Abstract
If M and N are type \(\mathrm {II}_1\) von Neumann factors with separable predual and one of them is embeddable in \(R^{\omega },\) then the tensor product \(M\overline{\otimes } N\) has Kadison’s Similarity Property (SP). In particular, \(M\overline{\otimes } L(\mathbf{F}_n)\) have the SP for all M, which, if M is full, represent the first examples of full type \(\mathrm {II}_1\) factors with the SP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arveson, W.B.: Subalgebras of \(C^*\)-algebras. Acta Math. 123, 141–224 (1969)
Brown, N.P.: Connes’ embedding problem and Lance’s WEP. Int. Math. Res. Not. 10, 501–510 (2004)
Brown, N.P., Ozawa, N.: \(C^*\)-Algebras and Finite-dimensional Approximations, Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence, RI (2008)
Choi, M.-D., Effros, E.G.: The completely positive lifting problem for \(C^*\)-algebras. Ann. Math. 104, 585–609 (1976)
Christensen, E.: Extensions of derivations. J. Funct. Anal. 27, 234–247 (1978)
Christensen, E.: Extensions of derivations II. Math. Scand. 50, 111–122 (1982)
Christensen, E.: Similarities of \(\rm {II}_1\) factors with property \(\Gamma \). J. Oper. Theory 15, 281–288 (1986)
Christensen, E.: Finite von Neumann algebra factors with property \(\Gamma \). J. Funct. Anal. 156, 366–380 (2001)
Haagerup, U.: Solution of the similarity problem for cyclic representations of \(C^*\)-algebras. Ann. Math. 118, 215–240 (1983)
Hadwin, D., Li, W.: The similarity degree of some \(C^*\)-algebras. Bull. Aust. Math. Soc. 89, 60–69 (2014)
Johanesova, M., Winter, W.: The similarity degree for \(Z\)-stable \(C^*\)-algebras. Bull. Lond. Math. Soc. 44, 1215–1220 (2012)
Kadison, R.V.: On the orthogonalization of operator representations. Am. J. Math. 77, 600–620 (1955)
Kadison, R.V.: Derivations of operator algebras. Ann. Math. 83, 280–293 (1966)
Kirchberg, E.: On nonsemisplit extensions, tensor products and exactness of group \(C^*\)-algebras. Invent. Math. 112, 449–489 (1993)
Kirchberg, E.: Commutants of unitaries in UHF algebras and functorial properties of exactness. J. Reine Angew. Math. 452, 39–77 (1994)
Kirchberg, E.: The derivation problem and the similarity problem are equivalent. J. Oper. Theory 36, 59–62 (1996)
Lance, E.C.: On nuclear \(C^*\)-algebras. J. Funct. Anal. 12, 157–176 (1973)
Li, W.: The similarity degree of approximately divisible \(C^*\)-algebras. Oper. Matrices 7, 425–430 (2013)
Ozawa, N.: About the QWEP conjecture. Int. J. Math. 15, 501–530 (2004)
Paulsen, V.I.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003)
Pisier, G.: Similarity problems and completely bounded maps. Springer Lecture Notes 1618 (1995)
Pisier, G.: The similarity degree of an operator algebra. St. Petersb. Math. J. 10, 103–146 (1999)
Pisier, G.: Remarks on the similarity degree of an operator algebra. Int. J. Math. 12, 403–414 (2001)
Pisier, G.: Similarity problems and length. Taiwan. J. Math. 5, 1–17 (2001)
Pisier, G.: A similarity degree characterization of nuclear \(C^*\)-algebras. Publ. Res. Inst. Math. Sci. (Kyoto) 42, 691–704 (2006)
Pop, F.: The similarity problem for tensor products of certain \(C^*\)-algebras. Bull. Aust. Math. Soc. 70, 385–389 (2004)
Pop, F.: Remarks on the completely bounded approximation property for \(C^*\)-algebras. Houst. J. Math. 43, 937–945 (2017)
Qian, W., Shen, J.: Similarity degree of a class of \(C^*\)-algebras. Int. Equ. Oper. Theory 84, 121–149 (2016)
Sakai, S.: Derivations of \(W^*\)-algebras. Ann. Math. 83, 273–279 (1966)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pop, F. Similarities of Tensor Products of Type \(\mathrm {II}_1\) Factors. Integr. Equ. Oper. Theory 89, 455–463 (2017). https://doi.org/10.1007/s00020-017-2406-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-017-2406-6