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On Extensions of Semigroups and Their Applications to Toeplitz Algebras

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Abstract

The paper deals with the normal extensions of cancellative commutative semigroups and the Toeplitz algebras for those semigroups. By the Toeplitz algebra for a semigroup S one means the reduced semigroup C*-algebra C*r(S). We study the normal extensions of cancellative commutative semigroups by the additive group ℤn of integers modulo n. Moreover, we assume that such an extension is generated by one element. We present a general method for constructing normal extensions of semigroups which contain no non-trivial subgroups. The Grothendieck group for a given semigroup and the group of all integers are involved in this construction. Examples of such extensions for the additive semigroup of non-negative integers are given. A criterion for a normal extension generated by an element to be isomorphic to a numerical semigroup is given in number-theoretic terms. The results concerning the Toeplitz algebras are the following. For a cancellative commutative semigroup S and its normal extension L generated by one element, there exists a natural embedding the semigroup C*-algebra C*r(S) into C*r(L). The semigroup C*-algebra C*r(L) is topologically ℤn-graded. The results in the paper are announced without proofs.

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References

  1. L. A. Coburn, “The C*-algebra generated by an isometry,” Bull. Am. Math. Soc. 73, 722–726 (1967).

    Article  MathSciNet  Google Scholar 

  2. L. A. Coburn, “The C*-algebra generated by an isometry. II,” Trans. Am. Math. Soc. 137, 211–217 (1969).

    MathSciNet  MATH  Google Scholar 

  3. R. G. Douglas, “On the C*-algebra of a one-parameter semigroup of isometries,” Acta Math. 128, 143–152 (1972).

    Article  MathSciNet  Google Scholar 

  4. G. J. Murphy, “Ordered groups and Toeplitz algebras,” J. Oper. Theory 18, 303–326 (1987).

    MathSciNet  MATH  Google Scholar 

  5. G. J. Murphy, “Toeplitz operators and algebras,” Math. Z. 208, 355–362 (1991).

    Article  MathSciNet  Google Scholar 

  6. X. Li, “Semigroup C*-algebras,” https://arxiv.org/pdf/1707.05940.pdf. Accessed 2019.

  7. E. V. Lipacheva and K. H. Hovsepyan, “Automorphisms of some subalgebras of the Toeplitz algebra,” Sib. Math. J. 57, 525–531 (2016).

    Article  MathSciNet  Google Scholar 

  8. E. V. Lipacheva and K. H. Hovsepyan, “The structure of C*-subalgebras of the Toeplitz algebra fixed with respect to a finite group of automorphisms,” Russ. Math. (Iz. VUZ) 59(6), 10–17 (2015).

    Article  MathSciNet  Google Scholar 

  9. M. A. Aukhadiev, S. A. Grigoryan, and E. V. Lipacheva, “Operator approach to quantization of semigroups,” Sb. Math. 205, 319–342 (2014).

    Article  MathSciNet  Google Scholar 

  10. S. A. Grigoryan, T. A. Grigoryan, E. V. Lipacheva, and A. S. Sitdikov, “C*-algebra generated by the path semigroup,” Lobachevskii J. Math. 37(6), 740–748 (2016).

    Article  MathSciNet  Google Scholar 

  11. S. A. Grigoryan, E. V. Lipacheva, and A. S. Sitdikov, “Nets of graded C*-algebras over partially ordered sets,” Algebra Anal. 30(6), 1–19 (2018).

    MathSciNet  MATH  Google Scholar 

  12. E. V. Lipacheva, “On a class of graded ideals of semigroup C*-algebras,” Russ. Math. 62(10), 37–46 (2018).

    Article  MathSciNet  Google Scholar 

  13. R. N. Gumerov, “On morms of operators generated by shift transformations arising in signal and image processing on meshes supplied with semigroups structures,” IOP Conf. Ser.: Mater. Sci. Eng. 158, 012042 (2016). https://doi.org/10.1088/1757-899X/158/1/012042/pdf. Accessed 2019.

    Article  Google Scholar 

  14. R. N. Gumerov, “Limit automorphisms of C*-algebras generated by isometric representations for semigroups of rationals,” Sib. Math. J. 59, 73–84 (2018).

    Article  MathSciNet  Google Scholar 

  15. R. N. Gumerov, E. V. Lipacheva, and T. A. Grigoryan, “On inductive limits for systems of C*-algebras,” Russ. Math. (Iz. VUZov) 62(7), 68–73 (2018).

