Texts: Consider the vector space A = C[0, 1] × C = {(f, α): f ∈ C[0, 1], α ∈ C} endowed with the product (f, α) • (g, β) := (fg, αg(0) + βf(0)), (f, α),(g, β) ∈ A, and the norm given by ||(f, α)|| := ||f||∞ + |α| for every (f, α) ∈ A.
(a) It is routine to show—you need not prove this—that A is an associative algebra and a Banach space. Show that A is an abelian Banach algebra with identity.
(b) Let h: A → C be a homomorphism and define h₁(f) := h((f, 0)) for every f ∈ C[0, 1]. Show that h₁ : C[0, 1] → C is a homomorphism.
(c) Show that the homomorphisms from A into C are precisely the maps (f, α) → h₁(f) for some homomorphism h₁ from C([0, 1]) into C.
(d) Give a concrete description of the maximal ideal space of A as a well-known topological space.
(e) For each x ∈ A, determine the spectrum σ(x) of x.
(f) Determine the (non-zero) radical of A.
(g) Is there an equivalent norm on A that makes it into a C∗-algebra?