Actions of arithmetic functions on matrices and corresponding representations. (English) Zbl 1309.46037

Cho, Ilwoo; Jorgensen, Palle

Ann. Funct. Anal. 5, No. 2, 90-117 (2014).

Summary: In this paper, we study a class of representations of arithmetic functions, and corresponding operator-theoretic and free probabilistic properties. We associate given arithmetic functions \(f\) to certain matrices \(\alpha_n(f )\).

MSC:

46L53	Noncommutative probability and statistics
46L54	Free probability and free operator algebras
47L15	Operator algebras with symbol structure
47L30	Abstract operator algebras on Hilbert spaces
47L55	Representations of (nonselfadjoint) operator algebras
05E15	Combinatorial aspects of groups and algebras (MSC2010)
46L10	General theory of von Neumann algebras
46L40	Automorphisms of selfadjoint operator algebras
11N99	Multiplicative number theory

Keywords:

arithmetic function; arithmetic algebra; linear functional; arithmetic prime probability space; Krein space; representation; convolution operator; multiplication operator