Geometric essence of "compact" operators on Hilbert C^*-modules

2018/10/05 by Troitsky, Evgenij · 1 citation
#46L08 #47B10 #47L80 #54E15 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1810.02792

Abstract

We introduce a uniform structure on any Hilbert C^*-module \mathcal N and prove the following theorem: suppose, F:\mathcal M→ \mathcal N is a bounded adjointable morphism of Hilbert C^*-modules over \mathcal A and \mathcal N is countably generated. Then F belongs to the Banach space generated by operators θ_x,y, θ_x,y(z):=x⟨ y,z⟩, x∈ \mathcal N, y,z∈ \mathcal M (i.e. F is \mathcal A-compact, or "compact") if and only if F maps the unit ball of \mathcal M to a totally bounded set with respect to this uniform structure (i.e. F is a compact operator).

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