# Noncommutative partial convexity via $\Gamma$-convexity

## TL;DR

This paper develops a theory of noncommutative convex sets defined by matrix polynomials constrained by a tuple of symmetric polynomials, extending classical convexity concepts into the free algebra setting.

## Contribution

It introduces the concept of $ ext{Gamma}$-convexity for free sets and proves a separation theorem characterizing these sets via linear pencils.

## Key findings

- $ ext{Gamma}$-convex sets are characterized by linear pencils.
- Established an Effros-Winkler Hahn-Banach separation theorem for $ ext{Gamma}$-convex sets.
- Provides a framework connecting classical convexity with noncommutative polynomial inequalities.

## Abstract

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $\Gamma$, a free set is called $\Gamma$-convex if it closed under isometric conjugation by isometries intertwining $\Gamma$. We establish an Effros-Winkler Hahn-Banach separation theorem for $\Gamma$-convex sets; they are delineated by linear pencils in the coordinates of $\Gamma$ and the variables $x$.