Positive Cones and L_p-Spaces for von Neumann Algebras

Abstract

The L_p-space L_p(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 L_p-norm. The Banach spaces L_p(M, η) for different reference vector η (with the same p) are isomorphic.
Any L_p element has a polar decomposition where the positive part L_p⁺(M, η) is defined to be either the intersection with the positive cone V_η^1/(2p) (for 2{≤}p{≤}∞) or the completion of the positive cone V_η^1/(2p) (for 1{≤}p<2). Any positive element has an interpretation as the (1/p)^th power ω^1/p of an ω∈M_*⁺ with its L_p-norm given by ||ω||^1/p.
Product of an L_p element and an L_q element is explicitly defined as an L_r element with r⁻¹=p⁻¹+q⁻¹ provided that 1{≤}r, and the Hölder inequality is proved.
The L_p-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 V_η^α for each α∈[0, 1/4].