The Bourgain Algebra of a Nest Algebra

In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = algN is a nest algebra, we show that each of the Bourgain algebras defined has the form (A + K) ∩ B, where B is the nest algebra corresponding to a certain subnest of N .We also characterize algebraically the second-order (and higher) Bourgain algebras of a nest algebra, showing for instance that the second-order two-sided Bourgain algebra coincides with the two-sided Bourgain algebra itself in this case. 1991 Mathematics Subject Classification: 47D25. Bourgain has shown ([2]) that, if X is a subspace of a C(K) space such that a certain set associated with X coincides with C(K), then X has the so-called Dunford-Pettis Property (DPP). This associated set was shown to be a norm-closed algebra by Cima, et al ([3], [4]),who labelled it the ‘Bourgain algebra’ of X , andhas since been studied for various spaces of functions by several authors(cf. [8], [9], [10]). In [7], this author defined an analogue of the Bourgain algebra for algebras of operators on a Hilbert space and computed this Bourgain algebra for certain examples of nest algebras as well as for the algebra of analytic Toeplitz operators on the Hardy space H of the unit circle.In this paper, we give an alternative formulation of the Bourgain algebrafor an operator algebra that better reflects the non-commutativity of the operator setting and provide complete algebraic characterizationsfor both the earlier and new formulations of the Bourgain algebra of a nest algebra. In the last section, we characterize the second-order, and higher, Bourgain algebras of a nest algebra, showing, for instance, thatthe second-order (two-sided) Bourgain algebra coincides with the Bourgain algebra itself.The author hereby expresses his