Topics in the structure and classification of C*-algebras and *-homomorphisms

Thesis

The results in this thesis concern the structure and classification of C*-algebras, with a particular focus on maps between C*-algebras. A unifying theme is the regularity property of Z-stability for C*-algebras (tensorially absorbing a copy of the Jiang-Su algebra Z), both how this property gives rise to structural features, and how one can relax the assumption of Z-stability in certain results to obtain uniqueness theorems from the weaker condition of strict comparison.

C*-algebras that are Z-stable have nice K-theoretical properties, and this fact is used to obtain uniqueness theorems for maps into such C*-algebras. We provide a new, shorter, and self-contained proof of K-stability for Z-stable C*-algebras, using Rørdam and Winter’s picture of Z.

A significant part of the thesis is devoted to proving uniqueness theorems for maps whose codomains are not necessarily Z-stable. One important example is the class of unital embeddings from a separable, nuclear C*-algebra into a II1 factor, which are well-known to be classified by traces in 2-norm by a result of Connes. We upgrade the uniqueness theorem in the norm topology, assuming in addition that the domain C*-algebra satisfies the UCT. We also prove uniqueness results for maps into ultraproducts of matrix algebras, which serve as a uniqueness counterpart to quasidiagonality. These results lie beyond the scope of the recent uniqueness theorems obtained from the abstract classification approach, as neither II1 factors nor ultraproducts of matrices are Z-stable.

The final part of the thesis follows a long-term strategy of proving uniqueness theorems for morphisms into C*-algebras, under the assumption of strict comparison alone, without Z-stability. The first step of the outline is established in the thesis. In prior work, maps into Z-stable C*-algebras are known to have a regularity property called property (SI), often obtained by extending maps from Ato the larger domain A⊗Z. We replace the Z-stability assumption by strict comparison and prove that such maps also have property (SI). As a consequence, the maps can be extended to the Z-stabilization of the domain C*-algebra. An essential ingredient is a new characterization of nuclearity in the separable setting, involving refined finite-dimensional approximations via pure states.

Authors: Hua, S

• DOI: 10.5287/ora-qaozrvowp
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Elliott classification program; K-theory; C*-algebras
• Subjects: Functional analysis