Markov measures and extended zeta functions

Abstract

In this paper we study a family of representations of the Cuntz algebras Op where p is a prime. These algebras are built on generators and relations. They are C*-algebras and their representations are a part of non-commutative harmonic analysis. Starting with specific generators and relations we pass to an ambient C*-algebra, for example in one of the Cuntz-algebras. Our representations are motivated by the study of frequency bands in signal processing: We construct induced measures attached to those representations which turned out to be related to a class of zeta functions. For a particular case those measures give rise to a class of Markov measures and q-Bernoulli polynomials. Our approach is amenable to applications in problems from dynamics and mathematical physics: We introduce a deformation parameter q, and an associated family of q-relations where the number q is a "quantum-deformation," and also a parameter in a scale of (Riemann-Ruelle) zeta functions. Our representations are used in turn in a derivation of formulas for this q-zeta function.

STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

“…In [19], this measure has been shown to converge to a q-zeta function related to Bernoulli convolution measures.…”

“…For general background on Fourier analysis and special functions, relevant to the present discussion, see [12]; noncommutative geometry [27]; graph algebras [30]; number theory and zeta functions [14]; representation theory [11,17,21,24]. Relevant references to the theory of invariant measures are [9,10,15,19].…”

Asymptotic Semicircular Laws Induced by p-Adic Number Fields ℚ p and C*-Algebras over Primes p

“…For number theory and free probability theory, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], respectively. In [23], weighted-semicircular elements, and semicircular elements induced by p-adic number fields Q p are considered by the author and Jorgensen, for each p ∈ P, statistically.…”

Banach-Space Operators Acting on Semicircular Elements Induced by p-Adic Number Fields over Primes p

“…Connections between primes and operators have been considered in different approaches (e.g., [1][2][3][4][5][6][7]). For instance, we consider relations between analysis on Q p , and (weighted-) semicircular elements, in [8][9][10].…”