Free nc function theory is an extension of the theory of holomorphic functions of several complex variables to the theory of functions on matrix tuples $Z=(Z_1,\cdots,Z_d)$ where $Z_i\in M_n(\mathbb{C})$ and $n$ is allowed to vary.

One can view such a tuple as a representation of $\mathbb{C}^d$ (viewed as a bimodule over $\mathbb{C}$). Here, a representation of a bimodule $X$ over an algebra $A$ is a pair $(\sigma,T)$ where $\sigma$ is a representation of $A$ and $T$ is a bimodule map $T(axb)=\sigma(a)T(x)\sigma(b)$.

An nc function is a function defined on such tuples $Z$ and takes values in $\cup_n M_n(\mathbb{C})$ which is graded and respects direct sums and similarity (equivalently, respects intertwiners).

In previous works we studied functions that are defined on the space of representations of a bimodule $E$ (more precisely, a correspondence) over a $W^*$-algebra $M$ (instead of the algebra $\mathbb{C}$), are graded and respect direct sums and similarities.

We referred to these functions as matricial functions.

Note that, while the representations of $\mathbb{C}$ are parameterized by $\mathbb{N}\cup \{\infty\}$, those of a general $W^*$-algebra form a more complicated category.

The classical correspondence between positive kernels and Hilbert spaces of functions has been recently extended by Ball, Marx and Vinnikov to nc completely positive kernels and Hilbert spaces of nc functions.

In this talk I will discuss a similar correspondence in our matricial context.

In place of a Hilbert space of functions we will get a $W^*$-correspondence whose elements are matricial functions.

I will also discuss what we get for an important class of kernels.

This is a joint work with Paul Muhly.