An Algebraic Matrix Spencer Theorem

Abstract

We develop an algebraic approach to matrix discrepancy based on the representation theory of finite-dimensional C${}^{\ast }$-algebras. As an application, we resolve a substantial structured special case of the Matrix Spencer conjecture. In particular, we show that for every family of contractions $A_1,\dots ,A_n$ that are contained in a finite-dimensional $C^{\ast }$-algebra $\mathcal{A}$ with ${\text{dim}}_{\mathbb{C}}(\mathcal{A})\lesssim n$, there exists signs $x\in \{\pm 1{\}}^n$ such that $\Vert {\sum }_{i=1}^n{x}_i{A}_i\Vert \le O(\sqrt{n})$. As a noteworthy special case, our main result also resolves the Group Spencer conjecture of (Bandeira'24). We furthermore prove that Matrix Spencer continues to hold for low-rank perturbations of matrix families coming from an $C^{\ast }$-algebra of small dimension.

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