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On the (Local) Lifting Property

Dominic Enders, Tatiana Shulman

Abstract

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 $\ast$-homomorphisms. We use it to prove that if $A$ and $B$ have the LP and $F$ is their finite-dimensional C*-subalgebra, then $A\ast_F B$ 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 $C^*(F_n\times F_n)$ is inductive limit of RFD C*-algebras with the LP. We prove that for a class of C*-algebras including $C^*(F_n\times F_n)$ and all contractible C*-algebras, the LLP is equivalent to Ext being a group. As byproduct of methods developed in the paper we generalize Kirchberg'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's characterization of RFD property of free products amalgamated over a finite-dimensional subalgebra and Blackadar's characterization of semiprojectivity of them.

On the (Local) Lifting Property

Abstract

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 and have the LP and is their finite-dimensional C*-subalgebra, then 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 is inductive limit of RFD C*-algebras with the LP. We prove that for a class of C*-algebras including and all contractible C*-algebras, the LLP is equivalent to Ext being a group. As byproduct of methods developed in the paper we generalize Kirchberg'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's characterization of RFD property of free products amalgamated over a finite-dimensional subalgebra and Blackadar's characterization of semiprojectivity of them.
Paper Structure (18 sections, 39 theorems, 94 equations)

This paper contains 18 sections, 39 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A characterization of the (L)LP in terms of lifting homomorphisms
  4. Free products
  5. Free products amalgamated over finite-dimensional subalgebras
  6. LP and LLP
  7. Some other lifting properties
  8. Soft tori have the LP
  9. Other properties of soft tori
  10. RFD
  11. Continuous field
  12. Local minima of commutators
  13. Very flexible stability
  14. Ext being a group versus the LLP
  15. Miscellaneous
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $A$ be separable, $B$$\sigma$-unital and $I$ an ideal in $B$. The following are equivalent: (i) any ccp map $A\to B/I$ lifts to a ccp map; (ii) for any $\ast$-homomorphism $f: A \to B/I$ there is a $\ast$-homomorphism $g: A \to M(B/I \otimes K)$ such that $f\oplus g$ lifts to a $\ast$-homomorphi

Theorems & Definitions (78)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 68 more