The structure of discontinuous homomorphisms from non-commutative C*-algebras

Abstract

Let X be a compact Hausdorff space, let C(X) denote the algebra of all continuous functions on X, let B be a Banach algebra, and let θ: → C(X) → B be a (possibly discontinuous) homomorphism with dense range. A classical theorem by W. G. Bade and P. C. Curtis ([2, Theorem 4.3]) describes in great detail the structure of θ we shall refer to this result as the Bade–Curtis theorem. Before we give a brief sketch of this theorem, we fix some notation. For Y ⊂ X let I(Y) and J(Y) denote the ideals of all functions in C(X) that vanish on Y and on a neighborhood of Y respectively; if Y = {x} for some x ɛ X, we write mx and Jx for I(Y) and J(Y) respectively. According to the Bade–Curtis theorem there is a finite set {x1,…, xn) ⊂ X, the so-called singularity set of θ, such that θ | ({x1, …, xn}) is continuous. As a consequence, the restriction of θ to the dense subalgebra of C (X) consisting of all those functions which are constant near each Xj (j = 1,…, n) is continuous, and extends to a continuous homomorphism θcont: C(X)→ B. Let θsing: = θ – θcont. Then θsing | I({x1,…, xn}) is a homomorphism onto a dense subalgebra of rad (B). θcont, and θsing are called the continuous and the singular part of θ respectively. Moreover, there are linear maps : C(X)⊒ B such that(i) (ii) is a homomorphism, andCondition (iii) forces the homomorphisms to map into rad(B); such homomorphisms are called radical homomorphisms.