Quasi-diagonal flows II

Abstract

Two similar notions defined for flows, quasi-diagonality and pseudo-diagonality, are shown to be equivalent; so approximately inner flows on a quasi-diagonal $C^{*}$-algebra are quasi-diagonal (not just pseudo-diagonal). We define a notion of MF flow which is weaker than quasi-diagonality and study equivalent conditions following Blackadar and Kirchberg's results on MF algebras and we characterize the dual flow of such on the crossed product as a dual MF flow. In the same spirit we introduce a notion of NF flow and show that NF flows are MF flows on nuclear $C^{*}$-algebras, or equivalently, quasi-diagonal flows on nuclear $C^{*}$-algebras. We also introduce a notion of strong quasi-diagonality (in parallel with strong quasi-diagonality versus quasi-diagonality for $C^{*}$-algebras), whose examples contain AF flows.