Quasidiagonality, AF-embeddability and the Blackadar-Kirchberg conjectures

Specialeforsvar ved Henning Olai Milhøj

Titel: Quasidiagonality, AF-embeddability and the Blackadar-Kirchberg conjectures

Abstract: This thesis examines the notions of quasidiagonality and AF-embeddability for $C^{\ast}$-algebras as well as the related Blackadar-Kirchberg conjectures. The $C^{\ast}$-algebraic approximation property of quasidiagonality is examined in detail, including permanence properties, obstructions as well as the representation theoretic formulation. The Tikuisis-White-Winter theorem, which states that, on separable, exact $C^{\ast}$-algebras satisfying the UCT, every faithful, amenable tracial state is quasidiagonal, is proved following the extension theoretic proof of Schafhauser. Some of the consequences of the theorem are studied, and connections to both the Blackadar-Kirchberg conjectures as well as Elliott's classification programme are established. Moreover, in a recent paper, Gabe showed that the Blackadar-Kirchberg conjectures hold true for traceless, exact $C^{\ast}$-algebras. A part of the necessary background information including the concept of primitive ideal spaces as well as Rørdam's ASH-algebra $\mathcal{A}_{[0,1]}$ will be examined, and Gabe's proof will be reproduced.

Vejleder: Mikael Rørdam
Censor: Wojciech Szymanski