Unperforated Pairs of Operator Spaces and Hyperrigidity of Operator Systems

Clouâtre, Raphaël

Canadian journal of mathematics, Tome 70 (2018) no. 6, pp. 1236-1260

Résumé

We study restriction and extension properties for states on ${{\text{C}}^{*}}$ -algebras with an eye towards hyperrigidity of operator systems. We use these ideas to provide supporting evidence for Arveson’s hyperrigidity conjecture. Prompted by various characterizations of hyperrigidity in terms of states, we examine unperforated pairs of self-adjoint subspaces in a ${{\text{C}}^{*}}$ -algebra. The configuration of the subspaces forming an unperforated pair is in some sense compatible with the order structure of the ambient ${{\text{C}}^{*}}$ -algebra. We prove that commuting pairs are unperforated and obtain consequences for hyperrigidity. Finally, by exploiting recent advances in the tensor theory of operator systems, we show how the weak expectation property can serve as a flexible relaxation of the notion of unperforated pairs.

Détail

DOI : 10.4153/CJM-2018-008-1

Mots-clés : 46L07, 46L30, 46L52, operator system, state, peak point, hyperrigidity conjecture

Bibliographie

[1] [1] Altomare, F., Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5 (2010), 92–164. Google Scholar

[2] [2] Archbold, R. J., Extensions of pure states and projections of norm one. II. Bull. London Math. Soc. 33 (2001), no. 1, 67–72. http://dx.doi.Org/10.1112/blms/33.1.67 Google Scholar

[3] [3] Arveson, W., Subalgebras of C* -algebras. Acta Math. 123 (1969), 141–224. http://dx.doi.org/10.1007/BF02392388 Google Scholar

[4] [4] Arveson, W., Subalgebras of C*-algebras. II. Acta Math. 128 (1972), no. 3-4, 271–308. http://dx.doi.Org/10.1007/BF02392166 Google Scholar

[5] [5] Arveson, W., An invitation to C*-algebras. Graduate Texts in Mathematics, 39, Springer-Verlag, New York-Heidelberg, 1976. Google Scholar

[6] [6] Arveson, W., The noncommutative Choquet boundary. J. Amer. Math. Soc. 21 (2008), no. 4,1065-1084. http://dx.doi.Org/10.1090/S0894-0347-07-00570-X Google Scholar

[7] [7] Arveson, W., The noncommutative Choquet boundary II: hyperrigidity. Israel J. Math. 184 (2011), 349–385. http://dx.doi.Org/10.1007/s11856-011-0071-z Google Scholar

[8] [8] Bishop, E. and de Leeuw, K., The representations of linear functionals by measures on sets of extreme points. Ann. Inst. Fourier. Grenoble 9 (1959), 305–331. Google Scholar

[9] [9] Blecher, D. P., The Shilov boundary of an operator space and the characterization theorems. J. Funct. Anal. 182 (2001), no. 2, 280–343. http://dx.doi.Org/10.1006/jfan.2000.3734 Google Scholar

[10] [10] Brown, N. P. and Ozawa, N., C*-algebras and finite-dimensional approximations. Graduate Studies in Mathematics, 88, American Mathematical Society, Providence, RI, 2008. http://dx.doi.Org/10.1090/gsm/088 Google Scholar

[11] [11] Choi, M. D. and Effros, E. G., Injectivity and operator spaces. J. Functional Analysis 24 (1977), no. 2, 156–209. Google Scholar

[12] [12] Clouâtre, R. and Hartz, M., Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity. J. Funct. Anal. 274 (2018) no. 6, 1690–1738. http://dx.doi.Org/10.1016/j.jfa.2O17.10.008 Google Scholar

[13] [13] Clouâtre, R. and Ramsey, C., A completely bounded non-commutative Choquet boundary for operator spaces. Int. Math. Res. Not. IMRN, rnx335. http://dx.doi.Org/10.1093/imrn/rnx335 Google Scholar

[14] [14] Davidson, K. R. and Kennedy, M., The Choquet boundary of an operator system. Duke Math. J. 164 (2015), no. 15, 2989–3004. http://dx.doi.org/10.1215/00127094-3165004 Google Scholar

[15] [15] Davidson, K. R. and Kennedy, M., Choquet order and hyperrigidity for function systems. 2016. arxiv:1608.02334 Google Scholar

[16] [16] Dixmier, J., C*-algebras. North-Holland Mathematical Library, 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Google Scholar

