The Correlation Numerical Range of a Matrix and Connes' Embedding Problem

TL;DR

This paper introduces a new numerical range for matrices based on correlation matrices and explores its properties, connecting it to Alain Connes' embedding problem, thus offering new insights into operator algebra theory.

Contribution

It defines a novel correlation-based numerical range for matrices and relates it to Connes' embedding problem, expanding the tools for analyzing operator algebras.

Findings

Introduced a new correlation numerical range for matrices.

Established properties of this new numerical range.

Linked the numerical range to Connes' embedding problem.

Abstract

We define a new numerical range of an n\timesn complex matrix in terms of correlation matrices and develop some of its properties. We also define a related numerical range that arises from Alain Connes' famous embedding problem.