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. sum N i,j=1 langleK(omega i ,omega j )y j ,y i ranglegeq0
for all omega 1 ,dots,omega N inOmega
, y 1 ,dots,y N inmathcalY
, 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.
