Abstract
In this note, we show that a von Neumann algebra M is injective if and only if the weak* similarity degree d*(M) ≤ 2.
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Brown, N. P., Ozawa, N.: C*-Algebras and Finite Dimensional Approximations, American Mathematical Society, Rhode Island, 2008
Christensen, E.: On non self-adjoint representations of operator algebras. Amer. J. Math., 103, 817–834 (1981)
Connes, A.: Classification of injective factors, Cases II1, II∞, IIIλ, λ ≠ 1. Ann. of Math., 104, 73–115 (1976)
Effros, E. G., Ruan, Z. J.: Operator Spaces, Oxford University Press, New York, 2000
Effros, E. G., Ruan, Z. J.: Operator space tensor products and Hopf convolution algebras. J. Operator Theory, 50, 131–156 (2003)
Elliott, G.: On approximately finite-dimensional von Neumann algebras. Math. Scand., 39, 91–101 (1976)
Elliott, G.: On approximately finite-dimensional von Neumann algebras, II. Canadian. Bull., 21, 415–418 (1978)
Haagerup, U.: Solution of the similarity problem for cyclic representations of C*-algebras. Ann. of Math., 118(2), 215–240 (1983)
Kadison, R.: On the orthogonalization of operator representations. Amer. J. Math., 77, 600–620 (1955)
Le Merdy, C.: The weak* similarity property on dual operator algebras. Integr. Equ. Oper. Theory, 37, 72–94 (2000)
Paulsen, V.: Completely bounded homomorphisms of operator algerbas. Proc. Amer. Math. Soc., 92, 225–228 (1984)
Pisier, G.: The similarity degree of an operator algebra. St. Petersburg Math. J., 10, 103–146 (1999)
Pisier, G.: Similarity Problems and Completely Bounded Maps, Second, expanded edition, Springer-Verlarg, Berlin, 2001
Pisier, G.: Introduction to Operator Space Theory, Cambridge University Press, Cambridge, 2003
Pisier, G.: A similarity degree charaterization of nuclear C*-algebras. Rubl. Res. Inst. Math. Sci., 42, 691–704 (2006)
Ruan, Z. J.: Subspaces of C*-algebras. J. Funct. Anal., 76, 217–230 (1988)
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Supported by the Zhejiang Qianjiang Talent Program in 2008, the Program for New Century Excellent Talents in University of Ministry of Education of China in 2010, the National Natural Science Foundation of China (Grant No. 11271321) and the Fundamental Research Funds of Zhejiang University
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Dong, Z., Zhao, Y.F. A weak* similarity degree characterization for injective von Neumann algebras. Acta. Math. Sin.-English Ser. 30, 1689–1697 (2014). https://doi.org/10.1007/s10114-014-3143-2
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DOI: https://doi.org/10.1007/s10114-014-3143-2