Completely positive kernels: the noncommutative correspondence setting

It is well known that a function K : Ω × Ω → L(Y) (where L(Y) is the set of all bounded linear operators on a Hilbert space Y) being (1) a positive kernel in the sense of Aronszajn (i.e. ∑N i,j=1〈K(ωi, ωj)yj , yi〉 ≥ 0 for all ω1, . . . , ωN ∈ Ω, y1, . . . , yN ∈ Y, and N = 1, 2, . . . ) is equivalent to (2) K being the reproducing kernel for a reproducing kernel Hilbert space H(K), and (3) K having a Kolmogorov decomposition K(ω, ζ) = H(ω)H(ζ)∗ for an operator-valued function H : Ω→ L(X ,Y) where X is an auxiliary Hilbert space. Recent work of the authors extended this result to the setting of free noncommutative functions (i.e. functions defined on matrices over a point set which respects direct sums and similarities) with the target set L(Y) of K replaced by L(A,L(Y)) where A is a C∗-algebra. In this talk, we discuss the next extension where the target set of K is replaced by L(A,La(E)) where A is a W ∗-algebra and La(E) is the set of adjointable operators on a Hilbert W ∗-module over a W ∗-algebra B. Various special cases of this result correspond to results of Kasparov, Murphy, and Szafraniec in the Hilbert C∗-module literature. Talk time: 07/19/2016 4:00PM— 07/19/2016 4:20PM Talk location: Cupples I Room 113 Special Session: State space methods in operator and function theory. Organized by J. Ball and S. ter Horst.