On the (Local) Lifting Property

The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We give a characterization of the (L)LP in terms of lifting -homomorphisms. We use it to prove that if A and B have the LP and F is their finite-dimensional C*-subalgebra, then AF B has the LP. This answers a question of Ozawa. We prove that Exel's soft tori have the LP. As a consequence we obtain that C^*(Fₙ× Fₙ) is inductive limit of RFD C*-algebras with the LP. We prove that for a class of C*-algebras including C^*(Fₙ× Fₙ) and all contractible C*-algebras, the LLP is equivalent to Ext being a group. As byproduct of methods developed in the paper we generalize Kirchebrg's theorem about extensions with the WEP, give short proofs of several, old and new, facts about soft tori, new unified proofs of Li and Shen characterization of RFD property of free products amalgamated over a finite-dimensional subalgebra and Blackadar's characterization of semiprojectivity of them, and we show that the class of C*-algebras with the LP is closed under cross products with amenable groups.