Notions of entropy for unitary representations

Abstract: Several notions of entropy are studied widely in the ergodic theory of probability-preserving actions of countable groups. These include the generalization of Kolmogorov--Sinai entropy for amenable groups using F\o lner sequences, percolative entropy entropy for general countable groups, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across the analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can generally be evaluated as Fuglede--Kadison determinants. The resulting formulas for those determinants can be seen as various non-commutative generalizations of Szeg\H{o}'s limit theorem for Toeplitz determinants. They also establish connections with Arveson's theory of subdiagonal subalgebras, and with some more recent exact entropy calculations in the ergodic theory of actions by automorphisms of compact Abelian groups.