Abstract. It is well known that every Boolean algebra of projections (on a Banach space) which is complete (resp. $\sigma $-complete), in the sense of W. Bade [1], is the range of some spectral measure (eg. on the Borel (resp. Baire) sets of its Stone space). For {\it equicontinuous} Boolean algebras which are complete or $\sigma $-complete the same is true in the setting of locally convex spaces. However, equicontinuity is unduly restrictive in practice. In this note we characterize precisely those Boolean algebras of projections acting in a (general) locally convex space which are the range of some spectral measure, thereby describing completely the intimate and subtle connection between Boolean algebras of projections and spectral measures.

AMS Subject Classification (1991): 47A67, 47B15

Received March 18, 1996. (Registered under 6111/2009.)