Positive Cones and LP-Spaces for Von Neumann Algebras

Abstract

The Lp-space Lp(M, η) for a von Neumann algebra M with reference to its cyclic and separating vector η in the standard representation Hilbert space H of M is constructed either as a subset of H (for 2≦p≦∞), or as the completion of H (for 1≦p<2) with an explicitly defined Lp-norm. The Banach spaces Lp(M, η) for different reference vector η (with the same p) are isomorphic. Any Lp element has a polar decomposition where the positive part Lp+(M, η) is defined to be either the intersection with the positive cone Vη1/(2p) (for 2≦p≦∞) or the completion of the positive cone Vη1/(2p) (for 1≦p<2). Any positive element has an interpretation as the (1/p)th power ω1/p of an ω∈M*+ with its Lp-norm given by ||ω||1/p. Product of an Lp element and an Lq element is explicitly defined as an Lr element with r-1=p-1+q-1 provided that 1≦r, and the Hölder inequality is proved. The Lp-space constructed here is isomorphic to those defined by Haagerup, Hilsum, and Kosaki. As a corollary, any normal state of M is shown to have one and only one vector representative in the positive cone Vηα for each α∈[0, 1/4]. © 1982, Research Institute forMathematical Sciences. All rights reserved.