A mistake has been corrected on Nov. 30 2020, results unchanged. Dec. 4 2020 version has minor improvements. Nov 26 2021 seems to be final version to appear in Journal of non-commutative geometry
We characterize the lifting property (LP) of a separable $C^*$-algebra $A$ by a property of its maximal tensor product with other $C^*$-algebras, namely we prove that $A$ has the LP if and only if for any family $(\{D_i\mid i\in I\}$ of $C^*$-algebras the canonical map $$ {\ell_\infty(\{D_i\}) \otimes_{\max} A}\to {\ell_\infty(\{D_i \otimes_{\max} A\}) }$$ is isometric. Equivalently, this holds if and only if $M \otimes_{\max} A= M \otimes_{\rm nor} A$ for any von Neumann algebra $M$.
The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between $\mathrm{C}^*$-algebras. Outside of the nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP. In this article, we show that the LLP holds for the algebraic contraction $\mathrm{C}^*$-algebras introduced by Hadwin and further studied by Loring and Shulman. We also show that the universal Pythagorean $\mathrm{C}^*$-algebras introduced by Brothier and Jones have the Lifting Property.
We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the development of a local theory of operator modules by introducing analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules we prove that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where…
We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C$^*$-algebras of countable groups with (relative) property (T). We derive that the full C$^*$-algebras of the groups $\mathbb Z^2\rtimes\text{SL}_2(\mathbb Z)$ and $\text{SL}_n(\mathbb Z)$, for $n\geq 3$, do not have the local lifting property (LLP). We also prove that the full C$^*$-algebras of a large class of groups $Γ$ with property (T), including those such that $\text{H}^2(Γ,\mathbb R)\not=0$ or $\text{H}^2(Γ,\mathbb ZΓ)\not=0$, do not have the lifting prop…
Obstructions to lifting cocycles on groupoids and the associated $C^*$-algebras
给定局部紧交换群$0 \to A \to B \to C \to 0$的短正合序列和局部紧Hausdorff广群$Γ$上的连续$C$ -值$1$ -上圈$ϕ$，我们构造了$Γ$的一个由$A$构成的扭曲，它是平凡的当且仅当$ϕ$提升.上循环决定了$\widehat{C}$到$\operatorname{Aut} C^*(Γ)$的一个强连续作用，并且证明了如果$Γ$是顺从的，则扭转的$C^*$ -代数同构于该作用的诱导代数.我们将我们的结果应用到一个广群所确定的局部有限覆盖的空间$X$和一个cocycle提供的Čech 1-cocycle的系数在层芽连续$\mathbb{T}$值函数。我们证明了所得到的twist的$C^*$ -代数是连续迹，并计算了它的Dixmier-Douady不变量。
Let $ξ:C^*(E)\to C^*(F)$ be a unital $*$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $*$-homomorphism $ϕ:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $ξ$ is homotopic to the map $\hatϕ:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hatϕ$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $ϕ$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz-Krieger algebras is …
Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.
