A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A -ALGEBRA

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<jats:title>Abstract</jats:title><jats:p>It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509413000066_inline1" /><jats:tex-math>$\mathrm{C}^*$</jats:tex-math></jats:alternatives></jats:inline-formula>-algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509413000066_inline2" /><jats:tex-math>$\mathrm{C}^*$</jats:tex-math></jats:alternatives></jats:inline-formula>-algebras and show that our method cannot produce a separable counterexample.</jats:p>

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