Notes on noncommutative LP and Orlicz spaces

Abstract

Since the pioneering work of Dixmier and Segal in the early 50’s, the theory of noncommutative LP-spaces has grown into a very refined and important theory with wide applications. Despite this fact there is as yet no self-contained peer-reviewed introduction to the most general version of this theory in print. The present work aims to fill this vacuum, in the process giving fresh impetus to the theory. The first part of the book presents: the introductory theory of von Neumann algebras – also including the slightly less common theory of generalized positive operators; the various notions of measurability, allowing the interpretation of unbounded affiliated operators as “quantum”" measurable functions, with the crucial notion of τ-measurability developed in more detail; Jordan *-morphisms (representing quantum measurable transformations) that behave well with regard to τ-measurability; and finally the different types of weights that occur naturally in the theory, before presenting a Radon-Nikodym theorem for such weights. The core, second part of the book is devoted to first developing the noncommutative theory of decreasing rearrangements, before using that technology to present the basic theory of LP and Orlicz spaces for semifinite algebras, and then the notion of crossed product, as well as the technology underlying it, indispensable for the theory of Haagerup LP-spaces for general von Neumann algebras. With this as a foundation, we are then finally ready to present the basic structural theory of not only Haagerup LP-spaces, but also Orlicz spaces for general von Neumann algebras.

Abelian von Neumann algebras, measure algebras and L^\infty-spaces

“…Similarly (ii) and the fact that * -isomorphisms preserve suprema immediately imply (iv) (e.g. see [11,Proposition 1.57]).…”

Quantum Fokker–Planck Dynamics

“…But since V (and therefore E V ) is bounded and everywhere defined, the sum E L+ V [ξ] = (−L − V )ξ 2 2 = (−L − V )ξ, ξ of these Dirichlet forms can then be shown to be yet another Dirichlet form, with domain F L . (Properly the sum of operators inside the inner product is here the so-called form sum, but in the present case the operator sum and form sum agree-see [29,Proposition 3.26] and the discussion preceding it.) It therefore follows that H q = L + V is the generator of a Markov semigroup.…”

“…Then, a → η(a) becomes a dense embedding of M into H ν . We similarly know that in the above setting, {ah 1/2 : a ∈ M} is a dense subspace of L 2 (M) [29,Theorem 7.45]. Now observe that for any a, b ∈ M, we have that a, b ν = ν(b * a) = tr(h 1/2 b * ah 1/2 ) = tr((bh 1/2 ) * (ah 1/2 )) = (ah 1/2 ), (bh 1/2 ) where the second inner product is the canonical inner product on L 2 (M).…”

“…Thus, the space L 2 (M) is clearly a copy of H ν with the densely defined map η(a) → ah 1/2 extending to a linear isometry. We shall therefore where convenient freely pass from the GNS to the Haagerup setting using the identification η(a) ↔ ah 1/2 For a recent account of the theory of non-commutative L p -spaces, we refer the reader to [29].…”

“…Let (M, L 2 (M), L 2 + (M), J) be the Haagerup-Terp standard form of the σ-finite von Neumann algebra M [29,Theorem 7.56]. In this context, h 1/2 is the cyclic and separating vector representing the faithful normal state ν on M in the sense that ν(x) = tr(h 1/2 xh 1/2 ) = xh 1/2 , h 1/2 for all x ∈ M. Within in this framework Cipriani and Zegarlinski [16] define a Dirichlet form with respect to the pair (M, ν) to be a quadratic, semicontinuous functional 2 for all ξ ∈ F. Let V and L, respectively, be as in Theorem 4.2 and Proposition 5.9.…”