Universal $C^∗$-algebras with the local lifting property
DOI:
https://doi.org/10.7146/math.scand.a-126018Abstract
The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between $\mathrm{C}^*$-algebras. Outside of the nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP\@. In this article, we show that the LLP holds for the algebraic contraction $\mathrm{C}^*$-algebras introduced by Hadwin and further studied by Loring and Shulman. We also show that the universal Pythagorean $\mathrm{C}^*$-algebras introduced by Brothier and Jones have the Lifting Property.
References
Akemann, C. A., and Pedersen, G. K., Ideal perturbations of elements in ${C}^*$-algebras, Math. Scand. 41 (1977), no. 1, 117–139. https://doi.org/10.7146/math.scand.a-11707
Akemann, C. A., Pedersen, G. K., and Tomiyama, J., Multipliers of ${C}^*$-algebras, J. Funct. Anal. 13 (1973), 277–301. https://doi.org/10.1016/0022-1236(73)90036-0
Blackadar, B., Shape theory for ${C}^*$-algebras, Math. Scand. 56 (1985), no. 2, 249–275. https://doi.org/10.7146/math.scand.a-12100
Blackadar, B., $K$-theory for operator algebras, second ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998.
Brenken, B. and Niu, Z., The ${C}^*$-algebra of a partial isometry, Proc. Amer. Math. Soc. 140 (2012), no. 1, 199–206. https://doi.org/10.1090/S0002-9939-2011-10988-2
Brothier, A., and Jones, V. F. R., Pythagorean representations of Thompson's groups, J. Funct. Anal. 277 (2019), no. 7, 2442–2469. https://doi.org/10.1016/j.jfa.2019.02.009
Brown, L. G., Stable isomorphism of hereditary subalgebras of ${C}^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348.
Brown, L. G., and Pedersen, G. K., ${C}^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. https://doi.org/10.1016/0022-1236(91)90056-B
Brown, N. P., and Ozawa, N., ${C}^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. https://doi.org/10.1090/gsm/088
Bunce, J. W., Models for $n$-tuples of noncommuting operators, J. Funct. Anal. 57 (1984), no. 1, 21–30. https://doi.org/10.1016/0022-1236(84)90098-3
Choi, M. D., The full ${C}^*$-algebra of the free group on two generators, Pacific J. Math. 87 (1980), no. 1, 41–48.
Choi, M. D., and Effros, E. G., Separable nuclear ${C}^*$-algebras and injectivity, Duke Math. J. 43 (1976), no. 2, 309–322.
Courtney, K. E., and Sherman, D., The universal ${C}^*$-algebra of a contraction, J. Operator Theory 84 (2020), no. 1, 153–184. https://doi.org/10.7900/jot
Eilers, S., Loring, T. A., and Pedersen, G. K., Stability of anticommutation relations: an application of noncommutative CW complexes, J. Reine Angew. Math. 499 (1998), 101–143.
Farenick, D., Kavruk, A. S., Paulsen, V. I., and Todorov, I. G., Characterisations of the weak expectation property, New York J. Math. 24A (2018), 107–135.
Gabe, J., Lifting theorems for completely positive maps, preprint, arXiv:1508.00389v3.
Hadwin, D., Lifting algebraic elements in ${C}^*$-algebras, J. Funct. Anal. 127 (1995), no. 2, 431–437. https://doi.org/10.1006/jfan.1995.1018
Hartz, M., Von Neumann's inequality for commuting weighted shifts, Indiana Univ. Math. J. 66 (2017), no. 4, 1065–1079. https://doi.org/10.1512/iumj.2017.66.6074
Ji, Z., Natarajan, A., Vidick, T., Wright, J., and Yuen, H., MIP*=RE, preprint, arXiv:2001.04383.
Junge, M., and Pisier, G., Bilinear forms on exact operator spaces and $mathcal{B}(mathcal{H})otimes mathcal{B}(mathcal{H})$, Geom. Funct. Anal. 5 (1995), no. 2, 329–363. https://doi.org/10.1007/BF01895670
Kasparov, G. G., Hilbert ${C}^*$-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), no. 1, 133–150.
Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group ${C}^*$-algebras, Invent. Math. 112 (1993), no. 3, 449–489. https://doi.org/10.1007/BF01232444
Kirchberg, E., Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math. 452 (1994), 39–77. https://doi.org/10.1515/crll.1994.452.39
Lance, C., On nuclear ${C}^*$-algebras, J. Funct. Anal. 12 (1973), 157–176. https://doi.org/10.1016/0022-1236(73)90021-9
Lance, E. C., Hilbert $C^* $-modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995, https://doi.org/10.1017/CBO9780511526206
Lin, H., Simple ${C}^*$-algebras with continuous scales and simple corona algebras, Proc. Amer. Math. Soc. 112 (1991), no. 3, 871–880. https://doi.org/10.2307/2048712
Loring, T. A., ${C}^*$-algebras generated by stable relations, J. Funct. Anal. 112 (1993), no. 1, 159–203. https://doi.org/10.1006/jfan.1993.1029
Loring, T. A., Lifting solutions to perturbing problems in ${C}^*$-algebras, Fields Institute Monographs, vol. 8, American Mathematical Society, Providence, RI, 1997. https://doi.org/10.1090/fim/008
Loring, T. A., and Shulman, T., Noncommutative semialgebraic sets and associated lifting problems, Trans. Amer. Math. Soc. 364 (2012), no. 2, 721–744. https://doi.org/10.1090/S0002-9947-2011-05313-4
Loring, T. A., and Shulman, T., Lifting algebraic contractions in ${C}^*$-algebras, Algebraic methods in functional analysis, 85–92, Oper. Theory Adv. Appl., vol. 233, Birkhäuser/Springer, Basel, 2014. https://doi.org/10.1007/978-3-0348-0502-5_6
Mingo, J. A., and Phillips, W. J., Equivariant triviality theorems for Hilbert ${C}^*$-modules, Proc. Amer. Math. Soc. 91 (1984), no. 2, 225–230. https://doi.org/10.2307/2044632
Olsen, C. L., and Pedersen, G. K., Corona ${C}^*$-algebras and their applications to lifting problems, Math. Scand. 64 (1989), no. 1, 63–86. https://doi.org/10.7146/math.scand.a-12248
Ozawa, N., On the lifting property for universal ${C}^*$-algebras of operator spaces, J. Operator Theory 46 (2001), no. 3, suppl., 579–591.
Ozawa, N., About the QWEP conjecture, Internat. J. Math. 15 (2004), no. 5, 501–530. https://doi.org/10.1142/S0129167X04002417
Ozawa, N., There is no separable universal ${II}_1$-factor, Proc. Amer. Math. Soc. 132 (2004), no. 2, 487–490. https://doi.org/10.1090/S0002-9939-03-07127-2
Paulsen, V. I., Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002.
Pedersen, G. K., ${C}^*$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.
Pisier, G., A non-nuclear ${C}^*$-algebra with the weak expectation property and the local lifting property, preprint.
Pisier, G., A simple proof of a theorem of Kirchberg and related results on ${C}^*$-norms, J. Operator Theory 35 (1996), no. 2, 317–335.
Popescu, G., Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. https://doi.org/10.2307/2001359
Rørdam, M., Ideals in the multiplier algebra of a stable ${C}^*$-algebra, J. Operator Theory 25 (1991), no. 2, 283–298.
Shulman, T., Lifting of nilpotent contractions, Bull. Lond. Math. Soc. 40 (2008), no. 6, 1002–1006. https://doi.org/10.1112/blms/bdn084
Shulman, T., Semiprojectivity of universal ${C}^*$-algebras generated by algebraic elements, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1363–1370. https://doi.org/10.1090/S0002-9939-2011-11144-4
Thom, A., Examples of hyperlinear groups without factorization property, Groups Geom. Dyn. 4 (2010), no. 1, 195–208. https://doi.org/10.4171/GGD/80
