Equivalence of Projections

Theorem: An AW*-algebra is the ring generated by its projections if and only if it has no abelian summand. Corollary: Every equivalence in an A W*-algebra may be implemented by a partial isometry in the ring generated by the projections of the algebra. The corollary is extended to certain finite Baer »-rings. An early proposition in the theory of von Neumann algebras is that every such algebra is the closed linear span of its projections. Dixmier observed that in a factorial algebra, the passage to the closure can be dispensed with if ring multiplication is also allowed: Every factorial von Neumann algebra is the algebra (sums, products and scalar multiples are the allowable operations) generated by its projections [2, Proposition 7]. Fillmore and Topping removed the restriction on the center as much as it can be removed: A von Neumann algebra is the algebra generated by its projections if and only if its abelian summand (if it has any) is finite-dimensional [8]; inspection of their arguments shows that the result is valid for A H/*-algebras. What about the ring (differences and products the allowable operations) generated by the projections? The answer is favorable, but abelian summands must be banished altogether: Theorem 1. An AW*-algebra is the ring generated by its projections if and only if it has no abelian summand. Corollary. Every equivalence in an A W*-aigebra may be implemented by a partial isometry in the ring generated by the projections of the algebra. The following remarks review the definitions and motivate the results. It is instructive to enlarge the setting somewhat; in the following remarks, A is a Baer *-ring [11], that is, an involutive ring such that the right annihilalor of each subset is the principal right ideal generated by a projection (=selfadjoint idempotent). 1. Projections e,fm A are said to be equivalent (relative to A), written e~f, if there exists weA such that w*w=e and ww*=f; such an element Received by the editors July 8, 197!. AMS 1969 subject classifications. Primary 4665.