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Derivations on the algebra of \(\tau\)-compact operators affiliated with a type I von Neumann algebra. (English) Zbl 1155.46038

Let \(B(H)\) be the algebra of all bounded linear operators on a Hilbert space \(H\) and let \(M\) be a von Neumann algebra in \(B(H)\) with a faithful normal semi-finite trace \(\tau\). A linear subspace \(D\) in \(H\) is said to be affiliated with \(M\) (denoted by \(D\eta M\)) if \(u(D)\subseteq D\) for any unitary operator \(u\) from the commutant \(M'\) of the algebra \(M\). A linear operator \(x\) on \(H\) with the domain \(D(x)\) is said to be affiliated with \(M\) (denoted as \(x\eta M\)) if \(u(D(x))\subseteq D(x)\) and \(ux(\xi)=xu(\xi)\) for all \(u\in M'\), \(\xi\in D(x)\). A linear subspace \(D\) in \(H\) is called \(\tau\)-dense if \(D\eta M\) and, given any \(\varepsilon>0\), there exists a projection \(p\) in \(M\) such that \(p(H)\subseteq D\) and \(\tau(p^\perp)\leq\varepsilon\). A closed linear operator \(x\) is said to be \(\tau\)-measurable (or totally measurable) with respect to the von Neumann algebra \(M\) if \(x\eta M\) and \(D(x)\) is \(\tau\)-dense in \(H\). The set of all \(\tau\)-measurable operators with respect to \(M\) is denoted by \(L(M,\tau)\). Moreover, let \(S_0(M,\tau)\) be the set of all operators \(x\in L(M,\tau)\) such that, given any \(\varepsilon>0\), there is a projection \(p\) in \(M\) with \(\tau(p^\perp)<\infty\), \(xp\in M\) and \(\| xp\| <\varepsilon\). In this paper, it is proved that if \(M\) is a von Neumann algebra of type I with center \(Z\), then any \(Z\)-linear derivation on the algebra \(S_0(M,\tau)\) is spatial and implemented by an element of \(L(M,\tau)\).

MSC:

46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
46L51 Noncommutative measure and integration
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References:

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