A crossed product approach to Orlicz spaces (Q2864125)

The author presents the first serious attempt for the construction of noncommutative Orlicz spaces for type III von Neumann algebras in the spirit of Haagerup's noncommutative \(L^p\)-spaces. After reminding the reader of the appropriate material on commutative Orlicz spaces, the `usual' way of defining the Banach function spaces for semifinite algebras \(M\) endowed with a f.n.s. trace \(\tau\) is presented, namely, \(L^\rho(M,\tau)=\{f\in M:\mu(f)\in L^\rho(0,\infty)\}\), where \(\rho\) is the Banach function norm, and \(\mu\) is the singular value function.NEWLINENEWLINEUnfortunately, this method cannot be generalized to the context of arbitrary von Neumann algebras. As the first step to such a generalization, the theory of Orlicz spaces is reformulated in the crossed product setting, which in the semifinite case means working with the tensor product \(M\otimes L^\infty(\mathbb{R})\). The main tool is the author's generalization of Haagerup's lemma that allows identifying the predual of the von Neumann algebra with a subspace of the space of measurable operators associated with the above tensor product. It uses the fundamental function \(\varphi\) of the Orlicz space in a crucial way, yielding an important formula for the Luxemburg norm corresponding to the Orlicz function \(\psi\): \(\|a\|_\psi=\mu_1(a\otimes \varphi_\psi(e^t)).\) As a consequence, one can define the Orlicz spaces inside the space of measurable operators affiliated with the tensor product, and obtain a proper duality theory for the spaces.NEWLINENEWLINEAfter this preparation, the definitions of Orlicz spaces for type III algebras are obtained in a natural way, and \(\mu_1\) becomes an obvious candidate for the norm. Unfortunately, the technical difficulties are overwhelming, and the results obtained show that \(\mu_1\) is only a quasinorm on the Orlicz spaces. Nevertheless, the topology given by the seminorm is exactly the one induced by the measure topology on the space \(\tilde A\), where \(A=M\rtimes_\sigma \mathbb{R}\); this phenomenon is known from the theory of Haagerup spaces. Moreover, the topology is normable if the upper Boyd index of the Orlicz space is strictly less then \(1\). The duality theory for the spaces is also close to the desired one.NEWLINENEWLINE The next part of the paper shows a proper context for interpolation.NEWLINENEWLINE First, the smallest and largest Orlicz spaces (denoted, resp., by \(L^{1\cap\infty}\) and \(L^{1+\infty}\)) are carefully examined, and then a way of building noncommutative function spaces using the \(K\)-method of real interpolation is indicated. The final part of the paper deals with the investigation of the technically easier case of \(\sigma\)-finite algebras, where stronger results are obtained, and with concluding remarks together with the indication of most important facts still to be established.NEWLINENEWLINETo sum up, this is an important paper with many excellent ideas and an agenda for future complements and improvements.