Compact and “compact” operators on standard Hilbert modules over $$W^*$$-algebras.(English)Zbl 1394.46047

The authors introduce a locally convex topology on the standard Hilbert module $$l^2(A)$$, where $$A$$ is a unital $$W^*$$-algebra such that any “compact” operator (that is, a member of the closed linear span of operators of the form $$\theta_{y,z}(x)=z\langle y,x\rangle$$, $$x,y,z\in l^2(A)$$) maps bounded sets (in the norm) into totally bounded sets in the introduced topology. In the special case where $$A=B(H)$$, they show that the converse is also true. Namely, any operator $$T$$ in the algebra of all adjointable operators on $$l^2(A)$$ that maps bounded sets into totally bounded sets is “compact”.

MSC:

 46L08 $$C^*$$-modules 47B07 Linear operators defined by compactness properties
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References:

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