x∗ | → 0. This shows that (A, | · |) is complete. 11 there exists c > 0 such that x∗ = |x| ≤ c x for all x ∈ A, as was to be shown. Let C ∞ [0, 1] denote the algebra of all infinitely many times differentiable functions on [0, 1].

3 L1 -algebras and Beurling algebras g−f 1,ω ω(x)|g(x) − f (x)|dx + = S 21 ω(x)|f (x)|dx. G\S We estimate the first integral on the right as follows: h(x) h(x) dx − η(x) ω(x) S h(x) + ω(x) − f (x) dx ω(x) S h(x) = |ω(x) − η(x)| dx S η(x) ω(x)|g(x) − f (x)|dx ≤ S ω(x) |h(x) − ω(x)f (x)|dx + S ≤ h ∞ δ |ω(x) − η(x)|dx S |h(x) − ω(x)f (x)|dx + S ≤ + |h(x) − ω(x)f (x)|dx. S It follows that g−f 1,ω ≤ + |h(x) − ω(x)f (x)|dx + S ω(x)|f (x)|dx G\S |h(x) − ω(x)f (x)|dx ≤ 2 . = + G This shows that Cc (G) is dense in L1 (G, ω).

20. Let A = B(l2 (N)) and let T ∈ A be the unilateral shift defined by (T x)1 = 0 and (T x)n = xn−1 for n ≥ 2 and x = (xn )n ∈ l2 (N). Show that σA (T ∗ T ) = σA (T T ∗). 21. Let A be the convolution algebra l1 (Z) and B the closed algebra consisting of all x = (xn )n ∈ l1 (Z) such that xn = 0 for all n < 0. Show that σA (δ1 ) = σB (δ1 ). 22. Let X = {z ∈ C : 1 ≤ |z| ≤ 2} and f (z) = z, z ∈ X. Let A be the smallest closed subalgebra of C(X) that contains 1 and f , and let B be the smallest closed subalgebra of C(X) that contains f and 1/f .

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