Free product von Neumann algebras of type 𝐼𝐼𝐼

Barnett, Lance

doi:10.1090/S0002-9939-1995-1224611-7

Free product von Neumann algebras of type $\textrm {III}$
HTML articles powered by AMS MathViewer

by Lance Barnett

Proc. Amer. Math. Soc. 123 (1995), 543-553

DOI: https://doi.org/10.1090/S0002-9939-1995-1224611-7

PDF | Request permission

Abstract:

In this paper we will show that most free products of von Neumann algebras with respect to nontracial states produce type ${\text {III}_\lambda }$ factors $(\lambda \ne 0)$. In addition, for all such $\lambda$, examples can be obtained with the component algebras being finite dimensional. Finally, conditions will be given to ensure that these free products will be full factors.

References

Daniel Avitzour, Free products of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 271 (1982), no. 2, 423–435. MR 654842, DOI 10.1090/S0002-9947-1982-0654842-1
Alain Connes, Une classification des facteurs de type $\textrm {III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115
A. Connes, Almost periodic states and factors of type $\textrm {III}_{1}$, J. Functional Analysis 16 (1974), 415–445. MR 0358374, DOI 10.1016/0022-1236(74)90059-7
Richard H. Herman and Masamichi Takesaki, States and automorphism groups of operator algebras, Comm. Math. Phys. 19 (1970), 142–160. MR 270167
Dusa McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443–461. MR 281018, DOI 10.1112/plms/s3-21.3.443
L. Pukánszky, Some examples of factors, Publ. Math. Debrecen 4 (1956), 135–156. MR 80894, DOI 10.5486/pmd.1956.4.3-4.05
Masamichi Takesaki, Conditional expectations in von Neumann algebras, J. Functional Analysis 9 (1972), 306–321. MR 0303307, DOI 10.1016/0022-1236(72)90004-3

Theory of operator algebras

Jun Tomiyama, On the projection of norm one in $W^*$-algebras. III, Tohoku Math. J. (2) 11 (1959), 125–129. MR 107825, DOI 10.2748/tmj/1178244633
Dan Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR 799593, DOI 10.1007/BFb0074909