Positive Cones and $L_p$-Spacesfor von Neumann Algebras

Source abstract

The L_p -space L_p (M, \eta) for a von Neumann algebra M with reference to its cyclic and separating vector \eta in the standard representation Hilbert space H of M is constructed either as a subset of H (for 2\leq p \leq \infty ), or as the completion of H (forl 1\leq p <2 ) with an explicitly defined L_p -norm. The Banach spaces L_p (M, \eta) for different reference vector \eta (with the same p ) are isomorphic. Any L_p element has a polar decomposition where the positive part L^+_p(M, \eta) is defined to be either the intersection with the positive cone V_\eta^{1/(2p)} (for 2\leq p \leq \infty ) or the completion of the positive cone V_\eta^{1/(2p)} (for 1\leq p <2 ). Any positive element has an interpretation as the (1/p) th power \omega^{1/p} of an \omega \in M^+_* with its L_p -norm given by \|\omega\|^{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\leq 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^\alpha_\eta for each \alpha\in [0, 1 /4 ] .