Quotients of L°° by Douglas Algebras and Best Approximation by Daniel H. Luecking and Rahman M. Younis

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 subalgebra 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 contained in all other Douglas algebras except 7700. After the Axler-Berg-JewellShields result, it was shown by one of us [10] that (77°° + C)/Hx is an M-ideal (see Received by the editors February 19, 1982. 1980 Mathematics Subject Classification. Primary 30H05, 46E15; Secondary 41A50.