Definable $\mathrm{K}$-homology of separable C*-algebras

可定义（上）同调、pro-torus刚性及（上）同调分类

Definable (co)homology, pro-torus rigidity, and (co)homological classification

本文引入可定义同调与上同调理论，作为Steenrod同调与Čech上同调的加强不变量，完全分类pro-tori及其补空间，并应用于界定若干数学分类问题的Borel复杂度。

Model structure on projective systems of $C^*$-algebras and bivariant homology theories

Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable $C^*$-algebras $\mathrm{Pro}(\mathtt{SC^*})$. The opposite of this model category models the $\infty$-category of pointed noncommutative spaces $\mathtt{N}\mathcal{S_*}$ defined by the third author. Our model structure on $\mathrm{Pro}(\mathtt{SC^*})$ extends the well-known category of fibrant objects structure on $\mathtt{SC^*}$. We show that the pro-category $\mathrm{Pro}(\mathtt{SC^*})$ also contains, as a full coreflective subcateg…

Twisted K-theory, K-homology and bivariant Chern-Connes type character of some infinite dimensional spaces

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate th…

Dixmier关于C*-代数谱的问题的解答

Solution to Dixmier's Problem about spectra of C*-algebras

本文解决了Dixmier关于单C*-代数谱的1967年问题，证明了在某些单C*-代数（如CAR代数与SL_3(Z)的约化交叉积）之间不存在Borel可定义的单射，并推广到可数amenable群与自由群的酉对偶，同时给出核非I型代数谱的Borel可定义双射存在性。

AF-embeddings of residually finite dimensional C*-algebras

It is shown that a separable exact residually finite dimensional C*-algebra with locally finitely generated (rational) even K-homology embeds in a uniformly hyperfinite C*-algebra.

Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…