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This paper is concerned with Kaplansky’s theory of Baer and Baer *-rings, specifically with the problem of classifying type I factors. We use the terminology in Kaplansky’s book [3]. Ornstein [9] calls the pair of dual vector spaces V, W splittable if in either V or W every closed subspace admits a closed complement such that the annihilators span the other space. An equivalent formulation is this: for every closed subspace M of I/ there exists an idempotent-with-adjoint E such that (V)E = M. Kaplansky remarks that, given a dual pair V, W, the ring of all linear operators on V that have adjoints on W is a Baer ring if and only if the pair V, W is splittable [3, p. 51. Kaplansky further states that a ring which is Baer, Zorn, and has no nilpotent ideals f 0 is a factor of type I if and only if it is primitive with a minimal one-sided ideal [3, p. 191. Kaplansky’s remarks may be combined to obtain a representation theorem for a class of type I Baer factors. For convenience we combine “Zorn and no nilpotent ideals f 0” into a single axiom |
