# Positive Cones and $L_{p}$-Spacesfor von Neumann Algebras

### Huzihiro Araki

Kyoto University, Japan### Tetsuya Masuda

Kyoto University, Japan

## 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 $H$ (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 $V_{η}$ (for $2≤p≤∞$ ) or the completion of the positive cone $V_{η}$ (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 $V_{η}$ for each $α∈[0,1/4]$.

## Cite this article

Huzihiro Araki, Tetsuya Masuda, Positive Cones and $L_{p}$-Spacesfor von Neumann Algebras. Publ. Res. Inst. Math. Sci. 18 (1982), no. 2, pp. 339–411

DOI 10.2977/PRIMS/1195183577