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A nonseparable amenable operator algebra which is not isomorphic to a $$C^{\ast}$$-algebra. (English) Zbl 1287.47057
Amenability is a concept for Banach algebras, generally viewed as a regularity property. For example: (i) a $$C^*$$-algebra is amenable (as a Banach algebra) if and only if it is nuclear; (ii) a locally compact group $$G$$ is amenable if and only if $$L^1(G)$$ is amenable (again, as a Banach algebra).
In this article, an “operator algebra” means an operator norm-closed subalgebra of the bounded operators on a Hilbert space. This is precisely the (concrete) definition of a $$C^*$$-algebra, except without the requirement of being $$^*$$-closed. Operator algebras are viewed as a subclass of Banach algebras, and an “isomorphism” is taken in this category, meaning that it is a bounded bijective homomorphism whose inverse is also bounded. It is well-known that there are operator algebras that aren’t isomorphic to $$C^*$$-algebras (such as algebras of upper-triangular matrices).
The regularity implicit in the concept of amenability has made it natural to ask: if an operator algebra $$A$$ is amenable (as a Banach algebra), is it isomorphic to a $$C^*$$-algebra? This was a long-standing open question asked by many people. In this article, it is shown that the answer is no: there exists an operator algebra which is amenable, yet is not isomorphic to a $$C^*$$-algebra.
This interesting counterexample sheds a lot of light on the problem. On the one hand, the algebra constructed is a subalgebra of $$\ell_\infty(\mathbb N,M_2)$$ and, in particular, is $$2$$-subhomogeneous. In this sense, the example is quite close to being abelian, in contrast to a recent result of L. Marcoux and A. Popov [“Abelian, amenable operator algebras are similar to $$C^\ast$$-algebras”, arXiv:1311.2982], that every abelian amenable operator algebra is isomorphic to a $$C^*$$-algebra.
On the other hand, the algebra constructed in this article is nonseparable. It arises as the preimage in $$\ell_\infty(\mathbb N,M_2)$$ of the operator algebra generated by a (non-$$*$$-)representation of an uncountable discrete abelian group $$\Gamma$$ in $$\ell_\infty(\mathbb N,M_2)/c_0(\mathbb N,M_2)$$. However, the authors show that, for any group represented in $$\ell_\infty(\mathbb N,M_2)/c_0(\mathbb N,M_2)$$, the preimage in $$\ell_\infty(\mathbb N,M_2)$$ is always isomorphic to a $$C^*$$-algebra, provided that the group is countable. The authors therefore leave open the tantalizing question of whether every separable amenable operator algebra is isomorphic to a $$C^*$$-algebra.

##### MSC:
 47L30 Abstract operator algebras on Hilbert spaces 46L05 General theory of $$C^*$$-algebras 03E75 Applications of set theory
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##### References:
 [1] Blecher, D. P. and Le Merdy, C.2004Operator Algebras and Their Modules—an Operator Space Approach. , Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford. · Zbl 1061.47002 [2] Choi, Y., On commutative, operator amenable subalgebras of finite von Neumann algebras, J. Reine Angew. Math., 678, 201-222, (2013) · Zbl 1282.46054 [3] Connes, A., On the cohomology of operator algebras, J. Funct. Anal., 28, 248-253, (1978) · Zbl 0408.46042 [4] Curtis, P. C. Jr and Loy, R. J. ‘A note on amenable algebras of operators’, Bull. Austral. Math. Soc.52, (1995), 327-329. · Zbl 0836.47034 [5] Farah, I. and Hart, B. ‘Countable saturation of corona algebras’, C. R. Math. Rep. Acad. Sci. Canada35, (2013), 35-56. · Zbl 1300.46047 [6] Gifford, J. A., Operator algebras with a reduction property, J. Aust. Math. Soc., 80, 297-315, (2006) · Zbl 1103.46031 [7] Haagerup, U., All nuclear $$\text{C}^*$$-algebras are amenable, Invent. Math., 74, 305-319, (1983) · Zbl 0529.46041 [8] Johnson, B. E.1972 In Cohomology in Banach Algebras, American Mathematical Society, Providence, RI. · Zbl 0256.18014 [9] Kirchberg, E., On subalgebras of the CAR-algebra, J. Funct. Anal., 129, 35-63, (1995) · Zbl 0912.46059 [10] Luzin, N. ‘O chastyah naturalp1nogo ryada’, Izv. AN SSSR, seriya mat.11 (5) (1947), 714-722 Available at http://www.mathnet.ru/links/55625359125306fbfba1a9a6f07523a0/im3005.pdf. [11] Monod, N.2001Continuous Bounded Cohomology of Locally Compact Groups. vol. 1758. Springer. · Zbl 0967.22006 [12] Monod, N. and Ozawa, N. ‘The Dixmier problem, lamplighters and Burnside groups’, J. Funct. Anal.258, (2010), 255-259. · Zbl 1184.43006 [13] Ormes, N. S., Real coboundaries for minimal Cantor systems, Pacific J. Math., 195, 453-476, (2000) · Zbl 1065.46503 [14] Pedersen, G. K.1990 ‘The corona construction’, In Operator Theory: Proceedings of the 1988 GPOTS-Wabash Conference, Indianapolis, IN, 1988, vol. 225, pp. 49-92. Longman Sci. Tech, Harlow. [15] Pisier, G.2001 ‘Similarity problems and completely bounded maps’, In Includes the Solution to The Halmos Problem, Second, expanded edition, vol. 1618,. Springer-Verlag, Berlin. · Zbl 0971.47016 [16] Pisier, G., Simultaneous similarity, bounded generation and amenability, Tohoku Math. J., 59, 2, 79-99, (2007) · Zbl 1160.46037 [17] Runde, V.2002 In Lectures on Amenability, vol. 1774, Springer-Verlag, Berlin. · Zbl 0999.46022 [18] Šeĭnberg, M. V., A characterization of the algebra $$C(\Omega )$$ in terms of cohomology groups, Uspekhi Mat. Nauk, 32, 203-204, (1977) · Zbl 0366.46041 [19] Talayco, D. E., Applications of cohomology to set theory I: Hausdorff gaps, Ann. Pure Appl. Logic, 71, 69-106, (1995) · Zbl 0824.03029 [20] Voiculescu, D., A note on quasi-diagonal $$\text{C}^*$$-algebras and homotopy, Duke Math. J., 62, 267-271, (1991) · Zbl 0833.46055 [21] Willis, G. A., When the algebra generated by an operator is amenable, J. Operator Theory, 34, 239-249, (1995) · Zbl 0855.46029
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