Weak Expectation
Property and Local
Lifting Property
Chapter 13
This chapter contains a potpourri of concepts and results which appear un-
related but turn out to have deep, harmonious connections. We begin with
two technical C -concepts (the local lifting property and the weak expecta-
tion property), proving tensorial characterizations in Corollary 13.2.5. The
last three sections explore connections with Kirchberg's QWEP conjecture
(see Section 13.3), surprising examples of nonsplit extensions, and the fun-
damental fact (due to Junge and Pisier) that lffi(7t) 8 lffi(7t) has more than
one C -norm.
13.1. The local lifting property
Definition 13.1.1. Let A be a C-algebra, J be a closed two-sided ideal in
a C -algebra B and Tr: B ----+ B / J be the quotient map. We say a c.c.p. map
cp: A ----+ B / J is liftable if there exists a c.c.p. map 'ljJ: A ----+ B such that
Tr o 'ljJ = cp. We say cp is locally liftable if for any finite-dimensional operator
system E c A, there exists a c.c.p. map 'ljJ: E----+ B such that Tr o 'ljJ ='PIE:
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