Ideal spaces of measurable operators affiliated to a semifinite von Neumann algebra. (English. Russian original) Zbl 06908372
Sib. Math. J. 59, No. 2, 243-251 (2018); translation from Sib. Mat. Zh. 59, No. 2, 309-320 (2018).
Summary: Suppose that \(\mathcal M\) is a von Neumann algebra of operators on a Hilbert space \(\mathcal H\) and \(\tau\) is a faithful normal semifinite trace on \(\mathcal M\). Let \(\mathcal E\), \(\mathcal F\) and \(\mathcal G\) be ideal spaces on \((\mathcal M, \tau)\). We find when a \(\tau\)-measurable operator \(X\) belongs to \(\mathcal E\) in terms of the idempotent \(P\) of \(\mathcal M\). The sets \(\mathcal E+\mathcal F\) and \(\mathcal E\cdot\mathcal F\) are also ideal spaces on \((\mathcal M,\tau)\); moreover, \(\mathcal E\cdot\mathcal F = \mathcal F\cdot\mathcal E\) and \((\mathcal E+\mathcal F)\cdot \mathcal G = \mathcal E\cdot \mathcal G+\mathcal F\cdot \mathcal G\). The structure of ideal spaces is modular. We establish some new properties of the \(L_1(\mathcal M,\tau)\) space of integrable operators affiliated to the algebra \(\mathcal M\). The results are new even for the \(\ast\)-algebra \(\mathcal M = \mathcal B(\mathcal H)\) of all bounded linear operators on \(\mathcal H\) which is endowed with the canonical trace \(\tau = \operatorname{tr}\).
Keywords:
Hilbert space; linear operator; von Neumann algebra; normal semifinite trace; measurable operator; compact operator; integrable operator; commutator; ideal spaceReferences:
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