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”.


46L08 \(C^*\)-modules
47B07 Linear operators defined by compactness properties
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