By Eberhard Kaniuth
Requiring just a simple wisdom of useful research, topology, complicated research, degree concept and team conception, this booklet presents a radical and self-contained advent to the idea 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 detailed periods of commutative Banach algebras, comparable to uniform algebras, workforce algebras and Beurling algebras, and tensor items. unique proofs and quite a few workouts are given. The ebook 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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Extra info for A Course in Commutative Banach Algebras
The function f also satisﬁes T g(x) = f (x)g(x) for all g ∈ C0 (X) and x ∈ X since (T g(x))2 = g(x)(T 2 g)(x) = 0 whenever g(x) = 0. Moreover, f ∞ ≤ T . In fact, given x ∈ X, by Urysohn’s lemma there exists g ∈ C0 (X) such that g(x) = 1 = g ∞ , and this implies |f (x)| = |T g(x)| ≤ T g ∞ ≤ T · g ∞ = T . It follows that the mapping f → Tf provides an isometric algebra isomorphism between C b (X) and the multiplier algebra of C0 (X). 14. Let G be a locally compact Abelian group. We determine the multiplier algebra of L1 (G).
Let A be a faithful Banach algebra. 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 ).
Proof. The mapping (f, a) → f a from L1 (G) × A into L1 (G, A) is bilinear. Hence there exists a unique linear mapping φ : L1 (G) ⊗ A → L1 (G, A) such that φ(f ⊗ a)(x) = f (x)a for all f ∈ L1 (G), a ∈ A and almost all x ∈ G. The map φ is a homomorphism since f (xy)g(y −1 )dy ab φ((f ∗ g) ⊗ (ab))(x) = G (f (xy)a)(g(y −1 )b)dy = G φ(f ⊗ a)(xy)φ(g ⊗ b)(y −1 )dy = G = (φ(f ⊗ a) ∗ φ(g ⊗ b))(x), for all f, g ∈ L1 (G), a, b ∈ A and almost all x ∈ G. For u = L1 (G) ⊗ A it follows that n φ(u) 1 fi ai (x) dx G i=1 i=1 n n fi (x)ai dx ≤ = G f i ⊗ ai ∈ n f i ai = = n i=1 |fi (x)| · ai dx G i=1 i=1 n fi = 1 ai .
A Course in Commutative Banach Algebras by Eberhard Kaniuth