STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

Abstract

We study Markov measures and p-adic random walks with the use of states on the Cuntz algebras O p . Via the Gelfand-Naimark-Segal construction, these come from families of representations of O p . We prove that these representations reflect selfsimilarity especially well. In this paper, we consider a Cuntz-Krieger type algebra where the adjacency matrix depends on a parameter q (q = 1 is the case of Cuntz-Krieger algebra). This is an ongoing work generalizing a construction of certain measures associated to random walks on graphs.2010 Mathematics subject classification: primary 47B47; secondary 81P15, 60G35, 33D50, 46C05, 46L52, 11D88, 33D80.

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].…”

W-Markov measures, transfer operators, wavelets and multiresolutions

“…One source of motivation for our present work is a number of recent papers dealing with generalized wavelet multiresolutions, see e.g., [5,32,38,39,46,53,61], and harmonic analysis on groupoids. While these themes may seem disparate, they are connected via a set of questions in operator algebra theory; see e.g., [26,43,44]. The positive operators considered here are in a general measure theoretic setting, but we stress that there is also a rich theory of positive integral operators is the metric space setting, often called Mercer operators, and important in the approach of Smale and collaborators to learning theory, see e.g., [20,59,66].…”

Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$

“…For more about number-theoretic motivations of our proceeding works, see e.g., [16], [17], [18], [19], [31] and [32]. And, for more about statistical analysis, see [1], [2], [3], [4], [5], [6], [15], [21], [22] and [25]. Also, for free probability theory, see e.g., [26], [27], [28], [29], [30], [24], [20], [33], [34] and [35].…”

“…Analysis on Q p . For more about p-adic analysis, see [31] and [32] (also, see [17] and [22]). Let Q p be the p-adic number fields for p ∈ P. Recall that Q p are the maximal p-norm-topology closures in the normed space (Q, |.| p ) of all rational numbers, where |.| p are the non-Archimedean norms, called p-norms on Q, for all p ∈ P.…”

“…Weighted-semicircularity. Let M p be the p-adic C * -algebra, and let S p be the boundary subalgebra (21) of M p , satisfying the structure theorem (22). Throughout this section, let's fix a prime p. Recall that if {P p,k } k∈Z are the generating projections (20) of S p , then…”