Clark theory in the Drury–Arveson space

We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges–Rovnyak type spaces H(b)H(b) contrastively contained in the Drury–Arveson space on the unit ball in CdCd. The Aleksandrov–Clark measures on the circle are replaced by a family of states on a certain noncommutative operator system, and the backward shift is replaced by a canonical solution to the Gleason problem in H(b)H(b). In addition we introduce the notion of a “quasi-extreme” multiplier of the Drury–Arveson space and use it to characterize those H(b)H(b) spaces that are invariant under multiplication by the coordinate functions.