Nuclear dimension of simple stably projectionless C*-algebras

We prove that Z-stable, simple, separable, nuclear, non-unital C*-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Z-stability for simple, separable, nuclear, non-elementary C*-algebras.Comment: 40 pages. Fixed a typo in the statement of Theorem 2.7. Analysis & PDE, to appea

W*-Bundles

This thesis collates, extends and applies the abstract theory of W*-bundles. Highlights include the standard form for W*-bundles, a bicommutant theorem for W*-bundles, and an investigation of completions, ideals, and quotients of W*-bundles. The Triviality Problem, whether all W*-bundles with fibres isomorphic to the hyperfinite II_1 factor are trivial, is central to this thesis. Ozawa's Triviality Theorem is presented, and property gamma and the McDuff property for W*-bundles are investigated thoroughly. Ozawa's Triviality Theorem is applied to some new examples such as the strict closures of Villadsen algebras and non-trivial C(X)-algebras. The solution to the Triviality Problem in the locally trivial case, obtained by myself and Pennig, is included. A theory of sub-W*-bundles is developed along the lines of Jones' subfactor theory. A sub-W*-bundle encapsulates a tracially continuous family of subfactors in a single object. The basic construction and the Jones tower are generalised to this new setting and the first examples of sub-W*-bundles are constructed

Classifying maps into uniform tracial sequence algebras

We classify $^*$-homomorphisms from nuclear $C^*$-algebras into uniform tracial sequence algebras of nuclear $\mathcal Z$-stable $C^*$-algebras via tracial data.Comment: M\"unster Journal of Mathematics, to appear. Prop 2.5 added, now 18 page

Uniform property Gamma

We further examine the concept of uniform property Gamma for C*-algebras introduced in our joint work with Winter. In addition to obtaining characterisations in the spirit of Dixmier's work on central sequence in II$_1$ factors, we establish the equivalence of uniform property Gamma, a suitable uniform version of McDuff's property for C*-algebras, and the existence of complemented partitions of unity for separable nuclear C*-algebras with no finite dimensional representations and a compact (non-empty) tracial state space. As a consequence, for C*-algebras as in the Toms-Winter conjecture, the combination of strict comparison and uniform property Gamma is equivalent to Jiang-Su stability. We also show how these ideas can be combined with those of Matui-Sato to streamline Winter's classification-by-embeddings technique.Comment: IMRN, to appear. Accepted version; 39 page

Tracially Complete C*-Algebras

We introduce a new class of operator algebras -- tracially complete C*-algebras -- as a vehicle for transferring ideas and results between C*-algebras and their tracial von Neumann algebra completions. We obtain structure and classification results for amenable tracially complete C*-algebras satisfying an appropriate version of Murray and von Neumann's property gamma for II_1 factors. In a precise sense, these results fit between Connes' celebrated theorems for injective II_1 factors and the unital classification theorem for separable simple nuclear C*-algebras. The theory also underpins arguments for the known parts of the Toms-Winter conjecture.Comment: 130 page

Locally Trivial W*-Bundles

We prove that a tracially continuous W$^*$-bundle $\mathcal{M}$ over a compact Hausdorff space $X$ with all fibres isomorphic to the hyperfinite II$_1$-factor $\mathcal{R}$ that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group $\mathrm{Aut}({\mathcal{R}})$ shown by Popa and Takesaki. There is no restriction on the covering dimension of $X$.Comment: 20 pages, this version will be published in the International Journal of Mathematic

Nuclear dimension of simple C*-algebras

We compute the nuclear dimension of separable, simple, unital, nuclear, Z-stable C∗-algebras. This makes classification accessible from Z-stability and in particular brings large classes of C∗-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme