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References

Published online by Cambridge University Press:  10 February 2020

Gilles Pisier
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Texas A & M University
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Tensor Products of C*-Algebras and Operator Spaces
The Connes–Kirchberg Problem
, pp. 470 - 481
Publisher: Cambridge University Press
Print publication year: 2020

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References

[1] Akemann, C., Anderson, J., and Pedersen, G., Triangle inequalities in operator algebras, Lin. Multi. Alg. 11 (1982), 167178.Google Scholar
[2] Akemann, C. and Ostrand, P., Computing norms in group C-algebras, Amer. J. Math. 98 (1976), 10151047.Google Scholar
[3] Anantharaman-Delaroche, C., Amenability and exactness for dynamical systems and their C-algebras, Trans. Amer. Math. Soc. 354 (2002), 41534178.Google Scholar
[4] Anantharaman, C. and Popa, S., “An introduction to II1-factors”, Cambridge University Press, Cambridge, to appear.Google Scholar
[5] Andersen, T. B., Linear extensions, projections, and split faces, J. Funct. Anal. 17 (1974), 161173.Google Scholar
[6] Anderson, J., Extreme points in sets of positive linear maps in B(H), J. Funct. Anal. 31 (1979), 195217.CrossRefGoogle Scholar
[7] Anderson, G., Guionnet, A., and Zeitouni, O., An introduction to random matrices, Cambridge University Press, Cambridge, 2010.Google Scholar
[8] Ando, H. and Haagerup, U., Ultraproducts of von Neumann algebras, J. Funct. Anal. 266 (2014), 68426913.Google Scholar
[9] Ando, H., Haagerup, U., and Winsløw, C., Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology, J. Reine Angew. Math. 715 (2016), 231250.Google Scholar
[10] Archbold, R. and Batty, C., C-tensor norms and slice maps, J. Lond. Math. Soc. 22 (1980), 127138.Google Scholar
[11] Arveson, W., Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578642.Google Scholar
[12] Arveson, W., Subalgebras of C-algebras, Acta Math. 123 (1969), 141–224. Part II. Acta Math. 128 (1972), 271308.CrossRefGoogle Scholar
[13] Arveson, W., Notes on extensions of C-algebras, Duke Math. J. 44 (1977), 329– 355.Google Scholar
[14] Arzhantseva, G. and Delzant, T., Examples of random groups, unpublished preprint, 2008.Google Scholar
[15] Avitsour, D., Free products of C-algebras, Trans. Amer. Math. Soc. 271 (1982), 423435.Google Scholar
[16] Bannon, J., Marrakchi, A., and Ozawa, N., Full factors and co-amenable inclusions, arXiv:1903.05395, 2019.Google Scholar
[17] Bekka, B., de la Harpe, P., and Valette, A., Kazhdan’s property (T), Cambridge University Press, Cambridge, 2008.Google Scholar
[18] Ben-Aroya, A. and Ta-Shma, A., Quantum expanders and the quantum entropy difference problem, arXiv:quant-ph/0702129, no. 3, 2007.Google Scholar
[19] Ben-Aroya, A., Schwartz, O., and Ta-Shma, A., Quantum expanders: motivation and constructions, Theory Comput. 6 (2010), 4779.CrossRefGoogle Scholar
[20] Berger, C. A., Coburn, L. A., and Lebow, A., Representation and index theory for C-algebras generated by commuting isometries, J. Funct. Anal. 27, no. 1 (1978), 5199.Google Scholar
[21] Bergh, J., On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), 775778.Google Scholar
[22] Bergh, J. and Löfström, J., Interpolation spaces: an introduction, Springer-Verlag, Berlin, 1976.Google Scholar
[23] Blackadar, B., Weak expectations and injectivity in operator algebras, Proc. Amer. Math. Soc. 68 (1978), 4953.Google Scholar
[24] Blackadar, B., Weak expectations and nuclear C-algebras, Indiana Univ. Math. J. 27 (1978), 10211026.Google Scholar
[25] Blackadar, B., Operator algebras: theory of C-algebras and von Neumann algebras, Encyclopaedia of mathematical sciences, 122, Springer-Verlag, Berlin, 2006.Google Scholar
[26] Blecher, D. P. and Labuschagne, L., Outers for noncommutative Hp revisited, Studia Math. 217 (2013), 265287.Google Scholar
[27] Blecher, D. P. and Le Merdy, C., Operator algebras and their modules: an operator space approach, Oxford University Press, Oxford, 2004.Google Scholar
[28] Blecher, D. and Paulsen, V., Explicit constructions of universal operator algebras and applications to polynomial factorization, Proc. Amer. Math. Soc. 112 (1991), 839850.Google Scholar
[29] Blecher, D., Ruan, Z. J., and Sinclair, A., A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188201.Google Scholar
[30] Boca, F., Free products of completely positive maps and spectral sets, J. Funct. Anal. 97 (1991), 251263.Google Scholar
[31] Boca, F., A note on full free product C∗-algebras, lifting and quasidiagonality, operator theory, operator algebras and related topics (Timis¸oara, 1996), 51–63, Theta Found., Bucharest, 1997.Google Scholar
[32] Bordenave, C. and Collins, B., Eigenvalues of random lifts and polynomial of random permutations matrices, Ann. of Math. 190 (2019), 811875.Google Scholar
[33] Bourgain, J., Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (1986), 221226.Google Scholar
[34] Bo˙zejko, M., Some aspects of harmonic analysis on free groups, Colloq. Math. 41 (1979), 265271.Google Scholar
[35] Bo˙zejko, M. and Fendler, G., Herz–Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Unione Mat. Ital. (6) 3-A (1984), 297302.Google Scholar
[36] Brown, L., Ext of certain free product C-algebras, J. Operator Theory 6 (1981), 135141.Google Scholar
[37] Brown, L., Invariant means and finite representation theory of C∗-algebras, Memoirs of the American Mathematical Society, 184, American Mathematical Society, Providence, RI, 2006.Google Scholar
[38] Brown, L. and Dykema, K., Popa algebras in free group factors, J. Reine Angew. Math. 573 (2004), 157180.Google Scholar
[39] Brown, N. P. and Ozawa, N., C-algebras and finite-dimensional approximations, Graduate studies in mathematics, 88, American Mathematical Society, Providence, RI, 2008.Google Scholar
[40] Burgdorf, S., Dykema, K., Klep, I., and Schweighofer, M., Addendum to “Connes’ embedding conjecture and sums of Hermitian squares” [Adv. Math. 217, no. 4 (2008) 1816–1837], Adv. Math. 252 (2014), 805811.Google Scholar
[41] de Cannière, J. and Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), 455500.CrossRefGoogle Scholar
[42] Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., and Valette, A., Groups with the Haagerup property. Gromov’s a-T-menability, Birkhäuser Verlag, Basel, 2001.Google Scholar
[43] Ching, W. M., Free products of von Neumann algebras, Trans. Amer. Math. Soc. 178 (1973), 147163.Google Scholar
[44] Choi, M. D., A Schwarz inequality for positive linear maps on C-algebras, Illinois J. Math. 18 (1974), 565574.Google Scholar
[45] Choi, M. D. and Effros, E., Nuclear C-algebras and the approximation property, Amer. J. Math. 100 (1978), 6179.Google Scholar
[46] Choi, M. D. and Effros, E., Nuclear C*-algebras and injectivity. The general case, Indiana Univ. Math. J. 26 (1977), 443446.Google Scholar
[47] Choi, M. D. and Effros, E., Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156209.Google Scholar
[48] Choi, M. D. and Effros, E., Separable nuclear C-algebras and injectivity, Duke Math. J. 43 (1976), 309322.Google Scholar
[49] Choi, M. D. and Effros, E., The completely positive lifting problem for C-algebras, Ann. Math. 104 (1976), 585609.Google Scholar
[50] Christensen, E., Effros, E., and Sinclair, A., Completely bounded multilinear maps and C-algebraic cohomology, Invent. Math. 90 (1987), 279296.Google Scholar
[51] Christensen, E. and Sinclair, A., On von Neumann algebras which are complemented subspaces of B(H), J. Funct. Anal. 122 (1994), 91102.Google Scholar
[52] Christensen, E. and Sinclair, A., Module mappings into von Neumann algebras and injectivity, Proc. Lond. Math. Soc. 71 (1995), 618640.Google Scholar
[53] Christensen, E. and Wang, L., Von Neumann algebras as complemented subspaces of B(H). Internat. J. Math. 25 (2014), 1450107, 9 pp.Google Scholar
[54] McClanahan, K., C-algebras generated by elements of a unitary matrix, J. Funct. Anal. 107 (1992), 439457.Google Scholar
[55] Cohn, P. M., Basic algebra, Springer, London, 2003.Google Scholar
[56] Collins, B. and Male, C., The strong asymptotic freeness of Haar and deterministic matrices, Ann. Sci. Éc. Norm. Supér. 47 (2014), 147163.Google Scholar
[57] Collins, B. and Dykema, K., A linearization of Connes’ embedding problem, New York J. Math. 14 (2008), 617641.Google Scholar
[58] Comfort, W.W., Negrepontis, S., The theory of ultrafilters, Springer, New York, 1974.Google Scholar
[59] Connes, A., Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann, Ann. Inst. Fourier (Grenoble) 24 (1974), 121155.Google Scholar
[60] Connes, A., A factor not anti-isomorphic to itself, Bull. Lond. Math. Soc. 7 (1975), 171174.Google Scholar
[61] Connes, A., Classification of injective factors. Cases II1, II, IIIλ, λ = 1, Ann. Math. (2) 104 (1976), 73115.Google Scholar
[62] Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups, Third edition. Springer-Verlag, New York, 1999.Google Scholar
[63] Courtney, K. and Sherman, D., “The universal C-algebra of a contraction”, arXiv:1811.04043, 2018, to appear.Google Scholar
[64] Cowling, M. and Haagerup, U., Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507549.Google Scholar
[65] Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.Google Scholar
[66] Cwikel, M. and Janson, S., Interpolation of analytic families of operators, Studia Math. 79 (1984), 6171.Google Scholar
[67] Davidson, K.. C-algebras by example, Fields Institute publication. Toronto, American Mathematical Society, Providence, RI, 1996.Google Scholar
[68] Davidson, K. and Kakariadis, E., A proof of Boca’s theorem, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), 869876.Google Scholar
[69] Davie, A. M., Matrix norms related to Grothendieck’s inequality, Banach spaces (Columbia, MO, 1984), Lecture Notes in Mathematics, 1166, Springer, Berlin, 1985.Google Scholar
[70] Devinatz, A., The factorization of operator valued analytic functions, Ann. Math. 73 (1961), 458495.Google Scholar
[71] Diestel, J., Fourie, J. H., and Swart, J., The metric theory of tensor products. Grothendieck’s résumé revisited, American Mathematical Society, Providence, RI, 2008.Google Scholar
[72] Dixmier, J., Les Algèbres d’Opérateurs dans l’Espace Hilbertien (Algèbres de von Neumann), Gauthier-Villars, Paris 1969. (In translation: von Neumann algebras, North-Holland, Amsterdam–New York 1981.)Google Scholar
[73] Douglas, R. and Howe, R., On the C-algebra of Toeplitz operators on the quarterplane, Trans. Amer. Math. Soc. 158 (1971), 203217.Google Scholar
[74] Douglas, R. and Pearcy, C., Von Neumann algebras with a single generator, Michigan Math. J. 16 (1969), 2126.Google Scholar
[75] Dykema, K. and Juschenko, K., Matrices of unitary moments, Math. Scand. 109 (2011), 225239.Google Scholar
[76] Dykema, K., Paulsen, V. and Prakash, J., Non-closure of the set of quantum correlations via graphs, Comm. Math. Phys. 365 (2019), 11251142.Google Scholar
[77] Effros, E. and Haagerup, U., Lifting problems and local reflexivity for C-algebras, Duke Math. J. 52 (1985), 103128.Google Scholar
[78] Effros, E., Junge, M., and Ruan, Z. J., Integral mappings and the principle of local reflexivity for noncommutative L1-spaces, Ann. Math. 151 (2000), 5992.Google Scholar
[79] Effros, E. and Lance, C., Tensor products of operator algebras, Adv. Math. 25 (1977), 134.CrossRefGoogle Scholar
[80] Effros, E. and Ruan, Z. J., Operator Spaces, Oxford University Press, Oxford, 2000.Google Scholar
[81] Elek, G. and Szabó, E., Hyperlinearity, essentially free actions and L2-invariants. The sofic property, Math. Ann. 332 (2005), 421441.Google Scholar
[82] Elliott, G., On approximately finite-dimensional von Neumann algebras, Math. Scand. 39 (1976), 91101.Google Scholar
[83] Elliott, G., On approximately finite-dimensional von Neumann algebras. II, Canad. Math. Bull. 21 (1978), 415418.Google Scholar
[84] Elliott, G. and Woods, E., The equivalence of various definitions for a properly infinite von Neumann algebra to be approximately finite dimensional, Proc. Amer. Math. Soc. 60 (1976), 175178.Google Scholar
[85] Eymard, P., L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181236.Google Scholar
[86] Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras I: stability, Bull. Lond. Math. Soc. 45 (2013), 825838.Google Scholar
[87] Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras III: elementary equivalence and II1 factors, Bull. Lond. Math. Soc. 46 (2014), 609628.Google Scholar
[88] Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras II: model theory, Israel J. Math. 201 (2014), 477505.Google Scholar
[89] Farah, I. and Shelah, S., A dichotomy for the number of ultrapowers, J. Math. Log. 10 (2010), 4581.Google Scholar
[90] Farenick, D., Kavruk, A., and Paulsen, V., C-algebras with the weak expectation property and a multivariable analogue of Ando’s theorem on the numerical radius, J. Operator Theory 70 (2013), 573590.Google Scholar
[91] Farenick, D., Kavruk, A., Paulsen, V., and Todorov, I., Characterisations of the weak expectation property, New York J. Math. 24A (2018), 107135.Google Scholar
[92] Farenick, D. and Paulsen, V., Operator system quotients of matrix algebras and their tensor products, Math. Scand. 111 (2012), 210243.Google Scholar
[93] Friedman, J., A proof of Alon’s second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc. 195, 910 (2008).Google Scholar
[94] Friedman, J., Joux, A., Roichman, Y., Stern, J., and Tillich, J.-P., The action of a few permutations on r-tuples is quickly transitive, Random Struct. Algo. 12 (1998), 335350.Google Scholar
[95] Fritz, T., Tsirelson’s problem and Kirchberg’s conjecture, Rev. Math. Phys. 24 (2012), 1250012, 67 pp.Google Scholar
[96] Gromov, M., Random walk in random groups, Geom. Funct. Anal. 13 (2003), 73146.Google Scholar
[97] Gross, L., A non-commutative extension of the Perron–Frobenius theorem, Bull. Amer. Math. Soc. 77 (1971), 343347.Google Scholar
[98] Grothendieck, A., Résumé de la théorie métrique des produits tensoriels topologiques, Boll. Soc. Mat. São-Paulo 8 (1953), 179. Reprinted in Resenhas 2 (1996), no. 4, 401–480.Google Scholar
[99] Guentner, E., Higson, N., and Weinberger, S., The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243268.Google Scholar
[100] Guichardet, A., Tensor products of C-algebras, Dokl. Akad. Nauk. SSSR 160 (1965), 986989.Google Scholar
[101] Guichardet, A., Tensor products of C∗-algebras (Part I. Finite tensor products. Part II. Infinite tensor products), Lecture Notes Series 12 and 13, Aarhus Universitet, 1969.Google Scholar
[102] Haagerup, U., The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271283.Google Scholar
[103] Haagerup, U., An example of a nonnuclear C-algebra, which has the metric approximation property, Invent. Math. 50 (1978–1979), 279293.Google Scholar
[104] Haagerup, U., Injectivity and decomposition of completely bounded maps, Operator algebras and their connections with topology and ergodic theory, 170– 222, Lecture Notes in Mathematics, 1132, Springer, Berlin, Heidelberg, 1985.Google Scholar
[105] Haagerup, U., A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space, J. Funct. Anal. 62 (1985), 160201.Google Scholar
[106] Haagerup, U., On convex combinations of unitary operators in C*-algebras, Mappings of operator algebras (Philadelphia, PA, 1988), 1–13, Progr. Math., 84, Birkhäuser Boston, Boston, MA, 1990.Google Scholar
[107] Haagerup, U., Self-polar forms, conditional expectations and the weak expectation property for C-algebras, Unpublished manuscript (1993).Google Scholar
[108] Haagerup, U., Group C-algebras without the completely bounded approximation property, J. Lie Theory 26 (2016), 861887.Google Scholar
[109] Haagerup, U., Knudby, S., and de Laat, T., A complete characterization of connected Lie groups with the approximation property, Ann. Sci. Éc. Norm. Supér. 49 (2016), 927946.Google Scholar
[110] Haagerup, U. and Kraus, J., Approximation properties for group C-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), 667699.Google Scholar
[111] Haagerup, U., Junge, M., and Xu, Q., A reduction method for noncommutative Lp-spaces and applications, Trans. Amer. Math. Soc. 362 (2010), 21252165.Google Scholar
[112] Haagerup, U., Kadison, R., and Pedersen, G., Means of unitary operators, revisited, Math. Scand. 100 (2007), 193197.Google Scholar
[113] Haagerup, U. and de Laat, T., Simple Lie groups without the approximation property, Duke Math. J. 162 (2013), 925964.Google Scholar
[114] Haagerup, U. and de Laat, T., Simple Lie groups without the approximation property II, Trans. Amer. Math. Soc. 368 (2016), 37773809.Google Scholar
[115] Haagerup, U. and Musat, M., Factorization and dilation problems for completely positive maps on von Neumann algebras, Comm. Math. Phys. 303 (2011), 555594.CrossRefGoogle Scholar
[116] Haagerup, U. and Musat, M., An asymptotic property of factorizable completely positive maps and the Connes embedding problem, Comm. Math. Phys. 338 (2015), 141176.Google Scholar
[117] Haagerup, U. and Pisier, G., Factorization of analytic functions with values in non-commutative L1-spaces, Canadian J. Math. 41 (1989), 882906.Google Scholar
[118] Haagerup, U. and Pisier, G., Bounded linear operators between C-algebras, Duke Math. J. 71 (1993), 889925.Google Scholar
[119] Haagerup, U. and Thorbjoernsen, S., Random matrices and K-theory for exact C-algebras, Doc. Math. 4 (1999), 341450 (electronic).Google Scholar
[120] Haagerup, U. and Thorbjørnsen, S., A new application of random matrices: Ext(C∗red(F2)) is not a group, Ann. Math. 162 (2005), 711775.Google Scholar
[121] Haagerup, U. and Winsløw, C., The Effros-Maréchal topology in the space of von Neumann algebras, Amer. J. Math. 120 (1998), 567617.Google Scholar
[122] Haagerup, U. and Winsløw, C., The Effros-Maréchal topology in the space of von Neumann algebras. II, J. Funct. Anal. 171 (2000), 401431.Google Scholar
[123] Hanche-Olsen, H. and Störmer, E., Jordan operator algebras, Pitman, Boston, 1984.Google Scholar
[124] Harcharras, A., On some “stability” properties of the full C-algebra associated to the free group F, Proc. Edinburgh Math. Soc. 41 (1998), 93116.Google Scholar
[125] Harmand, P., Werner, D., and Werner, W., M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, 1547 Springer-Verlag, Berlin, 1993.Google Scholar
[126] de la Harpe, P., Topics in geometric group theory, The University of Chicago Press, Second printing, with corrections and updates, Chicago, 2003.Google Scholar
[127] de la Harpe, P. and Valette, A., La propriété (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Astérisque 175 (1989), Soc. Math. France, Paris.Google Scholar
[128] Harris, S., A non-commutative unitary analogue of Kirchberg’s conjecture, Indiana Univ. Math. J. 68 (2019), 503536.Google Scholar
[129] Harris, S. and Paulsen, V., Unitary correlation sets, Integral Equations Operator Theory 89 (2017), 125149.Google Scholar
[130] Harrow, A., Quantum expanders from any classical Cayley graph expander, Quantum Inf. Comput. 8 (2008), 715721.Google Scholar
[131] Harrow, A. and Hastings, M., Classical and quantum tensor product expanders, Quantum Inf. Comput. 9 (2009), 336360.Google Scholar
[132] Hastings, M., Random unitaries give quantum expanders, Phys. Rev. A (3) 76 no. 3 (2007), 032315, 11 pp.Google Scholar
[133] Helson, H., Lectures on invariant subspaces, Academic Press, New York, 1964.Google Scholar
[134] Hiai, F. and Nakamura, Y., Distance between unitary orbits in von Neumann algebras, Pacific J. Math. 138 (1989), 259294.Google Scholar
[135] Itoh, S., Conditional expectations in C-crossed products, Trans. Amer. Math. Soc. 267 (1981), 661667.Google Scholar
[136] Jolissaint, P., A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), 311313.Google Scholar
[137] Junge, M. and Le Merdy, C., Factorization through matrix spaces for finite rank operators between C-algebras, Duke Math. J. 100, (1999), 299319.Google Scholar
[138] Junge, M., Navascues, M., Palazuelos, C., Peréz-García, D., Scholz, V.B., and Werner, R.F., Connes’ embedding problem and Tsirelson’s problem, J. Math. Phys. 52 (2011), 012102, 12 pp.Google Scholar
[139] Junge, M., Palazuelos, C., Perez-García, D., Villanueva, I., and Wolf, M. M., Operator Space theory: a natural framework for Bell inequalities, Phys. Rev. Lett. 104, 170405 (2010).Google Scholar
[140] Junge, M. and Palazuelos, C., Large violation of Bell inequalities with low entanglement, Comm. Math. Phys. 306 (2011), 695746.Google Scholar
[141] Junge, M. and Pisier, G., Bilinear forms on exact operator spaces and B(H) ⊗ B(H), Geom. Funct. Anal. 5 (1995), 329363.Google Scholar
[142] Kadison, R., Isometries of operator algebras, Ann. Math. 54 (1951), 325338.Google Scholar
[143] Kadison, R., A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. Math. 56 (1952), 494503.Google Scholar
[144] Kadison, R. and Pedersen, G., Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249266.Google Scholar
[145] Kadison, R. and Ringrose, J., Fundamentals of the theory of operator algebras, Vol. I, Birkhäuser Boston, Inc., Boston, MA, 1983.Google Scholar
[146] Kadison, R. and Ringrose, J., Fundamentals of the theory of operator algebras, Vol. II, Birkhäuser Boston, Inc., Boston, MA, 1992.Google Scholar
[147] Kadison, R. and Ringrose, J., Fundamentals of the theory of operator algebras, Vol. IV, Birkhäuser Boston, Inc., Boston, MA, 1992.Google Scholar
[148] Kassabov, M., Symmetric groups and expanders, Inv. Math. 170 (2007), 327354.Google Scholar
[149] Kavruk, A., Tensor products of operator systems and applications. Thesis (Ph.D.), University of Houston, 2011.Google Scholar
[150] Kavruk, A., Paulsen, V., Todorov, I., and Tomforde, M., Tensor products of operator systems, J. Funct. Anal. 261 (2011), 267299.Google Scholar
[151] Kavruk, A., Paulsen, V., Todorov, I., and Tomforde, M., Quotients, exactness, and nuclearity in the operator system category, Adv. Math. 235 (2013), 321360.Google Scholar
[152] Kavruk, A., The weak expectation property and Riesz interpolation, arXiv:1201.5414, 2012.Google Scholar
[153] Kavruk, A., Nuclearity related properties in operator systems, J. Operator Theory 71 (2014), 95156.Google Scholar
[154] Kirchberg, E., C-nuclearity implies CPAP, Math. Nachr. 76 (1977), 203212.Google Scholar
[155] Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group C-algebras, Invent. Math. 112 (1993), 449489.Google Scholar
[156] Kirchberg, E., Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math. 452 (1994), 3977.Google Scholar
[157] Kirchberg, E., Discrete groups with Kazhdan’s property T and factorization property are residually finite, Math. Ann. 299 (1994), 551563.Google Scholar
[158] Kirchberg, E., Exact C∗-algebras, tensor products, and the classification of purely infinite algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 943–954, Birkhäuser, Basel, 1995.Google Scholar
[159] Kirchberg, E., On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 3563.Google Scholar
[160] Kirchberg, E., On restricted perturbations in inverse images and a description of normalizer algebras in C-algebras, J. Funct. Anal. 129 (1995), 134.Google Scholar
[161]E. Kirchberg, Personal communication.Google Scholar
[162] Kirchberg, E. and Phillips, N. C., Embedding of exact C-algebras in the Cuntz algebra O2, J. Reine Angew. Math. 525 (2000), 1753.Google Scholar
[163] Klep, I. and Schweighofer, M., Connes’ embedding conjecture and sums of Hermitian squares, Adv. Math. 217 (2008), 18161837.Google Scholar
[164] Lafforgue, V. and De la Salle, M., Noncommutative Lp-spaces without the completely bounded approximation property, Duke Math. J. 160 (2011), 71116.Google Scholar
[165] Lance, C., On nuclear C-algebras, J. Funct. Anal. 12 (1973), 157176.Google Scholar
[166] Lehner, F., A characterization of the Leinert property, Proc. Amer. Math. Soc. 125 (1997), 34233431.Google Scholar
[167] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, vol. I, Sequence spaces, Springer Verlag, Berlin 1976.Google Scholar
[168] Lehner, F., Computing norms of free operators with matrix coefficients, Amer. J. Math. 121 (1999), 453486.Google Scholar
[169] Lieb, E., Convex trace functions and the Wigner–Yanase–Dyson conjecture, Adv. Math. 11 (1973), 267288.Google Scholar
[170] Lindenstrauss, J. and Rosenthal, H. P., The Lp spaces, Israel J. Math. 7 (1969), 325349.Google Scholar
[171] Loring, T., Lifting solutions to perturbing problems in C∗-algebras, Fields Institute Monographs, American Mathematical Society, Providence, RI, 1997.Google Scholar
[172] Lubotzky, A., Discrete groups, expanding graphs and invariant measures, Progress in Math, 125. Birkhäuser, 1994.Google Scholar
[173] Lubotzky, A., What is Property (τ)? Notices Amer. Math. Soc. 52 (2005), 626627.Google Scholar
[174] Lubotzky, A., Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. 49 (2012), 113162.Google Scholar
[175] Lubotzky, A., Phillips, R., and Sarnak, P., Hecke operators and distributing points on S2, I, Comm. Pure and Applied Math. 39 (1986), 149186.Google Scholar
[176] Mc Duff, D., Uncountably many II1 factors, Ann. Math. 90 (1969), 372377.Google Scholar
[177] Malcev, A. I., On isomorphic matrix representations of infinite groups of matrices (Russian), Mat. Sb. 8 (1940), 405–422 & Amer. Math. Soc. Transl. (2) 45 (1965), 1–18.Google Scholar
[178] Monod, N., Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. USA 110 (2013), 45244527.Google Scholar
[179] Nica, A., Asymptotically free families of random unitaries in symmetric groups, Pacific J. Math. 157 (1993), 295310.Google Scholar
[180] Oikhberg, T. and Ricard, É., Operator spaces with few completely bounded maps, Math. Ann. 328 (2004), 229259.Google Scholar
[181] Oikhberg, T. and Rosenthal, H. P., Extension properties for the space of compact operators, J. Funct. Anal. 179 (2001), 251308.Google Scholar
[182] Osajda, D., Small cancellation labellings of some infinite graphs and applications, arXiv:1406.5015, 2014.Google Scholar
[183] Osajda, D., Residually finite non-exact groups, Geom. Funct. Anal. 28 (2018), 509517.Google Scholar
[184] Ozawa, N., On the set of finite-dimensional subspaces of preduals of von Neumann algebras, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 309312.Google Scholar
[185] A non-extendable bounded linear map between C∗-algebras, Proc. Edinb. Math. Soc. (2) 44 (2001), 241248.Google Scholar
[186] Ozawa, N., On the lifting property for universal C-algebras of operator spaces, J. Operator Theory 46 no. 3, suppl. (2001), 579591.Google Scholar
[187] Ozawa, N., Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 691695.Google Scholar
[188] Ozawa, N., An application of expanders to B( 2) ⊗ (2003), 499–510.Google Scholar
[189] Ozawa, N., About the QWEP conjecture, Internat. J. Math. 15 (2004), 501530.Google Scholar
[190] Ozawa, N., Examples of groups which are not weakly amenable, Kyoto J. Math. 52 (2012), 333344.Google Scholar
[191] Ozawa, N., About the Connes embedding conjecture: algebraic approaches, Jpn. J. Math. 8 (2013), 147183.Google Scholar
[192] Ozawa, N., Tsirelson’s problem and asymptotically commuting unitary matrices, J. Math. Phys. 54 (2013), 032202, 8 pp.Google Scholar
[193] Ozawa, N. and Pisier, G., A continuum of C∗-norms on B(H)⊗B(H) and related tensor products, Glasgow Math. J. 58 (2016), 433443.Google Scholar
[194] Paterson, A., Amenability, American Mathematical Society, Mathematical Surveys and Monographs, 29, 1988.Google Scholar
[195] Paulsen, V., Completely bounded maps and dilations, Pitman Research Notes 146. Pitman Longman (Wiley) 1986.Google Scholar
