Von Neumann's inequality for commuting weighted shifts

We show that every multivariable contractive weighted shift dilates to a tuple of commuting unitaries, and hence satisfies von Neumann's inequality. This answers a question of Lubin and Shields. We also exhibit a closely related $3$-tuple of commuting contractions, similar to Parrott's example, which does not dilate to a $3$-tuple of commuting unitaries.Comment: 13 pages; minor change

von Neumann\u27s inequality for commuting weighted shifts

The failure of von Neumann\u27s inequality for three commuting contractions has been known since the seventies, thanks to examples of Kaijser-Varopoulos and Crabb-Davie. Nevertheless, this phenomenon is still not well understood. I will talk about a result which shows that von Neumann\u27s inequality holds for a particularly tractable class of commuting contractions, namely multivariable weighted shifts. This provides a positive answer to a question of Lubin and Shields from 1974. As an application, we see that there is no ``nice\u27\u27 Hilbert function space which is to commuting contractions as the Drury-Arveson space is to commuting row contractions

Finite dimensional approximations in operator algebras

A non-self-adjoint operator algebra is said to be residually finite dimensional (RFD) if it embeds into a product of matrix algebras. We characterize RFD operator algebras in terms of their matrix state space, and moreover show that an operator algebra is RFD if and only if every representation can be approximated by finite dimensional ones in the point weak operator topology. This is a non-self-adjoint version of a theorem of Exel and Loring for $C^*$-algebras. Moreover, we construct an example of an operator algebra for which approximation in the point strong operator topology is not possible. As a consequence, the maximal $C^*$-algebra generated by this operator algebra is not RFD. This answers questions of Clou\^atre and Ramsey and of Clou\^atre and Dor-On.Comment: 19 page