Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields

Research output: Preprint/documentation/reportPreprint

Abstract

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application of this result we prove that the cone of copositive matrices of size $n\geq5$ is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital $\mathbb{R}$-linear completely positive map $R\to R$ on a real closed field extension $R$ of $\mathbb{R}$.

Details

Original language Undefined Published - 13 Jun 2022
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Keywords

Keywords

• math.RA, math.LO, math.OA, math.OC