PROJECTIONS IN BANACH ALGEBRAS

There have been two main contributions to the theory of operator algebras on Hilbert space. In a series of five memoirs, Murray and von Neumann have made important strides toward the structure theory of the weakly closed case. In work begun by the Russian school, a study has been made of the more general case where merely uniform closure is assumed. In terminology suggested by Segal, we call these W*-algebras and C*-algebras respectively. A notable advantage of the C*-case is the existence of an elegant system of intrinsic postulates due to Gelfand and Neumark [2]; so one can, and does, study C*-algebras in an abstract fashion that pays no attention to any particular representation. A corresponding characterization of TV*-algebras is not known, but nevertheless several substitutes have been suggested. In [6] von Neumann postulated from the start a second topology behaving like the weak topology. Steen [8] assumed completeness relative to a topology induced by positive functionals. In the present paper the entire burden will be thrown upon a more algebraic, and in some sense more elementary assumption; briefly put, our postulate is the assumption of least upper bounds in the partially ordered set of projections-the precise axioms are given in ?2. We call the algebras in question ATV*algebras (the "A" suggesting "abstract"). This work is in essence a continuation of the study that was begun by Rickart