Abstract
The note is concerned with inductive sequences of Toeplitz algebras. The Toeplitz algebra is the \(C^{*}\)-subalgebra in the algebra of all bounded linear operators. This subalgebra is generated by the right shift operator on the Hilbert space of all square summable complex-valued functions defined on the additive semigroup of non-negative integers. We study the inductive sequences of Toeplitz algebras whose bonding \(\ast\)-homomorphisms are defined by arbitrary sequences of natural numbers. The inductive limits of such sequences are the reduced semigroup \(C^{*}\)-algebras generated by representations for semigroups of non-negative rational numbers. We consider the limit \(\ast\)-endomorphisms of these inductive limits. Such an endomorphism is induced by a morphism between two copies of the same inductive sequence of Toeplitz algebras. We give the necessary and sufficient conditions for these endomorphisms to be \(\ast\)-automorphisms of \(C^{*}\)-algebras. These criteria are formulated in algebraic, number-theoretical and functional terms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
R. N. Gumerov, ‘‘On finite-sheeted covering mappings onto solenoids,’’ Proc. Am. Math. Soc. 133, 2771–2778 (2005).
R. N. Gumerov, ‘‘On the existence of means on solenoids,’’ Lobachevskii J. Math. 17, 43–46 (2005).
S. A. Grigoryan and R. N. Gumerov, ‘‘On the structure of finite coverings of compact connected groups,’’ Topol. Appl. 153, 3598–3614 (2006).
S. A. Grigoryan, R. N. Gumerov, and A. V. Kazantsev ‘‘Group structure in finite coverings of compact solenoidal groups,’’ Lobachevskii J. Math. 6, 39–46 (2000).
R. N. Gumerov, ‘‘Weierstrass polynomials and coverings of compact groups,’’ Sib. Math. J. 54, 243–246 (2013).
R. N. Gumerov, ‘‘Characters and coverings of compact groups,’’ Russ. Math. (Iz. VUZ) 58 (4), 7–13 (2014).
R. N. Gumerov, ‘‘Coverings of solenoids and automorphisms of semigroup C*-algebras,’’ Uch. Zap. Kazan. Univ., Ser.: Fiz.-Mat. Nauki 160, 275–286 (2018).
A. Ya. Helemskii, Banach and Locally Convex Algebras (Oxford Sci., Clarendon, New York, 1993).
L. A. Coburn, ‘‘The C*-algebra generated by an isometry,’’ Bull. Am. Math. Soc. 73, 722–726 (1967).
R. G. Douglas, ‘‘On the C*-algebra of a one-parameter semigroup of isometries,’’ Acta Math. 128, 143–152 (1972).
G. J. Murphy, ‘‘Ordered groups and Toeplitz algebras,’’ J. Oper. Theory 18, 303–326 (1987).
X. Li, ‘‘Semigroup C*-algebras,’’ arxiv: 1707.05940 (2019).
E. V. Lipacheva and K. H. Hovsepyan, ‘‘Automorphisms of some subalgebras of the Toeplitz algebra,’’ Sib. Math. J. 57, 525–531 (2016).
R. N. Gumerov, ‘‘Limit automorphisms of C*-algebras generated by isometric representations for semigroups of rationals,’’ Sib. Math. J. 59, 73–84 (2018).
R. N. Gumerov, E. V. Lipacheva, and T. A. Grigoryan, ‘‘On inductive limits for systems of C*-algebras,’’ Russ. Math. (Iz. VUZ) 62 (7), 68–73 (2018).
R. N. Gumerov, ‘‘Inductive limits for systems of Toeplitz algebras,’’ Lobachevskii J. Math. 40 (4), 469–478 (2019).
E. V. Lipacheva, ‘‘Embedding semigroup C*-algebras into inductive limits,’’ Lobachevskii J. Math. 40 (5), 667–675 (2019).
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). https://doi.org/10.1007/s10773-019-04048-0
M. Rordam, F. Larsen, and N. Lausten, An Introduction to \(K\)-Theory for \(C^{*}\)-Algebras, Vol. 49 of London Math. Soc. Student Texts (Cambridge Univ. Press, Cambridge, 2000).
G. J. Murphy, \(C^{*}\)-Algebras and Operator Theory (Academic, New York, 1990).
R. N. Gumerov, ‘‘On norms 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). http://china.iopscience.iop.org/article/10.1088/1757-899X/158/1/012042/pdf. Accessed 2019.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by S. A. Grigoryan)
Rights and permissions
About this article
Cite this article
Gumerov, R.N. Inductive Sequences of Toeplitz Algebras and Limit Automorphisms. Lobachevskii J Math 41, 637–643 (2020). https://doi.org/10.1134/S1995080220040125
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080220040125