Representable E-theory for Co(X)-algebras

R和Z光滑交叉积的可表示K理论

Representable K-theory of Smooth Crossed Products by R and Z

证明了Frechet代数的Thom同构和Pimsner-Voiculescu正合列分别对R和Z的光滑交叉积成立.对于Banach代数与R和Z的L^{1}-交叉积也得到了同样的结果。

关于正投影与不可表示巴拿赫格代数的Wickstead猜想

Wickstead's conjecture on positive projections and non-representable Banach lattice algebras

本文证明了Wickstead关于正投影的猜想普遍成立，进而解决巴拿赫格代数的表示问题，还在Lean 4中形式化了相关结果。

On tensor products of operator modules

The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.

弱迹近似可表示作用

Weakly tracially approximately representable actions

本文针对有限阿贝尔群在无限维单的有单位元C*-代数上的逐点外自同构作用，建立弱迹Rokhlin性质与弱迹近似可表示性之间的对偶等价关系，推广了Izumi和Phillips的经典对偶结果。

Atiyah-Jänich theorem for $σ$-C*-algebras

K-theory for $ σ$-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips [{\it K-Theory} {\bf 3} (1989), 441--478]. We use his representable K-theory to show that the space of Fredholm modular operators with coefficients in an arbitrary unital $ σ$-C*-algebra $A$, represents the functor $X \mapsto {\rm RK}_{0}(C(X, A))$ from the category of countably compactly generated spaces to the category of abelian groups.

The tracial Rokhlin property for actions of finite groups on C*-algebras

We define "tracial" analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and of approximate innerness. We prove four analogs of related "nontracial" results. First, the crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. Second, an outer action of a finite abelian group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property if and only if i…

ZFC 中一个可分离表示的反例：Naimark 问题

A separably representable counterexample to Naimark's problem in ZFC

本文在 ZFC 中构造了一个单的、简单的、无限维 C*-代数，其所有非零不可约表示酉等价，但存在可分忠实表示，从而给出 Naimark 问题在 ZFC 中的反例，并揭示其与连续统假设的等价性。

Quasi-free actions of finite groups on the Cuntz algebra $\mathcal{O}_\infty$

We show that any faithful quasi-free actions of a finite group on the Cuntz algebra $\mathcal{O}_\infty$ are mutually conjugate, and that they are asymptotically representable.

Crossed product approach to equivariant localization algebras

The goal of this article is to provide a bridge between the gamma element method for the Baum--Connes conjecture (the Dirac dual-Dirac method) and the controlled algebraic approach of Roe and Yu (localization algebras). For any second countable, locally compact group G, we study the reduced crossed product algebras of the representable localization algebras for proper G-spaces. We show that the naturally defined forget-control map is equivalent to the Baum--Connes assembly map for any locally compact group G and for any coefficient G-C*-algebra B. We describe the gamma element method for the B…

Rokhlin actions of finite groups on UHF-absorbing C*-algebras

This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on C*-algebras satisfying a certain UHF-absorption condition. We show that given any finite group $G$ and a separable, unital C*-algebra $A$ that absorbs $M_{|G|^\infty}$ tensorially, one can lift any group homomorphism $G\to\operatorname{Aut}(A)/{\approx_u}$ to an honest Rokhlin action $γ$ of $G$ on $A$. Unitality may be dropped in favour of stable rank one or being stable. If $A$ belongs to a certain class of C*-algebras that is classifiable by a suitable invariant (e.g. $K$-theory)…

Finite cyclic group actions with the tracial Rokhlin property

We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of Z/2Z on the 2^{\infty} UHF algebra, whose crossed produc…

Quasimultipliers of Operator Spaces

We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the Banach-Stone theorem.

Operator algebra quantum homogeneous spaces of universal gauge groups

In this paper, we quantize universal gauge groups such as SU(\infty), as well as their homogeneous spaces, in the sigma-C*-algebra setting. More precisely, we propose concise definitions of sigma-C*-quantum groups and sigma-C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute it for the quantum homogeneous spaces associated to the universal gauge …

Finite dimensional irreducible representations and the uniqueness of the Lebesgue decomposition of positive functionals

We prove for an arbitrary complex $^*$-algebra $A$ that every topologically irreducible $^*$-representation of $A$ on a Hilbert space is finite dimensional precisely when the Lebesgue decomposition of representable positive functionals over $A$ is unique. In particular, the uniqueness of the Lebesgue decomposition of positive functionals over the $L^1$-algebras of locally compact groups provides a new characterization of Moore groups.