论文标题
$ c^{*} $的有限近似属性 - 模块III
Finite approximation properties of $C^{*}$-modules III
论文作者
论文摘要
我们介绍并研究了$ c^{*} $ - 代数$ a $的模块核维度的概念，该$ a $是$ c^*$ - 模块，另一个$ c^*$ - algebra $ \ algebra $ \ mathfrak a $具有兼容的动作。我们表明，如果$ a $为$ \ mathfrak a $ -nf，则$ a $的模块核尺寸为零。当$ \ mathfrak a $是$ c（x）$ - 代数带有简单纤维，带有$ x $ compact且完全断开连接时。我们还引入了一个模块分解等级的概念，并表明当$ \ mathfrak a $是Unital and Simples时，如果模块分解等级为$ a $是有限的，那么$ a $ a $ as $ \ mathfrak a $ qd。 We study the set $\mathcal T_\mathfrak A(A)$ of $\mathfrak A$-valued module traces on $A$ and relate the Cuntz semigroup of $A$ with lower semicontinuous affine functions on the set $\mathcal T_\mathfrak A(A)$.在此过程中，我们还证明了一个模块coi-effros升起定理。我们给出了一类示例的模块核维度的估计。
We introduce and study a notion of module nuclear dimension for a $C^{*}$-algebra $A$ which is $C^*$-module over another $C^*$-algebra $\mathfrak A$ with compatible actions. We show that the module nuclear dimension of $A$ is zero if $A$ is $\mathfrak A$-NF. The converse is shown to hold when $\mathfrak A$ is a $C(X)$-algebra with simple fibers, with $X$ compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when $\mathfrak A$ is unital and simple, if the module decomposition rank of $A$ is finite then $A$ is $\mathfrak A$-QD. We study the set $\mathcal T_\mathfrak A(A)$ of $\mathfrak A$-valued module traces on $A$ and relate the Cuntz semigroup of $A$ with lower semicontinuous affine functions on the set $\mathcal T_\mathfrak A(A)$. Along the way, we also prove a module Choi-Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.
