A characterization of weak (semi-)projectivity for commutative C*-algebras

Projectivity of Banach and $C^*$-algebras of continuous fields

We give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective $C^*$-algebras $\A$ defined by locally trivial continuous fields $\mathcal{U} = \{Ω,(A_t)_{t \in Ω},Θ\}$ such that each $C^*$-algebra $ A_{t}$ has a strictly positive element. For a commutative $C^*$-algebra $\D$ contained in ${\cal B}(H)$, where $H$ is a separable Hilbert space, we show that the condition of left projectivity of $\D$ is equivalent to the existence of a strictly positive element in $\…

A characterization of semiprojectivity for commutative C*-algebras

Given a compact, metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighborhood retract of dimension at most one. This confirms a conjecture of Blackadar. Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As further application of our findings we verify two conjectures of Loring and Blackadar in the commutative case, and we give a partial answer to the question, when a commutative C*-algebra is weakly (semi-)projective.

The Cuntz semigroup of unital commutative AI-algebras

We provide an abstract characterization for the Cuntz semigroup of unital commutative AI-algebras, as well as a characterization for abstract Cuntz semigroups of the form $\text{Lsc} (X,\overline{\mathbb{N}})$ for some $T_1$-space $X$. In our investigations, we also uncover new properties that the Cuntz semigroup of all AI-algebras satisfies.

A note on the Voiculescu's theorem for commutative C$^*$-algebras in semifinite von Neumann algebras

In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.

交换C* 代数的Cuntz半群的稳定性

Stability in the Cuntz semigroup of a commutative C*-algebra

证明了交换C* 代数的Cuntz半群的稳定性定理，这些定理类似于紧T2空间上拓扑向量丛的经典稳定性定理.然后给出了维数增长慢的单单位AH代数的几个应用：这类代数具有正元素的严格比较;它们的Cuntz半群是从Elliott不变量函式恢复的;低维连续函数在所有维数函数的空间中是稠密的，后者是Choquet单形.

Isometries on non-commutative (quantum) Lorentz spaces associated with semi-finite von Neumann algebras

In this article we characterize the extreme points of the unit ball of a non-commutative (quantum) Lorentz space associated with a semi-finite von Neumann algebra. This enables us to show that surjective isometries between non-commutative Lorentz spaces are projection disjointness preserving and finiteness preserving, which facilitates a characterization of the structure of these isometries.