an:06699472
Zbl 1373.46059
Dykema, Ken; Sukochev, Fedor; Zanin, Dmitriy
Algebras of log-integrable functions and operators
EN
Complex Anal. Oper. Theory 10, No. 8, 1775-1787 (2016).
00360488
2016
j
46L52 46H35 46E30 30H15
log-integrable functions; measurable operators; complete \(^*\)-algebras; Nevanlinna class
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 [\textit{W. Matuszewska} and \textit{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.
Ying-Fen Lin (Belfast)
Zbl 0202.39903