Discretization of C*-algebras

Functoriality of group algebras acting on $L^p$-spaces

We continue our study of group algebras acting on $L^p$-spaces, particularly of algebras of $p$-pseudofunctions of locally compact groups. We focus on the functoriality properties of these objects. We show that $p$-pseudofunctions are functorial with respect to homomorphisms that are either injective, or whose kernel is amenable and has finite index. We also show that the universal completion of the group algebra with respect to representations on $L^p$-spaces, is functorial with respect to quotient maps. As an application, we show that the algebras of $p$- and $q$-pseudofunctions on $\mathbb{…

A functorial approach to the C*-algebras of a graph

A functor from the category of directed trees with inclusions to the category of commutative C*-algebras with injective *-homomorphisms is constructed. This is used to define a functor from the category of directed graphs with inclusions to the category of C*-algebras with injective *-homomorphisms. The resulting C*-algebras are identified as Toeplitz graph algebras. Graph algebras are proved to have inductive limit decompositions over any family of subgraphs with union equal to the whole graph. The construction is used to prove various structural properties of graph algebras.

A double commutant theorem for operator algebras

Every unital nonselfadjoint operator algebra possesses canonical and functorial classes of faithful (even completely isometric) Hilbert space representations satisfying a double commutant theorem generalizing von Neumann's classical result. Examples and complementary results are given.

On discretization of C*-algebras

The C*-algebra of bounded operators on the separable infinite-dimensional Hilbert space cannot be mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra C(X) factors normally through $\ell^\infty(X)$. Consequently, there is no faithful functor discretizing C*-algebras to AW*-algebras, including von Neumann algebras, in this way.

Wieler 螺线管：非 Hausdorff 扩张性、Cuntz–Pimsner 模型与函子性质

Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties

本文研究由 Wieler 平稳逆极限诱导的 Fell 代数谱自映射，证明其构成紧致局部 Hausdorff 空间上的扩张满射局部同胚，并据此计算 Smale 空间稳定与不稳定 Ruelle 代数的 K-理论不变量。

Spatial discretization of Cuntz algebras

The (abstract) Cuntz algebra is generated by non-unitary isometries and has therefore no intrinsic finiteness properties. To approximate the elements of the Cuntz algebra by finite-dimensional objects, we thus consider a spatial discretization of this algebra by the finite sections method. For we represent the Cuntz algebra as a (concrete) algebra of operators on a Hilbert space and associate with each operator in this algebra the sequence of its finite sections. The goal of this paper is to examine the structure of the $C^*$-algebra which is generated by all sequences of this form. Our main r…