It is well known that for a function $$extract_itex$$K:OmegatimesOmegatomathcalL\left(mathcalY\right)$$/extract_itex$$ $$extract_itex$$K: Omega times Omega to mathcal{L}(mathcal{Y})$$/extract_itex$$ (where $$extract_itex$$mathcalL\left(mathcalY\right)$$/extract_itex$$ $$extract_itex$$mathcal{L}(mathcal{Y})$$/extract_itex$$ denotes the set of all bounded linear operators on a Hilbert space $$extract_itex$$mathcalY$$/extract_itex$$ $$extract_itex$$mathcal Y$$/extract_itex$$ ) the following are equivalent:

(a) $$extract_itex$$K$$/extract_itex$$ $$extract_itex$$K$$/extract_itex$$ is a positive kernel in the sense of Aronszajn (i.e. $$extract_itex$$su{m}_{i,j=1}^{N}langleK\left(omeg{a}_{i},omeg{a}_{j}\right){y}_{j},{y}_{i}ranglegeq0$$/extract_itex$$ $$extract_itex$${sum_{i,j=1}^N langle K(omega_i , omega_j) y_j, y_i rangle geq 0}$$/extract_itex$$ for all $$extract_itex$$omeg{a}_{1},dots,omeg{a}_{N}inOmega$$/extract_itex$$ $$extract_itex$$omega_1, dots, omega_N in Omega$$/extract_itex$$ , $$extract_itex$${y}_{1},dots,{y}_{N}inmathcalY$$/extract_itex$$ $$extract_itex$$y_1, dots, y_N in mathcal Y$$/extract_itex$$ , and $$extract_itex$$N=1,2,dots$$/extract_itex$$ $$extract_itex$$N=1,2,dots$$/extract_itex$$ ).

(b) $$extract_itex$$K$$/extract_itex$$ $$extract_itex$$K$$/extract_itex$$ is the reproducing kernel for a reproducing kernel Hilbert space $$extract_itex$$mathcalH\left(K\right)$$/extract_itex$$ $$extract_itex$$mathcal H (K)$$/extract_itex$$ .

(c) $$extract_itex$$K$$/extract_itex$$ $$extract_itex$$K$$/extract_itex$$ has a Kolmogorov decomposition: There exists an operator-valued function $$extract_itex$$H:OmegatomathcalL\left(mathcalX,mathcalY\right)$$/extract_itex$$ $$extract_itex$$H: Omega to mathcal{L}(mathcal X, mathcal Y)$$/extract_itex$$ (where $$extract_itex$$mathcalX$$/extract_itex$$ $$extract_itex$$mathcal X$$/extract_itex$$ is an auxiliary Hilbert space) such that $$extract_itex$$K\left(omega,zeta\right)=H\left(omega\right)H\left(zeta{\right)}^{\ast }$$/extract_itex$$ $$extract_itex$$K(omega, zeta)=H(omega)H(zeta)^*$$/extract_itex$$ .

In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative function theory with the target set $$extract_itex$$mathcalL\left(mathcalY\right)$$/extract_itex$$ $$extract_itex$$mathcal L ( mathcal Y)$$/extract_itex$$ of $$extract_itex$$K$$/extract_itex$$ $$extract_itex$$K$$/extract_itex$$ replaced by $$extract_itex$$mathcalL\left(mathcalA,mathcalL\left(mathcalY\right)\right)$$/extract_itex$$ $$extract_itex$$mathcal L (mathcal A, mathcal L (mathcal Y))$$/extract_itex$$ where $$extract_itex$$mathcalA$$/extract_itex$$ $$extract_itex$$mathcal A$$/extract_itex$$ is a $$extract_itex$${C}^{\ast }$$/extract_itex$$ $$extract_itex$$C^*$$/extract_itex$$ -algebra. In my talk, I will start with a brief introduction to free noncommutative function theory and follow up with a sketch of our proof. Afterwards, I will discuss some well-known results (e.g. Stinespring's dilation theorem for completely positive maps) which follow as corollaries and talk about more recent work.