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Non-Isomorphic Non-Hyperfinite Factors

Published online by Cambridge University Press:  20 November 2018

Wai-Mee Ching
Wai-Mee Ching*
Affiliation:
Louisiana State University, Baton Rouge, Louisiana; Courant Institute of Mathematical Sciences, New York, New York
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A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).

Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1293 - 1308
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Dixmier, J., Les algèbres d'opérateurs dans l'espace Hilbertien (Gauthiers-Villars, Paris, 1957).Google Scholar
2. Glimm, J., Type I C*-algebras, Ann. of Math. (2) 73 (1960), 572–612.Google Scholar
3. Murray, F. and von Neumann, J., On rings of operators, Ann. of Math. (2) 37 (1936), 116–229.Google Scholar
4. Murray, F. and von Neumann, J., On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716–808 Google Scholar
5. von Neumann, J., On rings of operators. III, Ann. of Math. (2) 41 (1940), 94–161 Google Scholar
6. von Neumann, J., On some algebraical properties of operator rings, Ann. of Math. (2) 44 (1943), 709–715.Google Scholar
7. Powers, R., Representations of uniformly hyperfinite algebras and their von Neumann rings, Ann. of Math. (2) 86 (1967), 138–171.Google Scholar
8. Pukanszky, L., Some examples of factors, Publ. Math. Debrecen 4 (1956), 135–156.Google Scholar
9. Sakai, S., On topological properties of W*-algebras, Proc. Japan Acad. 33 (1957), 439–444.Google Scholar
10. Schwartz, J., Two finite, non-hyperfinite non-isomorphic factors, Comm. Pure Appl. Math. 16 (1963), 19–26.Google Scholar
11. Schwartz, J., Non-isomorphism of a pair of factors of type III, Comm. Pure Appl. Math. 16 (1963), 111–120 Google Scholar
12. Segal, I., Non-commutative integration theory, Ann. of Math. (2) 57 (1953), 401–457.Google Scholar