CONTENTS INTRODUCTION v I. BACKGROUND (ordered abelian groups classification of unital ultramatricial 1 C-algebras and of unital complex AF C*-algebras) II. DESCRIPTION OF THE INVARIANT (introduction of t some basic 6 properties) III. FULLNESS OF THE INVARIANT (a functorial property of t ) 13 IV. FAITHFULNESS OF THE INVARIANT (UP TO INNER 21 AUTOMORPHISMS) (another functorial property of 1*) V. COMPLETENESS OF THE INVARIANT (classification of unital direct 27 limits of sequences of finite-dimensional semisimple IR-algebras via 1* and i , up to isomorphism and Morita equivalence) VI. PROPERTIES OF THE INVARIANT (some technical restrictions on 1*) 33 VII. APPLICATIONS (use of t to distinguish unital direct limits built from 40 matrices over only one or two of (R, C, IH) VIII. FORM OF THE INVARIANT (preliminary description of those diagrams 53 appearing as 1*(R)'s) IX. ALGEBRAS OF TYPE rh (exact characterization of those diagrams appearing 71 as 1 * of algebras of type rh) X. ALGEBRAS OF TYPE c (exact characterization of those diagrams appearing 81 as 1 * of algebras of type c) XI. MINIMALITY OF THE INVARIANT (examples showing that 1 * cannot 95 be reduced to a diagram using only two of the three KQ'S) XII. NON-UNITAL K0 (outline of K0 for algebras without unit classification 100 of non-unital ultramatricial (C-algebras and of non-unital complex AF C*-algebras) XIII. CLASSIFICATION OF NON-UNITAL DIRECT LIMITS (non-unital 109 analogs of Chapters V,VII,IX) XIV. REAL AF C*-ALGEBRAS (real AF C*-analogs of Chapters V,VII,IX,X, 125 XIH) XV. PERIOD TWO AUTOMORPHISMS OF COMPLEX AF 131 C*-ALGEBRAS (connection between such automorphisms and real AF C*-algebras of type c) XVI. REMARKS AND PROBLEMS 136 REFERENCES 141 INDEX OF DEFINITIONS 144 in

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