On Higher Order Bourgain Algebras of a Nest Algebra

Following earlier work in which we provided algebraic characterizations of the right, left, and two-sided Bourgain algebras, as well as the second order Bourgain algebras, associated with a nest algebra, we herein demonstrate that a given nest algebra has (essentially) at most six different third order Bourgain algebras, and that every fourth order (or higher) Bourgain algebra of the nest algebra coincides with one of at most third order. This puts the final touch on the description of Bourgain algebras of nest algebras. Following the work of Bourgain in [1], the notion of what is now referred to as the ‘Bourgain algebra’ associated with a subset X of a C(K) space was formalized by Cima et al. ([2], [3]), who, among numerous other things, showed it to be a norm-closed algebra. The Bourgain algebras of various function spaces have since been studied by several authors (cf. [7], [8]). In [5] and [6], working in the non-commutative setting of operators, we formulated analogous definitions, in topological terms, of right, left, and two-sided Bourgain algebras associated to a given algebra of operators, and provided algebraic characterizations of these in the case where the given operator algebra is a nest algebra. The second order Bourgain algebras associated to a nest algebra were characterized as well. In this note, we demonstrate that a given nest algebra has (essentially) at most six possible third order Bourgain algebras, and that every fourth order (or higher) Bourgain algebra of the nest algebra coincides with one of at most third order. This puts the final touch on the description of the Bourgain algebras associated with nest algebras.