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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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