Lifting Properties of Group C∗-Algebras and Operator Spaces

Specialeforsvar: Villads Ussing Bojesen

Titel: Lifting Properties of Group C∗-Algebras and Operator Spaces

Abstract: We study several important properties for C∗-algebras, namely the lifting property (LP), the local lifting property (LLP), and the weak expectation property (WEP). These properties have proved useful through the years, for example in relating various open problems such as the Connes Embedding Problem and Kirchberg’s QWEP conjecture. We give several reformulations of these properties, and study how they relate to each other as well as to other properties such as nuclearity. We discuss how they behave with respect to tensor products, and we also study these properties for group C∗-algebras. We prove an important theorem stating that the tensor product of C∗(F∞) and B(H) admits a unique C∗-norm, and we give tensorial characterisations of the WEP and the LLP. These results are due to Kirchberg. In Chapters 3, 4 and 5, we provide several important examples of C∗-algebras with or without the LLP: We show that B(ℓ2) fails the LLP, a result of Junge and Pisier, and we provide recent examples of groups whose full group C∗-algebras fail the LLP or the LP due to Ioana, Spaas and Wiersma. These groups have (relative) property (T), as well as non-trivial second cohomology group, and include the group SL3(Z). We present a recent construction of a non-nuclear C∗-algebra with both the WEP and the LLP due to Pisier, which is the first of its kind.

Vejledere:  Magdalena Musat, Pieter B. Spaas
Censor:      Jens Kaad, SDU