Residual finite dimensionality and representations of amenable operator algebras

The final publication is available at Elsevier via https://doi.org/10.1016/j.jmaa.2018.11.079. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable operator algebras, and we provide an affirmative answer for representations whose range is residually finite-dimensional. Furthermore, we show that weak-⁎ closed, amenable, residually finite-dimensional operator algebras are similar to ⁎-algebras, and in particular have the property that all their bounded representations are completely bounded. We prove our results for operator algebras having the so-called total reduction property, which is known to be weaker than amenability.Natural Sciences and Engineering Research Council of Canad