On some reflexive operator algebras constructed from two sets of closed operators and from a set of reflexive operator algebras

Abstract

0. Introduction. It is generally well-known that all the derivations of JΓ*-algebras are inner. Christensen [1] and Wagner [5] have proved that the same is true of nest and quasitriangular algebras. Furthermore, although Gilfeather, Hopenwasser and Larson [2] have shown that some CSL-algebras may have non-inner derivations, none of these derivations are implemented by bounded operators. The present paper extends the approach adopted in the earlier article [3] and considers a new method of constructing reflexive operator algebras si from two given sets of closed operators {F^ΊZi* {G^Zγ and from a given set of reflexive operator algebras {.SΓ} JLi (n can be a finite number or infinity). The structure of these algebras and their properties are very interesting. For example, one can show that, if certain conditions are applied to the operators {F^ and {(?<}, then the algebras sί are semi-simple and totally symmetric without, however, becoming C*-algebras [4]. These algebras also possess the following property: if A is reversible and belongs to J / , then A' also belongs to si. But in this paper we shall confine our discussion to two subjects: (i) Under what conditions on {Fi:} and {Gt} are the algebras s/ reflexive? (ϋ) What is the structure of Lat s/Ί Usually, when studing CSL-algebras, one considers the pairs ( j ^ , Latj/) in the same way as one considers the pairs (si, si') when studing W*-algebras. However, it has been suggested [3] that in the general case of operator algebras si it would be more useful to consider