By Eberhard Kaniuth

Requiring just a simple wisdom of practical research, topology, advanced research, degree thought and workforce concept, this ebook presents a radical and self-contained advent to the speculation 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 specific sessions of commutative Banach algebras, similar to uniform algebras, workforce algebras and Beurling algebras, and tensor items. specific proofs and a number of workouts are given. The e-book 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 conception dependent direction in harmonic analysis.

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**Example text**

Let X be a compact Hausdorﬀ space, Y a closed subspace of X and I the ideal {f ∈ C(X) : f |Y = 0}. Show that C(X)/I is isometrically isomorphic to C(Y ). 42. Let A be the space of all sequences f : N → C such that kf (k) → 0 as k → ∞. With pointwise multiplication and the norm f = sup{k|f (k)| : k ∈ N}, A is a Banach algebra. Show that the multiplier algebra of A is isometrically isomorphic to l∞ (N), where g ∈ l∞ (N) acts on A by pointwise multiplication. 43. 9. Show that the multiplier algebra of lp is isometrically isomorphic to l∞ .

Finally, we investigate tensor products of two commutative Banach algebras A and B, especially the projective tensor product A ⊗π B. 11). In fact, by using failure of the approximation property for Banach spaces one can construct semisimple commutative Banach algebras A and B such A ⊗π B is not semisimple. 1 Multiplicative linear functionals A linear functional ϕ on an algebra A is called multiplicative if ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ A. 2). We do not assume here that A is commutative. 1. Let A be a real or complex algebra with identity e, and let ϕ be a linear functional on A satisfying ϕ(e) = 1 and ϕ(x2 ) = ϕ(x)2 for all x ∈ A.

Clearly, ρ preserves the involution. Since ρ is surjective and the range of ψ contains ρ(L1 (G)⊗L1 (H)), ψ is also surjective. So φ = ψ −1 ◦ ρ : L1 (G) ⊗π L1 (H) → L1 (G × H) is the desired isometric ∗-isomorphism. It does not seem to be clear at all under what conditions the injective norm on the algebraic tensor product of two Banach algebras A and B is an algebra norm. The following proposition in particular shows that this is the case when B = C0 (X) for some locally compact Hausdorﬀ space X.