Abstract

The (local) lifting property was introduced by Kirchberg as a central theme in his 1993 paper which famously equated Connes’ Embedding Problem with a number of seemingly unrelated questions, such as the residual finite dimensionality of the direct product $F_2\times F_2$ of the free group. (This characterization later served as the bridge between Connes’ Embedding Problem and Tsirleson’s Problem in Quantum Information.) A C -algebra has the (local) lifting property ((L)LP) if every completely positive map from it into any quotient of C-algebras (locally) lifts. This straightforward (localized) projectivity for completely positive maps has deep and surprising equivalent formulations as well as obscure connections which are still not fully understood. In this talk, I will introduce the property, some of its equivalent formulations, its deep connections, and the frontiers of identifying classes of C*-algebras who have or lack it. .