Outer Conjugacy Classes of Trace Scaling Automorphisms of Stable UHF Algebras

Abstract

Connes [C] classified automorphisms of the II1 approximately finite-dimensional (AFD) factor R0;1 up to outer conjugacy. An automorphism of R0;1 determines a modulus or scaling factor on a trace ˆ . It follows from this classification and [CT] that automorphisms which do not leave a trace invariant are classified up to conjugacy by their modulus; for each 2 0; 1† there is up to conjugacy a unique automorphism of R0;1 which scales the trace by . Consequently, for such there is a unique factor of type III the Powers factor R whose associated type II1 factor is R0;1; necessarily, R ˆ R0;1 Z. A key idea for model building in Connes's work on the classification of automorphisms of AFD factors [C] and in subsequent work of Jones [J], Ocneanu [O], and Kawahigashi, Sutherland and Takesaki [KST] on amenable group actions is a non-commutative Rohlin lemma for an aperiodic automorphism of a finite von Neumann algebra. Recall that an automorphism of a von Neumann algebra M is said to be properly outer if the restriction to Me is outer for each non-zero invariant projection e, and aperiodic if every non-zero power is properly outer. Connes showed that an automorphism of M is properly outer if and only if for each non-zero projection e in M, and > 0, there exists a non-zero projection f in M such that kf f †k 0, there exist projections F1; ;Fn in M such that Pn jˆ1 Fj ˆ 1 and Fj† y Fj‡1 2 , j ˆ 1; . . . ; n, where Fn‡1 ˆ F1 and kxk2 ˆ x x† 1 2. MATH. SCAND. 83 (1998), 74^86