    Article  MathSciNet  Google Scholar 

  16. R. N. Gumerov, “Coverings of solenoids and automorphisms of semigroup C*-algebras,” Uch. Zap. Kazan. Univ., Ser.: Fiz.-Mat. Nauki 160, 275–286 (2018).

    MathSciNet  Google Scholar 

  17. R. N. Gumerov, “Inductive limits for systems of Toeplitz algebras,” Lobachevskii J. Math. 40(4), 469–478 (2019).

    Article  MathSciNet  Google Scholar 

  18. E. V. Lipacheva, “Embedding semigroup C*-algebras into inductive limits,” Lobachevskii J. Math. 40(4), 667–675 (2019).

    Article  MathSciNet  Google Scholar 

  19. R. N. Gumerov, E. V. Lipacheva, and T. A. Grigoryan, “On a topology and limits for inductive systems of C*-algebras,” Int. J. Theor. Phys. (2019, in press). https://doi.org/10.1007/s10773-019-04048-0

  20. S. Mac Lane, Homology, Classics in Mathematics (Springer, Berlin, Heidelberg, 1995).

    MATH  Google Scholar 

  21. A. Ya. Helemskii, Homology of Banach and Topological Algebras Vol. 41 of Mathematics and its Applications (Kluwer Academic, Springer, Netherlands, 1989).

    Google Scholar 

  22. A. H. Clifford, “Extensions of semigroups,” Trans. Am. Math. Soc. 68, 165–173 (1950).

    Article  MathSciNet  Google Scholar 

  23. L. Redei, “Die Verallgemeinerung der Schreierschen Erweiterungstheorie,” Acta Sci. Math. Szeged. 14, 252–273 (1952).

    MathSciNet  MATH  Google Scholar 

  24. L. M. Gluskin and I. L. Perepelicyn, “Normal extensions of semigroups,” Izv. Vyssh. Ucebn. Zaved., Mat. 12, 46–54 (1972).

    Google Scholar 

  25. L. M. Gluskin, “Normal extensions of commutative semigroups,” Sov. Math. (Iz. VUZ) 29(9), 16–27 (1985).

    MathSciNet  MATH  Google Scholar 

  26. B. V. Novikov, “Semigroup cohomologies: a survey,” Fundam. Prikl. Mat. 7(1), 1–18 (2001).

    MathSciNet  MATH  Google Scholar 

  27. J. M. G. Fell, “An extension of Mackey’s method to Banach *-algebraic bundles,” Mem. Am. Math. Soc. 90 (1969).

  28. R. Exel, “Amenability for Fell bundles,” J. Reine Angew. Math. 492, 41–73 (1997).

    MathSciNet  MATH  Google Scholar 

  29. R. Exel, Partial Dynamical Systems, Fell Bundles and Applications, Vol. 224 of Math. Surv. Monograph (Am. Math. Soc., Providence, RI, 2017)

    Book  Google Scholar 

  30. E. S. Lyapin, Semigroups (Fizmatgiz, Moscow, 1960) [in Russian].

    MATH  Google Scholar 

  31. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, No. 7, Part 1 of Mathematical Surveys and Monographs (Am. Math. Soc., Providence, RI, 1977).

    Google Scholar 

  32. J. C. Rosales and P. A. García-Sánchez, Numerical Semigroups, Vol. 20 of Developments in Mathematics (Springer, New York, Dordrecht, Heidelberg, London, 2009).

    Book  Google Scholar 

  33. I. Raeburn, “On graded C*-algebras,” Bull. Aust. Math. Soc. 97, 127–132 (2018).

    Article  MathSciNet  Google Scholar 

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Funding

The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project no. 1.13556.2019/13.1.

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Authors and Affiliations

  1. Chair of Higher Mathematics, Kazan Power Engineering University, Kazan, 420066, Russia

    S. A. Grigoryan & E. V. Lipacheva

  2. Chair of Mathematical Analysis, Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan, 420008, Russia

    R. N. Gumerov

Authors
  1. S. A. Grigoryan
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  2. R. N. Gumerov
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  3. E. V. Lipacheva
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Correspondence to S. A. Grigoryan, R. N. Gumerov or E. V. Lipacheva.

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Submitted by A. M. Elizarov

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Grigoryan, S.A., Gumerov, R.N. & Lipacheva, E.V. On Extensions of Semigroups and Their Applications to Toeplitz Algebras. Lobachevskii J Math 40, 2052–2061 (2019). https://doi.org/10.1134/S1995080219120059

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  • DOI: https://doi.org/10.1134/S1995080219120059

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