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

• Autorzy: Ilwoo Cho , Palle E. T. Jorgensen
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Directed Graphs; Graph Groupoids; Groupoid Dynamical Systems

Kategorie tematyczne

• 46L53: Noncommutative probability and statistics
• 05E15: Combinatorial aspects of groups and algebras
• 46L54: Free probability and free operator algebras
• 11R04: Algebraic numbers; rings of algebraic integers
• 46L40: Automorphisms
• 11G15: Complex multiplication and moduli of abelian varieties
• 46L10: General theory of von Neumann algebras
• 11R56: Ad\`ele rings and groups
• 11R47: Other analytic theory
• 11R09: Polynomials (irreducibility, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-02-19

zaakceptowano

2015-06-17

online

2015-07-08

Twórcy

autor

Ilwoo Cho

• St. Ambrose Univ., Dept. of Math. & Stat., 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803,
  U. S. A. / Univ. of Iowa, Dept. of Math., 14 McLean Hall, Iowa City, Iowa, 52242, U. S. A.

autor

Palle E. T. Jorgensen

• St. Ambrose Univ., Dept. of Math. & Stat., 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803,
  U. S. A. / Univ. of Iowa, Dept. of Math., 14 McLean Hall, Iowa City, Iowa, 52242, U. S. A.

Bibliografia

• [1] I. Cho, Operators Induced by Prime Numbers, Methods Appl. Math. Sci. 19, no. 4, (2013) 313 - 340.
• [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.
• [4] I. Cho, p-Adic Banach-Space Operators and Adelic Banach-Space Operators, Opuscula Math., 34, no. 1, (2014) 29 - 65.
• [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)
• [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)
• [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.
• [11] I. Cho, and P. E. T. Jorgensen, Actions of Arithmetic Functions on Matrices and Corresponding Representations, Ann. Funct. Anal., (2014) To Appear.
• [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.
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• [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.
• [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).

Identyfikatory

DOI

10.1515/spma-2015-0012