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Algebras of log-integrable functions and operators. (English) Zbl 1373.46059
Let \((\Omega, \nu)\) be a measure space, the function space \(\mathcal{L}_{\log}(\Omega, \nu)\) consists of measurable functions \(f\) such that \(\int_{\Omega} \log(1+ |f|)\, d \nu< \infty\). A non-commutative operator algebra version \(\mathcal{L}_{\log}(\mathcal{M}, \tau)\) over a von Neumann algebra \(\mathcal{M}\) with a normal, faithful, semifinite trace \(\tau\) is defined to be the set of all \(\tau\)-measurable operators affiliated with \(\mathcal{M}\) such that \(\tau(\log(1+ |T|))< \infty\). In this paper, the authors show that they are complete topological \(^*\)-algebras, with respect to the \(F\)-norms \(\|f\|_{\log}:= \int_{\Omega} \log(1+ |f|)\, d \nu\) and \(\|T\|_{\log}:= \tau(\log(1+ |T|))\), respectively. Note that the authors treat the commutative case \(\mathcal{L}_{\log}(\Omega, \nu)\) separately, although it is a special case of \(\mathcal{L}_{\log}(\mathcal{M}, \tau)\), they show that the space \(\mathcal{L}_{\log}(\Omega, \nu)\) is a non-locally convex generalised Orlicz space and that the \(F\)-norm \(\|\cdot\|_{\log}\) is equivalent to the one given in [W. Matuszewska and W. Orlicz, Stud. Math. 21, 107–115 (1961; Zbl 0202.39903)]. The authors also explore connections with the Nevanlinna class of holomorphic functions on the open unit disk.

46L52 Noncommutative function spaces
46H35 Topological algebras of operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
30H15 Nevanlinna spaces and Smirnov spaces
Full Text: DOI arXiv
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