Quotients of L ∞ by Douglas Algebras and Best Approximation

Abstract

We show that L°°/A is not the dual space of any Banach space when A is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in L°° has a nonunique best approximation in A. We therefore also show that the Douglas algebra Hx + Lf, when F is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi (6) and independently obtained by Marshall and Zame. 1. Background and introduction. Let L°° be the usual space of (equivalence classes of) bounded measurable functions on the unit circle F. Let H°° denote the subalge- bra of L°° consisting of those functions whose Poisson extensions to the open unit disk 7) are analytic. We let X denote the maximal ideal space of L00 and identify L°° with the space of continuous complex-valued functions on X. We furnish L°° with the essential supremum norm which we merely denote || • ||. Then 7700 is a Banach subalgebra of L°° and if A is any closed algebra with 7/°° E A E L°°, we let M(A) denote the maximal ideal space of A. Elements of A may be identified with functions on M(A). In particular, functions in 77°° may be considered as functions on any one of 7), F, X or M(HX), and we do not distinguish notationally between these interpretations. We make use of the Chang-Marshall Theorem (4 and 11) which states that any closed subalgebra A of L00 which contains H°° is generated as a closed algebra by 77°° together with the set {b: b is a Blaschke product in 7700 and b E A}. Such algebras are commonly called Douglas algebras. The reader will need a familiarity with such concepts from the theory of uniform algebras as representing measures, peak sets and weak peak sets. For uniform algebras see the book of Gamelin (5). For basic facts about 77°° and M(H°°) see (7, 12 and 14). The subject of best approximation in Douglas algebras got its start with a theorem of Axler, Berg, Jewell and Shields (2, Theorem 4) which states that every function / E F°° has a best approximation in 77°° + C = {h + g: h E Hx, g is continuous on F). That is, there exists a function/* £ 7700 + C such that II / — f* = dist(/, Hx + C). From (12) we know Hx + C is a Douglas algebra and is