An Algebraic Matrix Spencer Theorem

Abstract: We develop an algebraic approach to matrix discrepancy based on the representation theory of finite-dimensional C$^*$-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 $A_1,\ldots,A_n$ that are contained in a finite-dimensional $C^*$-algebra $\mathcal A$ with $\text{dim}{\mathbb C} (\mathcal A) \lesssim n$, there exists signs $x\in{\pm1}^n$ such that $|\sum{i=1}^n x_i A_i| \le O(\sqrt n)$. 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 $C^*$-algebra of small dimension.