SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

Gilles PISIER

doi:10.1007/s10473-021-0616-0

Acta mathematica scientia, Series B >

2021 , Vol. 41 >Issue 6: 2055 - 2085

DOI: https://doi.org/10.1007/s10473-021-0616-0

Articles

SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

Gilles PISIER

Expand

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

Received date: 2021-04-06

Revised date: 2021-09-24

Online published: 2021-12-27

Fold

Abstract

We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of $M$ $$Id_M=vu:M{\buildrel u\over\longrightarrow} B(H) {\buildrel v\over\longrightarrow} M$$ with $u$ normal, unital, positive and $v$ completely contractive. As a corollary, if $M$ has a separable predual, $M$ is isomorphic (as a Banach space) to $B(\ell_2)$. For instance this applies (rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since $B(H)$ fails the approximation property (due to Szankowski) there are $M$'s (namely $B(H)^{**}$ and certain finite examples defined using ultraproducts) that are not seemingly injective. Moreover, for $M$ to be seemingly injective it suffices to have the above factorization of $Id_M$ through $B(H)$ with $u,v$ positive (and $u$ still normal).

Key words： von Neumann algebra; injectivity; positive approximation property

Cite this article

Gilles PISIER . SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS[J]. Acta mathematica scientia, Series B, 2021 , 41(6) : 2055 -2085 . DOI: 10.1007/s10473-021-0616-0