[17] [17] Dritschel, M. A. and McCullough, S. A., Boundary representations for families of representations of operator algebras and spaces. J. Operator Theory 53 (2005), no. 1, 159–167. Google Scholar

[18] [18] Farenick, D. R., Extremal matrix states on operator systems. J. London Math. Soc. (2) 61 (2000), no. 3, 885-892. http://dx.doi.Org/10.1112/S0024610799008613 Google Scholar

[19] [19] Fuller, A. H., Hartz, M., and Lupini, M., Boundary representations of operator spaces, and compact rectangular matrix convex sets. 2016. arxiv:1610.05828 Google Scholar

[20] [20] Gamelin, T. W., Uniform algebras. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. Google Scholar

[21] [21] Hamana, M., Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 773–785. http://dx.doi.org/10.2977/prims/1195187876 Google Scholar

[22] [22] article M. Hamana, Triple envelopes and Silov boundaries of operator spaces. Math. J. Toyama Univ. 22 (1999), 77–93. Google Scholar

[23] [23] Hamhalter, J., Multiplicativity of extremal positive maps on abelian parts of operator algebras. J. Operator Theory 48 (2002), no. 2, 369–383. Google Scholar

[24] [24] Kadison, R. V. and Singer, I. M., Extensions of pure states. Amer. J. Math. 81 (1959), 383–400. http://dx.doi.Org/10.2307/2372748 Google Scholar

[25] [25] Kavruk, A. S., The Weak expectation property and Riesz interpolation. 2012. arxiv:1201.5414 Google Scholar

[26] [26] Kennedy, M. and Shalit, O. M., Essential normality, essential norms and hyperrigidity. J. Funct. Anal. 268 (2015), no. 10, 2990–3016. http://dx.doi.Org/10.1016/j.jfa.2015.03.014 Google Scholar

[27] [27] Kleski, C., Korovkin-type properties for completely positive maps. Illinois J. Math. 58 (2014), no. 4, 1107–1116. Google Scholar

[28] [28] Korovkin, P. P., On convergence of linear positive operators in the space of continuous functions. (Russian) Doklady Akad. Nauk SSSR (N.S.) 90 (1953), 961–964. Google Scholar

[29] [29] Lance, C., On nuclear C*-algebras. J. Functional Analysis 12 (1973), 157–176. http://dx.doi.Org/10.1016/0022-1236(73)90021-9 Google Scholar

[30] [30] Marcus, A. W., Spielman, D. A., and Srivastava, N., Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem. Ann. of Math. (2) 182 (2015), no. 1, 327–350. http://dx.doi.Org/10.4007/annals.2015.1 82.1.8 Google Scholar

[31] [31] Muhly, P. S. and Solel, B., An algebraic characterization of boundary representations. In: Nonselfadjoint operator algebras, operator theory, and related topics, Oper. Theory Adv. Appl., 104, Birkhâuser, Basel, 1998, pp. 189–196. Google Scholar

[32] [32] Namboodiri, M. N. N., Pramod, S., Shankar, P., and Vijayarajan, A. K., Quasi hyperrigidity and weak peak points for non-commutative operator systems. 2016. arxiv:161O.O2165 Google Scholar

[33] [33] Paulsen, V., Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, 78, Cambridge University Press, Cambridge, 2002. Google Scholar

[34] [34] Phelps, R. R., Lectures on Choquet's theorem. Second éd., Lecture Notes in Mathematics, 1757, Springer-Verlag, Berlin, 2001. http://dx.doi.Org/10.1007/b76887 Google Scholar

[35] [35] Rudin, W., Function theory in the unit ball ofC”. Classics in Mathematics, Springer-Verlag, Berlin, 2008. Google Scholar

[36] [36] SzNagy, B., Foias, C., Bercovici, H., and Kérchy, L., Harmonic analysis of operators on Hilbert space. Universitext, Springer, New York, 2010. http://dx.doi.org/10.1007/978-1-4419-6094-8 Google Scholar

[37] [37] Saskin, J. A., The MH'man-Choquet boundary and the theory of approximations. Funkcional. Anal. i Prilozen. 1 (1967), no. 2, 95–96. Google Scholar

[38] [38] Webster, C. and Winkler, S., The Krein-Milman theorem in operator convexity. Trans. Amer. Math. Soc. 351 (1999), no. 1, 307–322. http://dx.doi.org/10.1090/S0002-9947-99-02364-8 Google Scholar