Pick interpolation for a uniform algebra

Abstract

I am reporting on joint work with Brian Cole and Keith Lewis. Let A be a uniform algebra on a compact Hausdorff space X, i.e. let A be an algebra of continuous complex-valued functions on X, closed under uniform convergence on X, separating points and containing the constants. LetM denote the maximal ideal space of A. Gelfand’s theory gives that X may be embedded in M as a closed subset and each f in A has a natural extension to M as a continuous function. Set ‖f‖ = maxX |f |. We consider the following interpolation problem: choose n points M1, . . . ,Mn in M. Put I = {g ∈ A | g(Mj) = 0, 1 ≤ j ≤ n} , and form the quotient-algebra A/I. A/I is a commutative Banach algebra which is algebraically isomorphic to C under coordinatewise multiplication. For f ∈ A, [f ] denotes the coset of f in A/I and ‖[f ]‖ denotes the quotient norm. We put, for w = (w1, . . . , wn) in C, D = {w ∈ C | ∃ f ∈ A such that f(Mj) = wj , 1 ≤ j ≤ n, and ‖[f ]‖ ≤ 1} . Our problem is to describe D. It is easy to see that D is a closed subset of the closed unit polydisk ∆ in C and has non-void interior. It turns out that D has the following property which we call hyperconvexity. We write ‖ ‖∆k for the supremum norm on ∆. Let P be a polynomial in k variables and choose k points w′, w′′, . . . , w in C. We apply P to this k-tuple of points, using the algebra structure in C. Then P (w′, w′′, . . . , w) = (P (w′ 1, w ′′ 1 , . . . , w (k) 1 ), P (w ′ 2, w ′′ 2 , . . . , w (k) 2 ), . . .) ∈ C .