In this paper we suggest an approach for constructing an $$L_1$$ L 1 -type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we introduce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an $$L_1$$ L 1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent $$L_1$$ L 1 -type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that $$L_1$$ L 1 -type space is isometrically isomorphic to the predual of von Neumann algebra in a natural way. We give a small list of alternate definitions of the seminorm, and a special definition for the case of semifinite von Neumann algebra, in particular. We study order properties of $$L_1$$ L 1 -type space, and demonstrate the connection between semifinite normal weights and positive elements of this space. At last, we construct a similar L-space for the positive element of C*-algebra, and study the connection between this L-space and the $$L_1$$ L 1 -type space in case when this C*-algebra is a von Neumann algebra.
Positivity – Springer Journals
Published: May 18, 2016
It’s your single place to instantly
discover and read the research
that matters to you.
Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
All the latest content is available, no embargo periods.
“Whoa! It’s like Spotify but for academic articles.”@Phil_Robichaud