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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.

MSC:
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
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