# Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Ilwoo Cho; Palle E. T. Jorgensen

Special Matrices (2015)

- Volume: 3, Issue: 1, page 123-154, electronic only
- ISSN: 2300-7451

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topIlwoo Cho, and Palle E. T. Jorgensen. "Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs." Special Matrices 3.1 (2015): 123-154, electronic only. <http://eudml.org/doc/270904>.

@article{IlwooCho2015,

abstract = {In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we consider free probability on AG.},

author = {Ilwoo Cho, Palle E. T. Jorgensen},

journal = {Special Matrices},

keywords = {Directed Graphs; Graph Groupoids; Groupoid Dynamical Systems; directed graphs; graph groupoids; groupoid dynamical systems},

language = {eng},

number = {1},

pages = {123-154, electronic only},

title = {Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs},

url = {http://eudml.org/doc/270904},

volume = {3},

year = {2015},

}

TY - JOUR

AU - Ilwoo Cho

AU - Palle E. T. Jorgensen

TI - Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

JO - Special Matrices

PY - 2015

VL - 3

IS - 1

SP - 123

EP - 154, electronic only

AB - In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we consider free probability on AG.

LA - eng

KW - Directed Graphs; Graph Groupoids; Groupoid Dynamical Systems; directed graphs; graph groupoids; groupoid dynamical systems

UR - http://eudml.org/doc/270904

ER -

## References

top- [1] I. Cho, Operators Induced by Prime Numbers, Methods Appl. Math. Sci. 19, no. 4, (2013) 313 - 340. Zbl1332.46064
- [2] I. Cho, Graph Groupoids and Partial Isometries, ISBN: 978-3-8383-1397-9, (2009) Lambert Academic Press
- [3] I. Cho, Classification on Arithmetic Functions and Corresponding Free-Moment L-Functions, Bulletin Korea Math. Soc., (2015) To Appear. Zbl1329.11113
- [4] I. Cho, p-Adic Banach-Space Operators and Adelic Banach-Space Operators, Opuscula Math., 34, no. 1, (2014) 29 - 65. Zbl06291546
- [5] I. Cho, Fractals on Graphs, ISBN: 978-3-639-19447-0, (2009) Verlag with Dr. Muller
- [6] I. Cho, Operations on Graphs, Groupoids, and Operator Algebras, ISBN: 978-8383-5271-8, (2010) Lambert Academic Press.
- [7] I. Cho, C -Valued Functions Induced by Graphs, Compl. Anal. Oper. Theo., DOI:10.1007/s11785-014-0368-0, (2014).
- [8] I. Cho, and P. E. T. Jorgensen, An Application of Free Probability to Arithmetic Functions, Compl. Anal. Oper. Theo., DOI: 10.1007/s11785-014-0378-y, (2014) Zbl06496671
- [9] I. Cho and P. E. T. Jorgensen, Krein-Space Representation of Arithmetic Functions Determined by Primes, Alg. Rep. Theo, DOI: 10.1007/s11785-014-9473-z, (2014) Zbl1305.05233
- [10] I. Cho, and P. E. T. Jorgensen, Krein-Space Operators Induced by Dirichlet Characters, Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, (2014) 3 - 33. Zbl1322.11065
- [11] I. Cho, and P. E. T. Jorgensen, Actions of Arithmetic Functions on Matrices and Corresponding Representations, Ann. Funct. Anal., (2014) To Appear. Zbl1309.46037
- [12] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol 1, ISBN: 978-981-02-0880-6, (1994) World Scientific. Zbl0812.46076
- [13] D. Bump, Automorphic Forms and Representations, Cambridge Studies in Adv. Math., 55, ISBN: 0-521-65818-7, (1996) Cambridge Univ. Press.
- [14] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, Vol. 1, ISBN: 0-8218-1140-1, (2002) Published by Amer. Math. Soc.
- [15] J. P. S. Kung, M. R. Murty, and G-C Rota, On the Ré dei Zeta Function, J. Number Theo., 12, (1980) 421 - 436. Zbl0446.05003
- [16] P. Flajolet and R. Sedgewick, Analytic Combinatorics, ISBN: 978-0-521-89806-5, (2009) Cambridge Univ. Press.
- [17] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Amer. Math. Soc. Mem., vol 132, no. 627, (1998). Zbl0935.46056

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