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Algebras in Analysis
April 5, 2019 18:05–19:35, Moscow, Lomonosov Moscow State University, room 13-20.

Geometric essence of “compact” operators on Hilbert $C^*$-modules

E. V. Troitskii

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Abstract: We introduce a uniform structure on any Hilbert $C^*$-module $N$ and prove the following theorem: suppose, $F:M\to N$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $A$ and $N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $\theta_{x,y}$, $\theta_{x,y}(z):=x\langle y,z\rangle$, $x\in N$, $y,z\in M$ (i.e. $F$ is $A$-compact, or “compact”) if and only if $F$ maps the unit ball of $M$ to a totally bounded set with respect to this uniform structure (i.e. $F$ is a compact operator).
The talk is based on the preprint https://arxiv.org/abs/1810.02792.

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