Abstract
Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO’s) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO’s. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.
Similar content being viewed by others
References
Voiculescu D. Symmetries of some reduced free product C*-algebras. Lect Notes in Math, Vol 1132. Berlin: Springer-Verlag, 1985. 556–588
Voiculescu D, Dykema K J, Nica A. Free Random Variables. CRM Monograph Series No.1. Provindence: Amer Math Soc, 1992
Ching W M. Free products of von Neumann algebras. Trans of Amer Math Soc, 1973, 178: 147–163
Avitzour D. Free products of C*-algebras. Trans Amer Math Soc, 1982, 271: 423–435
Redulescu F. The fundamental group of the von Neumann algebra of the free group with infinitely many generators. J Amer Math Soc, 1992, 5(3): 513–532
Voiculescu D. The analogues of entropy and Fisher’s information measure in free probability III, Absence of Cartan Subalgebras. Geom Funct Anal, 1994, 6(1): 172–199
Ge L. Application of Free entropy to finite von Neumann algebras, I. Amer J Math, 1997, 119: 467–485
Ge L. Application of Free entropy to finite von Neumann algebras, II. Ann of Math, 1998, 147: 143–157
Ge L, Popa S. On some decomposition properties for factors of type II 1. Duke Math J, 1998, 94: 79–101
Effros E G, Ruan Z J. Operator Spaces. London Math Soc. Monographs New Series 23. Oxford: Clarendon Press, 2000
Ruan Z J. Subspaces of C*-algebras. J Funct Anal, 1988, 76: 217–230
Paulsen V. Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, Vol. 78. Combridge: Cambridge University press, 2002
Pisier G. An introduction to operator space theory. London Mathematical Society Lecture Notes No. 294. Cambridge: Cambridge University Press, 2003
Junge M, Pisier G. Bilinear forms on exact operator spaces and B(H) ⊗ B(H). Geom and Funct Anal, 1995, 5: 329–363
Ledoux M, Talagrand M. Probability in Banach Spaces. Berlin Heidelberg: Springer-Verlag, 1991
Effros E G. New directions and problems in operator space theory. A speech in workshop on Free probability and non-commutative Banach Space theory. Berkeley, 2001
Pestov V. Operator Spaces and Residually finite-dimensional C*-algebras. J Funct Anal, 1994, 123: 308–317
Effros E, Ozawa N, Ruan Z J. On injectivity and nuclearity for operator spaces. Duke Math J, 2001, 110(3): 489–521
Kaur M, Ruan Z J. Local properties of ternary ring of operators and their linking C*-algebras. J Funct Analy, 2002, 195: 262–305
Ruan Z J. Type decomposition and Rectangular AFD property for W*-TRO’s. Canad J Math, 2004, 56(4): 843–870
Hamana M. Triple envelopes and Sillov boundaries of operator spaces. Math J Toyama Univer, 1999, 22: 299–319
Kadison R V, Ringrose J. Fundamentals of the Theory of Operator Algebras, I, II. Graduate Studies in Mathematics, Vol. 15, 16. Providence: Amer Math Soc, 1997
Davidson K. C*-algebras by examples. Fields Institute Monograph Series Vol. 6. Providence: Amer Math Soc, 1996
Kye S H, Ruan Z J. On the local lifting property of operator spaces. J Funct Anal, 1999, 168: 355–379
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gao, M. Certain free products of operator spaces. SCI CHINA SER A 49, 800–819 (2006). https://doi.org/10.1007/s11425-006-0800-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11425-006-0800-7