Real structure in unital separable simple C *-algebras with tracial rank zero and with a unique tracial state

P Stacey
2006 New York Journal of Mathematics New York J. Math   unpublished
Let A be a simple unital C *-algebra with tracial rank zero and with a unique tracial state and let Φ be an involutory *-antiautomorphism of A. It is shown that the associated real algebra A Φ = {a ∈ A : Φ(a) = a * } also has tracial rank zero. Let A be a unital simple separable C *-algebra with tracial rank zero and suppose that A has a unique tracial state. If Φ is an involutory *-antiautomorphism of A, then it is clear that the associated real algebra A Φ = {a ∈ A : Φ(a) = a * } is unital
more » ... a * } is unital and simple with a unique tracial state, but it is not clear that it has tracial rank zero, even when A is approximately finite-dimensional. The purpose of the present note is to show that techniques recently developed by Phillips [14] and Osaka and Phillips [12], [13] can be used to show that A Φ does have tracial rank zero. This raises the possibility of classifying all real structures in the algebras under consideration by developing a real analogue of Lin's classification [10] of C *-algebras of tracial rank zero. Previously all classifications of real structures in non-type I simple C *-algebras, such as [2], [3], [15] for AF algebras and [5], [16] for irrational rotation algebras, have assumed very specific forms for the real algebras. The key step in showing that A Φ has tracial rank zero is to show that Φ has the tracial Rokhlin property, defined below, as introduced in Definition 1.
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