A characterization of absolutely dilatable Schur multipliers.(English)Zbl 07807344
Summary: Let \(M\) be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let \(T : M \to M\) be a contraction. We say that \(T\) is absolutely dilatable if there exist another von Neumann algebra \(M^\prime\) equipped with a nsf trace, a \(w^\ast\)-continuous trace preserving unital \(\ast\)-homomorphism \(J : M \to M^\prime\) and a trace preserving \(\ast\)-automorphism \(U : M^\prime \to M^\prime\) such that \(T^k = E U^k J\) for all integer \(k \geq 0\), where \(E : M^\prime \to M\) is the conditional expectation associated with \(J\). Given a \(\sigma\)-finite measure space \((\Omega, \mu)\), we characterize bounded Schur multipliers \(\phi \in L^\infty(\Omega^2)\) such that the Schur multiplication operator \(T_\phi : B(L^2(\Omega)) \to B(L^2(\Omega))\) is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra \(N\) with a separable predual, equipped with a normalized normal faithful trace \(\tau_N\), and a \(w^\ast\)-measurable essentially bounded function \(d : \Omega \to N\) such that \(\phi(s, t) = \tau_N(d (s)^\ast d(t))\) for almost every \((s, t) \in \Omega^2\).
MSC:
| 47A20 | Dilations, extensions, compressions of linear operators |
| 46L06 | Tensor products of \(C^*\)-algebras |
| 46E40 | Spaces of vector- and operator-valued functions |
Cite
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