Positive Cones and -Spacesfor von Neumann Algebras

  • Huzihiro Araki

    Kyoto University, Japan
  • Tetsuya Masuda

    Kyoto University, Japan


The -space for a von Neumann algebra with reference to its cyclic and separating vector in the standard representation Hilbert space of is constructed either as a subset of (for ), or as the completion of (forl ) with an explicitly defined -norm. The Banach spaces for different reference vector (with the same ) are isomorphic.

Any element has a polar decomposition where the positive part is defined to be either the intersection with the positive cone (for ) or the completion of the positive cone (for ). Any positive element has an interpretation as the th power of an with its -norm given by .

Product of an element and an element is explicitly defined as an element with provided that , and the Hölder inequality is proved.

The -space constructed here is isomorphic to those defined by Haagerup, Hilsum, and Kosaki.

As a corollary, any normal state of is shown to have one and only one vector representative in the positive cone for each .

Cite this article

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

DOI 10.2977/PRIMS/1195183577