It is well known that for a function
denotes the set of all bounded linear operators on a Hilbert space
) the following are equivalent:
is a positive kernel in the sense of Aronszajn (i.e.
is the reproducing kernel for a reproducing kernel Hilbert space
has a Kolmogorov decomposition: There exists an operator-valued function
is an auxiliary Hilbert space) such that
In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative function theory with the target set
-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.