The functional calculus for commuting row contractions

A commuting row contraction is a d-tuple of commuting operators T1, . . . , Td such that ∑d i=1 TiT ∗ i ≤ I. Such operators have a polynomial functional calculus which extends to a norm closed algebra of multipliers Ad on Drury-Arveson space. We characterize those row contractions which admit an extension of this map to a weak-∗ continuous functional calculus on the full multiplier algebra. In particular, we show that completely non-unitary row contractions are always absolutely continuous, in direct parallel with the case of a single contraction. This is based on the detailed structure of the dual space of Ad. Finally, we consider refinements of this question for row contractions that are annihilated by a given ideal. This is joint work with Raphaël Clouâtre.