It is well known that for a function K:OmegatimesOmegatomathcalL(mathcalY) (where mathcalL(mathcalY) denotes the set of all bounded linear operators on a Hilbert space mathcalY ) the following are equivalent:

(a) K is a positive kernel in the sense of Aronszajn (i.e. sumi,j=1NlangleK(omegai,omegaj)yj,yiranglegeq0 for all omega1,dots,omegaNinOmega , y1,dots,yNinmathcalY , and N=1,2,dots ).

(b) K is the reproducing kernel for a reproducing kernel Hilbert space mathcalH(K) .

(c) K has a Kolmogorov decomposition: There exists an operator-valued function H:OmegatomathcalL(mathcalX,mathcalY) (where mathcalX is an auxiliary Hilbert space) such that K(omega,zeta)=H(omega)H(zeta) .

In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative function theory with the target set mathcalL(mathcalY) of K replaced by mathcalL(mathcalA,mathcalL(mathcalY)) where mathcalA is a C -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.