Wild AW * ‐Factors and Kaplansky‐Rickart Algebras

was to examine the algebraic properties of W*-algebras, to consider C*-algebras satisfying a suitable selection of these algebraic properties and then see which properties of W*-algebras (alias von Neumann algebras) still held good for these more general algebras. Despite the great success achieved in fulfilling this program, the theory of AW*-algebras has frequently been criticised on the grounds that, apart from the well-known commutative AW*-algebras which are not W*-algebras and the AW*-algebras constructed from them, the only known examples of AW*-algebras are the original von Neumann algebras. Indeed, Kaplansky proved the beautiful theorem that an AW*-algebra of Type I is a W*-algebra if, and only if, its centre is a W*-algebra. From this Kaplansky was led to the natural conjecture [15] that whenever an AW*-algebra has a trivial centre than it is a W*-algebra. Quite recently, Dyer [8] showed this to be false by an ingenious and elaborate ad hoc construction of a counterexample. One aim of this paper is to provide some evidence for the belief that AW *-factors which are not W*-algebras arise, not just as freaks, but as natural objects which are forced on our attention. We shall investigate a special class of AW*-algebras, the Kaplansky-Rickart algebras (or, more shortly, KR-algebras). To each separable C*-algebra s/ a unique KR-algebra s& is associated in a natural way. Roughly speaking, s$ is a Dedekind-cut completion of s4 in the same way as the real numbers are a Dedekind-cut completion of the rationals (precise definitions are given later). Every KR-algebra is a monotone complete, countably decomposable AW*-algebra which also enjoys some very special properties, for example, its space of pure states is separable. The KR-algebras are precisely the regular c-completions [27] of separable C*-algebras. It turns out that a KR-algebra can only be a W*-algebra if it is a countable direct sum of algebras of the form JSf (H), where H is separable. The general theory of KR-algebras leads easily to a counterexample to Kap-lansky 's conjecture. Let s4 be any separable, simple, infinite-dimensional C*-algebra with identity and let s£ be the associated KR-algebra. Then s$ is an AW*-factor which is as far as possible from being a W*-algebra—for the only positive normal functional on s£ is the zero functional. 1. Kaplansky-Rickart algebras For simplicity, all C*-algebras considered will be assumed to have units. Capital script letters, " sf ", " 2& " will denote C*-algebras and the corresponding spaces …