Abstract
We consider the reduced semigroup \(C^{*}\)-algebras for monoids with the cancellation property. If there exists a surjective semigroup homomorphism from a monoid onto a group then the corresponding semigroup \(C^{*}\)-algebra can be endowed with the structure of a Banach module over its \(C^{*}\)-subalgebra. For a such monoid, we give conditions under which this Banach module is free.
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REFERENCES
L. A. Coburn, ‘‘The \(C^{*}\)-algebra generated by an isometry,’’ Bull. Am. Math. Soc. 73, 722–726 (1967).
L. A. Coburn, ‘‘The \(C^{*}\)-algebra generated by an isometry. II,’’ Trans. Am. Math. Soc. 137, 211–217 (1969).
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).
G. J. Murphy, ‘‘Toeplitz operators and algebras,’’ Math. Z. 208, 355–362 (1991).
X. Li, ‘‘Semigroup \(C^{*}\)-algebras,’’ in Operator Algebras and Applications, The Abel Symposium 2015, Ed. by T. M. Carlsen, N. S. Larsen, S. Neshveyev, and Ch. Skau (Springer, Cham, 2016), p. 191.
M. A. Aukhadiev, S. A. Grigoryan, and E. V. Lipacheva, ‘‘A compact quantum semigroup generated by an isometry,’’ Russ. Math. (Iz. VUZ) 55, 78 (2011).
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).
E. V. Lipacheva and K. H. Hovsepyan, ‘‘The structure of invariant ideals of some subalebras of Toeplitz algebra,’’ J. Contemp. Math. Anal. 50 (2), 70–79 (2015).
S. A. Grigorian and E. V. Lipacheva, ‘‘On the structure of \(C^{*}\)-algebras generated by representations of an elementary inverse semigroup,’’ Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 158, 180–193 (2016).
S. A. Grigoryan, T. A. Grigoryan, E. V. Lipacheva, and A. S. Sitdikov, ‘‘\(C^{*}\)-algebra generated by the path semigroup,’’ Lobachevskii J. Math. 37, 740–748 (2016).
E. V. Lipacheva, ‘‘On a class of graded ideals of semigroup C*-algebras,’’ Russ. Math. (Iz. VUZ) 62 (10), 37–46 (2018).
E. V. Lipacheva, ‘‘Embedding semigroup C*-algebras into inductive limits,’’ Lobachevskii J. Math. 40, 667–675 (2019).
S. A. Grigoryan, E. V. Lipacheva, and A. S. Sitdikov, ‘‘Nets of graded \(C^{*}\)-algebras over partially ordered sets,’’ SPb. Math. J. 30, 901–915 (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. 60, 499–511 (2021).
S. A. Grigoryan, R. N. Gumerov, and E. V. Lipacheva, ‘‘On extensions of semigroups and their applications to Toeplitz algebras,’’ Lobachevskii J. Math. 40, 2052–2061 (2019).
R. N. Gumerov and E. V. Lipacheva, ‘‘Topological grading of semigroup C*-algebras,’’ Herald of the Bauman Moscow State Technical University, Series Natural Sciences 90 (3), 44–55 (2020).
E. V. Lipacheva, ‘‘On graded semigroup \(C^{*}\)-algebras and Hilbert modules,’’ Proc. Steklov Inst. Math. 313, 120–130 (2021).
R. Exel, Partial Dynamical Systems, Fell Bundles and Applications, vol. 224 of Math. Surv. Monograph (Am. Math. Soc., Providence, RI, 2017).
A. Ya. Helemskii, Banach and Locally Convex Algebras (Oxford Science, Clarendon, New York, 1993).
A. Ya. Helemskii, Lectures and Exercises in Functional Analysis (AMS, Providence, RI, 2006).
ACKNOWLEDGMENTS
The author thanks Professor S.A. Grigoryan for drawing her attention to the subject under consideration.
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Lipacheva, E.V. A Semigroup \(\boldsymbol{C}^{\mathbf{*}}\)-Algebra Which Is a Free Banach Module. Lobachevskii J Math 42, 2386–2391 (2021). https://doi.org/10.1134/S1995080221100152
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DOI: https://doi.org/10.1134/S1995080221100152