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.

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