A C*-metric algebra consists of a unital C*-algebra and a Leibniz Lip-norm on the C*-algebra. We show that if the Lip-norms concerned are lower semicontinuous, then any unital *-homomorphism from a C*-metric algebra to another one is necessarily Lipschitz. We come to the result that the free product of two unital completely Lipschitz contractive *-homomorphisms from upper related C*-metric algebras coming from *-filtrations to those which are lower related is a unital Lipschitz *-homomorphism.
Let A be a unital AF-algebra(simple or non-simple) and let α be an automorphism of A.Suppose that α has certain Rokhlin property and A is α-simple.Suppose also that there is an integer J≥1 such that αJ0=idK0(A).The author proves that A α Z has tracial rank zero.
