Games in the matrix

TL;DR

This work studies the asymptotic behavior of complex matrix algebras through a blend of logic and operator algebra. It introduces EF-like games on graphs and norm-based analogs for matrices to compare finite structures, connecting short-game indistinguishability to first-order theory and 0-1 laws. By moving from strict algebraic equalities to norm-based distances, it explores what matrix algebras can and cannot be distinguished in the large, and relates these ideas to permutation groups, soficity, and Ulam stability. The paper then situates these finite observations in the broader context of ultraproducts, showing how the limit theories of matrix algebras interface with deep set-theoretic questions such as CH, and ends by highlighting Popa’s question on tracial ultraproducts as a guiding, overarching goal. The results sketch a landscape where logical, algebraic, and set-theoretic methods converge to illuminate the structure of large matrix algebras and their ultraproducts.

Abstract

This is about algebras of complex $n\times n$ matrices. Do these algebras look similar for all large $n$? This paper is intended for general audience.

Figures (2)

• Figure 1: The position after five innings. Since $m$ and $l$ are very large, we zoomed out so much that the vertices blended into two continuous circles. The corresponding short (red) arcs are of the same length, while the long (blue) arcs are simply sufficiently long. No matter what vertex the Challenger plays, we can respond by choosing a vertex in the corresponding arc that is either at the right distance or sufficiently far from the corresponding ends.
• Figure 2: The projection to the line $x=y$ and the rotation by $\pi/2$ around the origin as applied to some points randomly chosen in the plane.

Theorems & Definitions (9)

• Theorem 1.1
• proof
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1