[196] Paulsen, V., Completely bounded maps and operator algebras, Cambridge University Press, Cambridge, 2002.Google Scholar
[197] Paulsen, V. and Suen, C.-Y., Commutant representations of completely bounded maps, J. Operator Theory 13 (1985), 87101.Google Scholar
[198] Pestov, V., Operator spaces and residually finite-dimensional C-algebras, J. Funct. Anal. 123 (1994), 308317.Google Scholar
[199] Pier, J. P., Amenable locally compact groups, Wiley Interscience, New York, 1984. B( 2), J. Funct. Anal. 198Google Scholar
[200] Pisier, G., Factorization of linear operators and the geometry of Banach spaces, CBMS (Regional Conferences of the A.M.S.) no. 60 (1986), Reprinted with corrections 1987.Google Scholar
[201] Pisier, G., Remarks on complemented subspaces of von Neumann algebras, Proc. Royal Soc. Edinburgh 121 A (1992), 14.Google Scholar
[202] Pisier, G., Espace de Hilbert d’opérateurs et interpolation complexe, Comptes Rendus Acad. Sci. Paris, Série I 316 (1993), 4752.Google Scholar
[203] Pisier, G., Projections from a von Neumann algebra onto a subalgebra, Bull. Soc. Math. France 123 (1995), 139153.Google Scholar
[204] Pisier, G., A simple proof of a theorem of Kirchberg and related results on C-norms, J. Operator Theory 35 (1996), 317335.Google Scholar
[205] Pisier, G., The operator Hilbert space OH, complex interpolation and tensor norms, Memoirs Amer. Math. Soc. 122 no. 585 (1996), 1103.Google Scholar
[206] Pisier, G.. Quadratic forms in unitary operators, Linear Algebra Appl. 267 (1997), 125137.Google Scholar
[207] Pisier, G., Similarity problems and completely bounded maps. Second, Expanded Edition, Springer Lecture Notes, 1618 (2001).Google Scholar
[208] Pisier, G., Introduction to operator space theory, Cambridge University Press, Cambridge, 2003.Google Scholar
[209] Pisier, G., Remarks on B(H) ⊗ B(H), Proc. Indian Acad. Sci. (Math. Sci.) 116 (2006), 423428.Google Scholar
[210] Pisier, G., Grothendieck’s theorem, past and present, Bull. Amer. Math. Soc. 49 (2012), 237323.Google Scholar
[211] Pisier, G., Random matrices and subexponential operator spaces, Israel J. Math. 203 (2014), 223273.Google Scholar
[212] Pisier, G., Quantum expanders and geometry of operator spaces, J. Europ. Math. Soc. 16 (2014), 11831219.Google Scholar
[213] Pisier, G., On the metric entropy of the Banach–Mazur compactum, Mathematika 61 (2015), 179198.Google Scholar
[214] Pisier, G., Martingales in Banach spaces, Cambridge University Press, Cambridge, 2016.Google Scholar
[215] Pisier, G. and Xu, Q., Non-commutative Lp-spaces, Handbook of the geometry of Banach spaces, vol. II, North-Holland, Amsterdam, 2003.Google Scholar
[216] Popa, S., On the Russo–Dye theorem, Michigan Math. J. 28 (1981), 311315.Google Scholar
[217] Popa, S., A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras, J. Operator Theory 16 (1986), 261272.Google Scholar
[218] Popa, S., Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math. 111 (1993), 375405.Google Scholar
[219] Pusz, W. and Woronowicz, S. L., Form convex functions and the WYDL and other inequalities, Lett. Math. Phys. 2 (1977/78), 505512.Google Scholar
[220] Pusz, W. and Woronowicz, S. L., Functional calculus for sesquilinear forms and the purification map, Rep. Mathematical Phys. 8 (1975), 159170.Google Scholar
[221] Pytlik, T. and Szwarc, R., An analytic family of uniformly bounded representations of free groups, Acta Math. 157 (1986), 287309.Google Scholar
[222] Rădulescu, F., A comparison between the max and min norms on C(Fn) ⊗ C(Fn), J. Operator Theory 51 (2004), 245253.Google Scholar
[223] Rădulescu, F., Combinatorial aspects of Connes’s embedding conjecture and asymptotic distribution of traces of products of unitaries, Operator Theory 20, 197205, Theta Ser. Adv. Math. 6, Theta, Bucharest, 2006.Google Scholar
[224] Rankin, R. A., Modular forms and functions, Cambridge University Press, Cambridge, 1977.Google Scholar
[225] Rieffel, M., Induced representations of C-algebras, Adv. Math. 13 (1974), 176257.Google Scholar
[226] Sakai, S., C-algebras and W-algebras, Springer-Verlag, New York, 1971.Google Scholar
[227] Sarnak, P., What is an expander? Notices Amer. Math. Soc. 51 (2004), 762763.Google Scholar
[228] Selberg, A., On the estimation of Fourier coefficients of modular forms, Proceedings of the Symposium Pure Mathematics, Vol. VIII, American Mathematical Society, Providence, RI, 1965, pp. 115.Google Scholar
[229] Sherman, D., On cardinal invariants and generators for von Neumann algebras, Canad. J. Math. 64 (2012), 455480.Google Scholar
[230] Slofstra, W., The set of quantum correlations is not closed, Forum Math. Pi 7 (2019), e1, 41 pp.Google Scholar
