Non-commutative shifts and crossed products

When A is a unital simple AF C∗-algebra and has a unique tracial state, it is shown that the crossed product of the two-sided infinite tensor product by the shift is a tracially AF C∗-algebra. A similar result is given to the crossed product of a certain non-unital two-sided infinite tensor product by the shift. Applying a far-reaching classification result of such C∗-algebras by H. Lin, we obtain an example of a one-parameter automorphism group on some AF C∗-algebra which is not approximately inner, a counter-example to the AF version of the so-called Powers-Sakai conjecture [23].