Positive Cones and $L_p$-Spacesfor von Neumann Algebras

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 $2\le p\le \infty$), or as the completion of $H$ (forl $1\le p<2$) 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 $2\le p\le \infty$ ) or the completion of the positive cone $V_{\eta }^{1/(2p)}$ (for $1\le p<2$). Any positive element has an interpretation as the $(1/p)$th power ${\omega }^{1/p}$ of an $\omega \in M_{\ast }^{+}$ with its $L_p$ -norm given by $\Vert \omega {\Vert }^{1/p}$ .

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 $1\le r$, 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 $V_{\eta }^{\alpha }$ for each $\alpha \in [0,1/4]$.