References

[1] Anantharaman C, Popa S. An Introduction to II₁-Factors. Cambridge Univ Press, to appear
[2] Andersen T B. Linear extensions, projections, and split faces. J Funct Anal, 1974, 17:161-173
[3] Ando T. Closed range theorems for convex sets and linear liftings. Pacific J Math, 1973, 44:393-410
[4] Ando T. A theorem on nonempty intersection of convex sets and its application. J Approximation Theory, 1975, 13:158-166
[5] Arveson W. Notes on extensions of C^*-algebras. Duke Math J, 1977, 44:329-355
[6] Bekka B, de la Harpe P, Valette A. Kazhdan's Property (T). Cambridge:Cambridge University Press, 2008
[7] Blecher D P, Le Merdy C. Operator Algebras and Their Modules:An Operator Space Approach. Oxford:Oxford Univ Press, 2004
[8] Blecher D, Paulsen V. Tensor products of operator spaces. J Funct Anal 1991, 99:262-292
[9] Blower G. The Banach space B(l²) is primary. Bull London Math Soc, 1990, 22:176-182
[10] Brown N P, Ozawa N. C^*-algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics, 88. Providence, RI:American Mathematical Society, 2008
[11] de Cannière J, Haagerup U. Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer J Math, 1985, 107:455-500
[12] Choi M D. A Schwarz inequality for positive linear maps on C^*-algebras. Illinois J Math, 1974, 18:565-574
[13] Choi M D, Effros E. The completely positive lifting problem for C^*-algebras. Ann of Math, 1976, 104:585-609
[14] Choi M D, Effros E. Separable nuclear C^*-algebras and injectivity. Duke Math J, 1976, 43:309-322
[15] Choi M D, Effros E. Lifting problems and the cohomology of C^*-algebras. Canadian J Math, 1977, 29:1092-1111
[16] Choi M D, Effros E. Nuclear C^*-algebras and injectivity:The general case. Indiana Univ Math J, 1977, 26:443-446
[17] Choi M D. Effros E. Injectivity and operator spaces. J Funct Anal, 1977, 24:156-209
[18] Christensen E, Sinclair A. Completely bounded isomorphisms of injective von Neumann algebras. Proc Edinburgh Math Soc, 1989, 32(2):317-327
[19] Connes A. Classification of injective factors. Cases II₁, II_∞, III_λ, λ≠1. Ann Math, 1976, 104(2):73-115
[20] Connes A, Jones V. Property T for von Neumann algebras. Bull London Math Soc, 1985, 17:57-62
[21] Davidson K. C^*-algebras by Example. Fields Institute Monogrphs. Providence RI:Amer Math Soc, 1996
[22] Dixmier J. Les Algèbres d'Opérateurs dans l'Espace Hilbertien (Algèbres de von Neumann). Paris:Gauthier-Villars, 1969(in translation:von Neumann Algebras, North-Holland, Amsterdam-New York 1981)
[23] Effros E, Lance C. Tensor products of operator algebras. Adv Math, 1977, 25:1-34
[24] Effros E, Ruan Z J. Operator Spaces. Oxford:Oxford Univ Press, 2000
[25] Effros E, Størmer E. Positive projections and Jordan structure in operator algebras. Math Scand, 1979, 45:127-138
[26] Friedman Y, Russo B. Solution of the contractive projection problem. J Funct Anal, 1985, 60:56-79
[27] Haagerup U. The standard form of von Neumann algebras. Math Scand, 1975, 37:271-283
[28] Haagerup U. An example of a nonnuclear C^*-algebra, which has the metric approximation property. Invent Math, 1978/79, 50:279-293
[29] Haagerup U. Injectivity and decomposition of completely bounded maps//Lecture Notes in Math 1132. Springer, 1985:170-222
[30] Haagerup U. Group C^*-algebras without the completely bounded approximation property. J Lie Theory, 2016, 26:861-887
[31] Haagerup U, Pisier G. Bounded linear operators between C^*-algebras. Duke Math J, 1993, 71:889-925
[32] Harmand P, Werner D, Werner W. M-ideals in Banach spaces and Banach algebras. Lecture Notes in Mathematics 1547. Berlin:Springer-Verlag, 1993
[33] Harris L A. Bounded symmetric homogeneous domains in infinite dimensional spaces//Lecture Notes in Mathematics 364. Berlin:Springer-Verlag, 1974:13-40
[34] Harris L A. A generalization of C^*-algebras. Proc London Math Soc, 1981, 42:331-361
[35] Ji Z, Natarajan A, Vidick T, Wright J, Yuen H. MIP ^*=RE. arxiv, January, 2020
[36] Jolissaint P, Valette A. Normes de Sobolev et convoluteurs bornés sur L₂(G). Ann Inst Fourier (Grenoble), 1991, 41:797-822
[37] Junge M, Le Merdy C. Factorization through matrix spaces for finite rank operators between C^*-algebras. Duke Math J, 1999, 100:299-319
[38] Kavruk A. Nuclearity related properties in operator systems. J Operator Theory, 2014, 71:95-156
[39] Kirchberg E. On nonsemisplit extensions, tensor products and exactness of group C^*-algebras. Invent Math, 1993, 112:449-489
[40] Kirchberg E. Commutants of unitaries in UHF algebras and functorial properties of exactness. J Reine Angew Math 1994, 452:39-77
[41] Oikhberg T. Subspaces of maximal operator spaces. Integral Equations Operator Theory, 2004, 48:81-102
[42] Oikhberg T, Rosenthal H P. Extension properties for the space of compact operators. J Funct Anal, 2001, 179:251-308
[43] Ozawa N. Local Theory and Local Reflexivity for Operator Spaces[D]. Texas A&M University, 2001
[44] Ozawa N. On the lifting property for universal C^*-algebras of operator spaces. J Operator Theory, 2001, 46(3):579-591
[45] Ozawa N. About the QWEP conjecture. Internat J Math, 2004, 15:501-530
[46] Paulsen V. The maximal operator space of a normed space. Proc Edinburgh Math Soc, 1996, 39:309-323
[47] Paulsen V. Completely Bounded Maps and Operator Algebras. Cambridge:Cambridge Univ Press, 2002
[48] Paulsen V, Todorov I, Tomforde M. Operator system structures on ordered spaces. Proc Lond Math Soc, 2011, 102:25-49
[49] Pisier G. Projections from a von Neumann algebra onto a subalgebra. Bull Soc Math France, 1995, 123:139-153
[50] Pisier G. The operator Hilbert space OH, complex interpolation and tensor norms. Memoirs Amer Math Soc, 1996, 122(585):1-103
[51] Pisier G. Introduction to Operator Space Theory. Cambridge:Cambridge University Press, 2003
[52] Pisier G. Tensor Products of C^*-algebras and Operator Spaces, The Connes-Kirchberg Problem. Cambridge University Press, 2020 Available at:https://www.math.tamu.edu/pisier/TPCOS.pdf
[53] Robertson A G, Smith R. Liftings and extensions of maps on C^*-algebras. J Operator Theory, 1989, 21:117-131
[54] Robertson A G, Wassermann S. Completely bounded isomorphisms of injective operator systems. Bull London Math Soc, 1989, 21:285-290
[55] Sakai S. C^*-algebras and W ^*-algebras. New York, Heidelberg:Springer-Verlag, 1971
[56] Szankowski A. B(H) does not have the approximation property. Acta Math, 1981, 147:89-108
[57] Takesaki M. Theory of Operator Algebras. vol I. Berlin, Heidelberg, New York:Springer-Verlag, 1979
[58] Takesaki M. Theory of Operator Algebras. vol III. Berlin, Heidelberg, New York:Springer-Verlag, 2003
[59] Tomiyama J. On the projection of norm one in W ^*-algebras. Proc Japan Acad, 1957, 33:608-612
[60] Vesterstrøm J. Positive linear extension operators for spaces of affine functions. Israel J Math, 1973, 16:203-211
[61] Wassermann S. On tensor products of certain group C^*-algebras. J Funct Anal, 1976, 23:239-254
[62] Wassermann S. Injective W ^*-algebras. Proc Cambridge Phil Soc, 1977, 82:39-47

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References