Positive Cones and Lₚ-Spacesfor von Neumann Algebras

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 (forl 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_η1/(2p) (for 2≤ p ≤ ∞ ) or the completion of the positive cone V_η1/(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 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^α_η for each α∈ [0, 1 /4 ] .