# ROOM TWO: Complete Pick spaces---what are they, and why are they interesting?

Let $(H,k)$ be a reproducing kernel Hilbert space on a set $X$. For $\Lambda \subseteq X$,

let $H_\Lambda = \vee \{ k_\lambda : \lambda \in \Lambda \}$, and $P_\Lambda$ be projection

onto $H_\Lambda$. There is a map $\pi$ from the multiplier algebra of $H$ to that of $H_\Lambda$ given by

$ \pi : M_\phi \mapsto P_\Lambda M_\phi P_\Lambda$.

$(H,k)$ is called a complete Pick space if, for every $\Lambda \subseteq X$, the map $\pi$ is a complete quotient map

(a quotient map means it has an isometric cross-section; complete means the same is true for all the

induced maps on the matrix algebras over the multiplier algebra).

Examples of complete Pick spaces include the Hardy space, the Dirichlet space and the Drury-Arveson space.

They were originally studied because much of the theory of Nevanlinna-Pick interpolation carries through to these spaces.

Recently it has turned out that many other properties of $H^\infty({\mathbb D})$ that were thought to be

function theoretic also carry over to complete Pick spaces,

for example Carleson's characterization of

interpolating sequences, and the inner-outer factoring of functions.

Interestingly, the proofs are not just operator theoretic, but sometimes rely on non-commutative analysis.

This is joint work with Alexandru Aleman, Michael Hartz and Stefan Richter.