Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-28T12:34:31.743Z Has data issue: false hasContentIssue false
  • English
  • Français

On Maximal Abelian Subrings of Factors of Type II

Published online by Cambridge University Press:  20 November 2018

Lajos Pukánszky
Lajos Pukánszky*
Affiliation:
RIAS, Baltimore
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Although we possess a fairly complete knowledge of the abelian subrings of rings of operators in a Hilbert space which are algebraically isomorphic to the ring of all bounded operators of a finite or infinite dimensional unitary space, that is of factors of Type I, very little is known of abelian subrings of factors of Type II1. In (1), Dixmier investigated several properties of maximal abelian subrings of factors of Type II. It turned out that their structure differs essentially from that of maximal abelian subrings of factors of Type I. He showed the existence of maximal abelian subrings in approximately finite factors, possessing the property that every inner*-automorphism carrying this subring into itself is necessarily implemented by a unitary operator of this subring. These maximal abelian subrings are called singular. In addition, he constructed a IIi factor containing two singular abelian subrings which cannot be connected by an inner *automorphism of this ring.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 289 - 296
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Dixmier, J., Sous-anneaux abêliens maximaux dans les facteurs de type fini, Ann. Math., 59 (1957), 279–286.Google Scholar
2. Murray, F.J. and von Neumann, J., On rings of operators, Ann. Math., 37 (1936), 116–229.Google Scholar
3. Murray, F.J. and von Neumann, J. On rings of operators IV, Ann. Math., U (1943), 716–808.Google Scholar
4. Segal, I.E., Decompositions of operator algebras II, Mem. Amer. Math. Society, no. 9 (1951).Google Scholar