Gregory Marx (BGU)

Monday, November 27, 2017, 16:00 – 17:00, -101

Abstract:

It is well known that a function $K: \Omega \times \Omega \to \mathcal{L}(\mathcal{Y})$ (where $\mathcal{L}(\mathcal{Y}$) is the set of all bounded linear operators on a Hilbert space$\mathcal Y$) being (1) a positive kernel in the sense of Aronszajn (i.e. $\sum_{i,j=1}^N \langle K(\omega_i , \omega_j) y_j, y_i \rangle \geq 0$ for all $\omega_1, \dots, \omega_N \in \Omega$, $y_1, \dots, y_N \in \mathcal Y$, and $N=1,2,\dots$) is equivalent to (2) $K$ being the reproducing kernel for a reproducing kernel Hilbert space $\mathcal H (K)$, and (3) $K$ having a Kolmogorov decomposition $K(\omega, \zeta)=H(\omega)H(\zeta)^*$ for an operator-valued function $H: \Omega \to \mathcal{L}(\mathcal X, \mathcal Y)$ where $\mathcal X$ is an auxiliary Hilbert space.

In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative functions with the target set $\mathcal L ( \mathcal Y)$ of $K$ replaced by $\mathcal L (\mathcal A, \mathcal L (\mathcal Y))$ where $\mathcal A$ is a $C^*$-algebra. In my talk, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring’s dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.