INTRODUCTION Let R be a direct limit of a countable sequence of finite-dimensional semisimple real algebras (i.e., a countable-dimensional locally finite-dimensional-serrnsimple IR-algebra), or the C*-norm-closure of such a direct limit (i.e., an approximately finite-dimensional, or AF, real C*-algebra). (During this introduction, we shall refer to the first case as "the ring case", and to the second as "the C*-algebra case".) Motivated by (i) Elliott's classification of direct limits of countable sequences of finite-dimensional semisimple complex algebras and complex AF C*-algebras [5], (ii) classical results classifying involutions on finite-dimensional semisimple complex algebras, and (iii) the classification by Handelman and Rossmann of automorphisms of period two on the algebras appearing in (i) (cf. [13], [14], and Section XV of the present paper), we study the real algebras described above and completely classify them, up to isomorphism, Morita equivalence, or stable isomorphism. We also show how our classification easily distinguishes various types of algebras within the given classes, and we partially solve the problem of determining exactly which values are attained by the invariants used in classifying these algebras. In the case of (i), Elliott's invariant (which completely classifies up to Morita equivalence) is equivalent to KQ of the algebra, viewed as a partially ordered abelian group. The analogous invariant that we use in the case of real algebras is the diagram i(R): K0(R) K0(R ® C) - K0(R ® H) where I denotes the real quaternions, the KQ'S are viewed as partially ordered abelian groups, and the connecting maps are KQ of the corresponding algebra maps R-»R®(D --»R ® IH, viewed as positive homomorphisms between the ordered KQ groups. While we would like to say that i(R) is a complete Morita equivalence invariant for the algebras we study, there is a conspicuous lack in the literature of a development of Morita equivalence either for rings without unit or for real C*-algebras. Consequently, in the non-unital ring case we ignore Morita equivalence, while in the C*-algebra case we replace Morita equivalence by stable isomorphism, based on analogy with the result of Brown, Green, and Rieffel that separable complex C*-algebras are strongly Morita equivalent if and only if they are stably isomorphic [3, Theorem 1.2]. v
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