[231] Slofstra, W., A group with at least subexponential hyperlinear profile, arXiv:1806.05267, 2018.Google Scholar
[232] Smith, R. R., Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991), 156175.Google Scholar
[233] Størmer, E., On the Jordan structure of C-algebras, Trans. Amer. Math. Soc. 120 (1965), 438447.Google Scholar
[234] Størmer, E., Multiplicative properties of positive maps, Math. Scand. 100 (2007), 184192.Google Scholar
[235] Størmer, E., Positive linear maps of operator algebras, Springer, Heidelberg, 2013.Google Scholar
[236] Suen, C-Y., Completely bounded maps on C*-algebras, Proc. Amer. Math. Soc. 93 (1985), 8187.Google Scholar
[237] Szwarc, R., An analytic series of irreducible representations of the free group, Ann. Inst. Fourier 38 (1988), 87110.Google Scholar
[238] Takesaki, M., A note on the cross-norm of the direct product of C-algebras, Kodai Math. Sem. Rep. 10 (1958), 137140.Google Scholar
[239] Takesaki, M., Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249310.Google Scholar
[240] Takesaki, M., Theory of Operator algebras, vol. I, Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
[241] Takesaki, M., Theory of Operator algebras, vol. II, Springer-Verlag, Berlin, Heidelberg, New York, 2003.Google Scholar
[242] Takesaki, M., Theory of Operator algebras, vol. III, Springer-Verlag, Berlin, Heidelberg, New York, 2003.Google Scholar
[243] Tao, T., Expansion in finite simple groups of Lie type, American Mathematical Society, Providence, RI, 2015.Google Scholar
[244] Thom, A., Examples of hyperlinear groups without factorization property, Groups Geom. Dyn. 4 (2010), 195208.Google Scholar
[245] Tomiyama, J., Tensor products and projections of norm one in von Neumann algebras, Lecture Notes, University of Copenhagen, 1970.Google Scholar
[246] Tomiyama, J., Tensor products and approximation problems of C-algebras, Publ. Res. Inst. Math. Sci. 11 (1975/76), 163183.Google Scholar
[247] Tomiyama, J., On the product projection of norm one in the direct product of operator algebras. Tôhoku Math. J. 11, no. (1959) 305313.Google Scholar
[248] Tomiyama, J., On the projection of norm one in W-algebras. III, Tôhoku Math. J. (2) 11 (1959) 125129.Google Scholar
[249] Tonge, A., The complex Grothendieck inequality for 2 × 2 matrices, Bull. Soc. Math. Grèce (N.S.) 27 (1986), 133136.Google Scholar
[250] Trott, S., A pair of generators for the unimodular group, Canad. Math. Bull. 5 (1962), 245252.Google Scholar
[251] Tsirelson, B.S., Quantum generalizations of Bell’s inequality, Lett. Math. Phys. 4 (1980), 93100.Google Scholar
[252] Valette, A., Minimal projections, integrable representations and property (T), Arch. Math. (Basel) 43 (1984), 397406.Google Scholar
[253] Voiculescu, D., Property T and approximation of operators, Bull. London Math. Soc. 22 (1990), 2530.Google Scholar
[254] Voiculescu, D., Dykema, K., and Nica, A., Free Random Variables, American Mathematical Society, Providence, RI, 1992.Google Scholar
[255] Wassermann, S., On tensor products of certain group C-algebras, J. Funct. Anal. 23 (1976), 239254.Google Scholar
[256] Wassermann, S., Injective W-algebras, Proc. Cambridge Phil. Soc. 82 (1977), 3947.Google Scholar
[257] Wassermann, S., A pathology in the ideal space of L(H) ⊗ L(H), Indiana Univ. Math. J. 27 (1978), 10111020.Google Scholar
[258] Wassermann, S., Exact C-algebras and related topics, Lecture Notes Series, 19. Seoul National University, Seoul, 1994.Google Scholar
[259] Wassermann, S., C-algebras associated with groups with Kazhdan’s property T, Ann. Math. 134 (1991), 423431.Google Scholar
[260] Werner, D., Some lifting theorems for bounded linear operators, Functional analysis (Essen, 1991), 279291, Lecture Notes in Pure and Applied Mathematics, 150, Dekker, New York, 1994.Google Scholar
[261] Willig, P., On hyperfinite W∗-algebras, Proc. Amer. Math. Soc. 40 (1973), 120122.Google Scholar
[262] Wilson, J. S., On characteristically simple groups, Math. Proc. Cambridge Philos. Soc. 80 (1976), 1935.Google Scholar
[263] Woronowicz, S., Selfpolar forms and their applications to the C-algebra theory, Rep. Mathematical Phys. 6 (1974), 487495.Google Scholar
[264] Zhang, C., Representation and geometry of operator spaces, Ph.D. thesis, University of Houston, 1995.Google Scholar
[265] Zippin, M., The separable extension problem, Israel J. Math. 26 (1977), 372387.Google Scholar
[266] Zippin, M., Extension of bounded linear operators, Handbook of the geometry of Banach spaces, Vol. 2, 1703–1741, North-Holland, Amsterdam, 2003.Google Scholar

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