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

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 AΦ = {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 AΦ = {a ∈ A : Φ(a) = 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 AΦ 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 AΦ has tracial rank zero is to show that Φ has the tracial Rokhlin property, defined below, as introduced in Definition 1.1 of [14]. Definition 1. Let A be a stably finite simple unital C∗-algebra and let Φ be an involutory ∗-antiautomorphism of A. Then Φ has the tracial Rokhlin property if for every finite set F ⊂ A, every > 0, every N ∈ N and every nonzero positive element x ∈ A there are mutually orthogonal projections e0, e1 ∈ A such that: (1) ‖Φ(e0)− e1‖ < . (2) ‖eja− aej‖ < for 0 ≤ j ≤ 1 and all a ∈ F . (3) The projection 1−e0−e1 is Murray–von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x. (4) For every 0 ≤ j ≤ 1 there are N mutually orthogonal projections f1, . . . , fN ≤ ej , Received September 8, 2005. Mathematics Subject Classification. 46L35,46L40,46L05.