Continuity of the support of a state

Author:
Esteban Andruchow

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3565-3570

MSC (2000):
Primary 46L30, 46L05, 46L10

Published electronically:
July 2, 2002

MathSciNet review:
1920035

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite von Neumann algebra and a projection. It is well known that the map which assigns its support projection to a positive normal functional of is not continuous. In this note it is shown that if one restricts to the set of positive normal functionals with support equivalent to a fixed , endowed with the norm topology, and the set of projections of is considered with the strong operator topology, then the support map is continuous. Moreover, it is shown that the support map defines a homotopy equivalence between these spaces. This fact together with previous work implies that, for example, the set of projections of the hyperfinite II factor, in the strong operator topology, has trivial homotopy groups of all orders .

**1.**E. Andruchow, A. Varela, Homotopy of vector states, preprint.**2.**Ola Bratteli and Derek W. Robinson,*Operator algebras and quantum statistical mechanics. Vol. 1*, Springer-Verlag, New York-Heidelberg, 1979. 𝐶*- and 𝑊*-algebras, algebras, symmetry groups, decomposition of states; Texts and Monographs in Physics. MR**545651****3.**Richard V. Kadison and John R. Ringrose,*Fundamentals of the theory of operator algebras. Vol. II*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR**859186****4.**Sorin Popa and Masamichi Takesaki,*The topological structure of the unitary and automorphism groups of a factor*, Comm. Math. Phys.**155**(1993), no. 1, 93–101. MR**1228527**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L30,
46L05,
46L10

Retrieve articles in all journals with MSC (2000): 46L30, 46L05, 46L10

Additional Information

**Esteban Andruchow**

Affiliation:
Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J. M. Gutierrez entre J.L. Suarez y Verdi, (1613) Los Polvorines, Argentina

Email:
eandruch@ungs.edu.ar

DOI:
https://doi.org/10.1090/S0002-9939-02-06745-X

Keywords:
State space,
support projection

Received by editor(s):
September 1, 2000

Published electronically:
July 2, 2002

Communicated by:
David R. Larson

Article copyright:
© Copyright 2002
American Mathematical Society