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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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