Generalizing the amenable Szegő-type theorem to broader C*-algebras

Formulate and prove a generalization of Theorem A (the Szegő-like limit theorem over amenable groups) to positive functionals on an abstract class of C*-algebras that extends beyond group C*-algebras of countable amenable groups. Identify an appropriate structural hypothesis replacing amenability and establish the corresponding determinant-limit formula.

Background

Theorem A provides a Szegő-type determinant limit for matrix-valued positive definite functions on amenable groups. Extending this to non-group C*-algebras would require an appropriate substitute for amenability (e.g., nuclearity or quasidiagonality), but the author notes the lack of a precise formulation. A concrete problem statement is subsequently posed to motivate research directions.