Skip to main content
Log in

On the purification of factor states

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let\(\mathfrak{A}\) be aC*-algebra and\(\mathfrak{A}^ \circ \) be an opposite algebra. Notions of exact andj-positive states of\(\mathfrak{A}^ \circ \)\(\mathfrak{A}\) are introduced. It is shown, that any factor state ω of\(\mathfrak{A}\) can be extended to a pure exactj-positive state\(\tilde \omega \) of\(\mathfrak{A}^ \circ \)\(\mathfrak{A}\). The correspondence ω→\(\tilde \omega \) generalizes the notion of the purifications map introduced by Powers and Størmer. The factor states ω1 and ω2 are quasi-equivalent if and only if their purifications\(\tilde \omega _1 \) and\(\tilde \omega _2 \) are equivalent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Araki, H.: A latties of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys.4, 1345–1362 (1963).

    Google Scholar 

  2. Dixmier, J.: Les algebras d'operateurs dans l'espace Hilbertien. Paris: Gauthier-Villars 1969.

    Google Scholar 

  3. Murray, F. J., von Neumann, J.: On rings of operators I. Ann. Math.37, 116–229 (1936).

    Google Scholar 

  4. Powers, R. T., Størmer, E.: Free states of the canonical anticommutation relations. Commun. math. Phys.16, 1–33 (1970).

    Google Scholar 

  5. Takesaki, M.: Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Mathematics128. Berlin-Heidelberg-New York: Springer 1970.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Woronowicz, S.L. On the purification of factor states. Commun.Math. Phys. 28, 221–235 (1972). https://doi.org/10.1007/BF01645776

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01645776

Keywords

Navigation