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Applications of the complex interpolation method to a von Neumann algebra: non-commutative \(L^ p\)-spaces.(English)Zbl 0604.46063

The paper includes a new construction of non-commutative \(L^ p\)-spaces (for \(1<p<\infty)\) associated with a von Neumann algebra which is not necessarily semi-finite. The construction is essentially different from that in the papers by Connes, Haagerup.
The \(L^ p\)-spaces are defined as interpolation spaces between the von Neumann algebra and its predual. The uniform convexity of \(L^ p\)-spaces and the duality between \(L^ p\) and \(L^ q\) are proved by making use of the complex interpolation method and Tomita-Takesaki modular theory.
Then, these \(L^ p\)-spaces are compared with Haagerup \(L^ p\)-spaces.
Reviewer: A.Bartoszewicz

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46M35 Abstract interpolation of topological vector spaces

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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