The L^p-spaces of a von Neumann Algebra
by
Tony Falcone
Illinois State University
Coauthors: Masamichi Takesaki (UCLA)

It has become commonplace to think of von Neumann algebras as ``non-commutative L<sup>\infty</sup>-spaces.'' Hence, it is a natural question to consider an appropriate definition of the associated ``non-commutative L^p-spaces.'' This issue has been addressed previously by Haagerup, Kosaki, Izumi, Terp, et al., but we have had occasion to revisit it in the context of our construction of the quantum flow of weights on a von Neumann algebra. Specifically, we have been able to demonstrate that there exists a natural (and in fact canonical) construction for L^p-spaces which agrees (upon specialization) with all previous descriptions. In particular, our point of view provides a unified framework which enables one to see that the descriptions due to Haagerup and Kosaki are in fact equivalent; moreover, our vantage allows one to view these L^p-spaces not as isolated Banach spaces, but rather as part of a continuum of spaces which are naturally ``attached'' to each von Neumann algebra.

Date received: April 30, 1999