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

Special Matrices (2015)

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

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Abstract

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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.

How to cite

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Ilwoo 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

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9. [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. [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. [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
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15. [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
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17. [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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