TAILIEUCHUNG - Báo cáo toán học: "Infinite traces of AF-algebras and characters of $U\left(\infty\right)$ "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Infinite traces of AF-algebras and characters of $U\left(\infty\right)$ . | Copyright by INCREST 1983 J. OPERATOR THEORY 9 1983 205-236 INFINITE TRACES OF AF-ALGEBRAS AND CHARACTERS OF t oo ROBERT p. BOYER INTRODUCTION The representation theory of approximately finite-dimensional AF c -alge-bras was vigorously developed by Strătilă and Voiculescu. Their main objective was a study of the unitary representations of the unitary group t oo the direct limit of the classical unitary groups U n such a study had been previously suggested by Kirillov 7 . Having introduced an AF-algebra 41 AịUịpoỴ such that the factor representations of t7 oo naturally correspond to those of SI Strătilă and Voiculescu proceeded to investigate various families of factor representations of U po . In particular they discovered certain traceable factor normal representations 11 and subsequently VoicIliescu 13 found a list of finite factor representations of U pó which he conjectured to be complete. The purpose of our work was to determine all the normal representations of U tx and to examine to what extent the canonical bijection established by Pukanszky and Green see 9 6 between the primitive ideal space and the quasi-equivalence classes of normal representations of the c of a connected locally compact group holds for t7 oo . In order to achieve this goal we had to extend the dynamical system characterization of finite traces of AF-algebras of Strãtilă and Voiculescu to infinite traces roughly speaking we show that any infinite trace is fully determined by its values on the Bratteli diagram 2 of the AF-algebra. This is carried out in Section 1. In Section 2 afrer giving an explicit description of the Bratteli diagram of 41 and its primitive ideals and quotients we show how the solution of a problem posed by I. J. Schoenberg 10 in 1948 and solved by A. Edrei 5 in 1953 may be used to demonstrate the completeness of Voiculescu s list of finite characters . finite factor traces . We then show in Section 3 by a detailed analysis that a primitive ideal of .

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