Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-22T18:23:57.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal AW*-algebras

Published online by Cambridge University Press:  14 November 2011

J. D. Maitland Wright
J. D. Maitland Wright
Affiliation:
Department of Mathematics, University of Reading
Get access Rights & Permissions [Opens in a new window]

Synopsis

In recent years it has become clear that AW*-algebras can be much more pathological and unlike von Neumann algebras than was originally expected. When AW*-algebras are monotone complete, then the work of Kadison and Pederson shows that a particularly smooth and elegant theory can be developed. A technically weaker requirement on an AW*-algebra is that it be “normal”. This condition, which says that the lattice of projections is embedded in a well-behaved way in the partially ordered set of all self-adjoint elements, can sometimes be used as a substitute for monotone completeness. In this note we prove that when an AW*-algebra is of finite type (that is x*x = 1 implies xx* = 1) then it is normal.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 1-2 , 1980 , pp. 137 - 141
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Berberian, S. K.. Baer *-rings (Berlin: Springer, 1972).CrossRefGoogle Scholar
2Kaplansky, I.. Projections in Banach algebras. Ann. of Math. 53 (1951), 235–249.CrossRefGoogle Scholar
3Kaplansky, I.. Algebras of Type I. Ann. of Math. 56 (1952), 460–472.CrossRefGoogle Scholar
4Kaplansky, I.. Modules over operator algebras. Amer. J. Math. 75 (1953), 839–858.CrossRefGoogle Scholar
5Kaplansky, I.. Rings of operators (New York: Benjamin, 1968).Google Scholar
6Kadison, R. V. and Pederson, G. K.. Equivalence in operator algebras. Math. Scand. 27 (1970), 205–222.CrossRefGoogle Scholar
7Saito, K.. On the embedding as a double commutator in a Type I AW*-algebra II. Tôhoku Math. J. 26 (1974), 333–339.Google Scholar
8Saito, K.. AW*-algebras with monotone convergence property and examples by Takenouchi and Dyer. Tôhoku Math. J. 31 (1979), 31–40.CrossRefGoogle Scholar
9Widom, H.. Embedding in algebras of Type I. Duke Math. J. 23 (1956), 309–324.CrossRefGoogle Scholar
10Wright, J. D. M.. On AW*-algebras of finite type. J. London Math. Soc. 12 (1976), 431–439.CrossRefGoogle Scholar