Abstract

Quite recently a criterion for the \(\mathcal{A}\)-compactness of an ajointable operator \(F\colon {\mathcal M} \to\mathcal{N}\) between Hilbert \(C^*\)-modules, where \(\mathcal{N}\) is countably generated, was obtained. Namely, a uniform structure (a system of pseudometrics) in \(\mathcal{N}\) was discovered such that \(F\) is \(\mathcal{A}\)-compact if and only if \(F(B)\) is totally bounded, where \(B\subset {\mathcal M} \) is the unit ball.

We prove that (1) for a general \(\mathcal{N}\), \(\mathcal{A}\)-compactness implies total boundedness, (2) for \(\mathcal{N}\) with \(\mathcal{N}\oplus K\cong L\), where \(L\) is an uncountably generated \(\ell_2\)-type module, total boundedness implies compactness, and (3) for \(\mathcal{N}\) close to be countably generated, it suffices to use only pseudometrics of “frame-like origin” to obtain a criterion for \(\mathcal{A}\)-compactness.

中文翻译：

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Hilbert $$C^*$$ -Modules 中的紧凑算子和统一结构

摘要

最近，Hilbert \(C^*\)之间的一个可连接算子\(F\colon {\mathcal M} \to\mathcal{N}\)的\(\mathcal{A}\) -紧凑性的标准-模块，其中\(\mathcal{N}\)是可数生成的。即，发现\(\mathcal{N}\)中的统一结构（伪度量系统）使得\(F\)是\(\mathcal{A}\) -compact 当且仅当\(F( B)\)是完全有界的，其中\(B\subset {\mathcal M} \)是单位球。

我们证明（1）对于一般的\(\mathcal{N}\)，\(\mathcal{A}\) -compactness 意味着总有界性，（2）对于\(\mathcal{N}\)与\( \mathcal{N}\oplus K\cong L\)，其中\(L\)是一个不可数生成的\(\ell_2\)型模块，总有界意味着紧凑性，并且（3）对于\(\mathcal{N }\)接近于可数生成，仅使用“类似框架的原点”的伪度量就足以获得\(\mathcal{A}\) -compactness 的标准。