Trivial extensions of semigroups and semigroup $C^*$-algebras

E. V. Lipacheva

Ufa mathematical journal, Tome 14 (2022) no. 2, pp. 67-77

Résumé

The object of the study in the paper is reduced semigroup $C^*$-algebras for left cancellative semigroups. Such algebras are a very natural object because it is generated by isometric shift operators belonging to the image of the left regular representation of a left cancellative semigroup. These operators act in the Hilbert space consisting of all square summable complex-valued functions defined on a semigroup. We study the question on functoriality of involutive homomorphisms of semigroup $C^*$-algebras, that is, the existence of the canonical embedding of semigroup $C^*$-algebras induced by an embedding of corresponding semigroups. In order to do this, we investigate the reduced semigroup $C^*$-algebras associated with semigroups involved in constructing normal extensions of semigroups by groups. At the same time, in the paper we consider one of the simplest classes of extensions, namely, the class of so-called trivial extensions. It is shown that if a semigroup $L$ is a trivial extension of the semigroup $S$ by means of a group $G$, then there exists the embedding of the reduced semigroup $C^*$-algebra $C^*_r(S)$ into the $C^*$-algebra $C^*_r(L)$ which is induced by an embedding of the semigroup $S$ into the semigroup $L$.In the work we also introduce and study the structure of a Banach $C^*_r(S)$-module on the underlying space of the reduced semigroup $C^*$-algebra $C^*_r(L)$. To do this, we use a topological grading for the $C^*$-algebra $C^*_r(L)$ over the group $G$. In the case when a semigroup $L$ is a trivial extension of a semigroup $S$ by means of a finite group, we prove the existence of the structure of a free Banach module over the reduced semigroup $C^*$-algebra $C^*_r(S)$ on the underlying Banach space of the semigroup $C^*$-algebra $C^*_r(L)$.We give examples of extensions of semigroups and reduced semigroup $C^*$-algebras for a more complete characterization of the issues under consideration and for revealing connections with previous results.

Détail

Keywords: cancellative semigroup, normal extension of a semigroup, trivial extension of a semigroup, reduced semigroup $C^*$-algebra, embedding a semigroup $C^*$-algebra, Banach module, free module.

Bibliographie

[1] L.A. Coburn, “The $C^*$-algebra generated by an isometry”, Bull. Amer. Math. Soc., 73:5 (1967), 722–726

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

[3] R.G. Douglas, “On the $C^*$-algebra of a one-parameter semigroup of isometries”, Acta Math., 128:3 (1972), 143–151

[4] G.J. Murphy, “Ordered groups and Toeplitz algebras”, J. Oper. Theory, 18:2 (1987), 303–326

[5] G.J. Murphy, “Toeplitz operators and algebras”, Math. Z., 208:3 (1991), 355–362

[6] A. Nica., “$C^*$-algebras generated by isometries and Wiener –Hopf operators”, J. Operator Theory, 27:1 (1992), 17–52

[7] M. Laca, I. Raeburn, “Semigroup crossed products and the Toeplitz algebras of nonabelian groups”, J. Funct. Anal., 139:2 (1996), 415–440

[8] X. Li, “Semigroup $C^*$-algebras and amenability of semigroups”, J. Functional Analysis, 262:10 (2012), 4302–4340

[9] S.A. Grigoryan, T.A. Grigoryan, E.V. Lipacheva, A. S. Sitdikov, “$C^*$-Algebra generated by the path semigroup”, Lobachevskii J. of Math., 37:6 (2016), 740–748

[10] S.A. Grigoryan, E.V. Lipacheva, A.S. Sitdikov, “Nets of graded $C^*$-algebras over partially ordered sets”, St. Petersburg Math. J., 30:6 (2019), 901–915

[11] R.N. Gumerov, E.V. Lipacheva, “Topological Grading of Semigroup C*-Algebras”, Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 90:3 (2020), 44–55

[12] E.V. Lipacheva, “On graded semigroup $C^*$-algebras and Hilbert modules”, Proc. Steklov Inst. Math., 313 (2021), 120–130

[13] E.V. Lipacheva, “A Semigroup $C^*$-algebra which is a Free Banach Module”, Lobachevskii J. of Math., 42:10 (2021), 2386–2391

[14] B.V. Novikov, “Semigroup cohomologies: a survey”, Fund. Prikl. Matem., 7:1 (2001), 1–18 (in Russian)

[15] A.H. Clifford, “Extensions of semigroups”, Trans. Amer. Math. Soc., 68:2 (1950), 165–173

[16] L. Rédei, “Die Verallgemeinerung der Schreierschen”, Erweiterungstheorie Acta Sci. Math. Szeged, 14 (1952), 252–273

[17] L.M. Gluskin, I.L. Perepelitsyn, “Normal extensions of semi-groups”, Izv. VUZov. Matem., 1972, no. 12, 46–54 (in Russian)

[18] L.M. Gluskin, “Normal extensions of commutative semigroups”, Russian Math., 29:9 (Iz. VUZ)), 16–27

[19] I.S. Berdnikov, R.N. Gumerov, E.V. Lipacheva, “On the Stone-Cech Compactification Functor and the Normal Extensions of Monoids”, Lobachevskii J. of Math., 42:10 (2021), 2295–2305

[20] S.A. Grigoryan, R.N. Gumerov, E.V. Lipacheva, “On Extensions of Semigroups and Their Applications to Toeplitz Algebras”, Lobachevskii J. of Math., 40:12 (2019), 2052–2061

[21] R.N. Gumerov, “Normal extensions of semigroups and embeddings of semigroup $C^*$-algebras”, Trudy MFTI, 12:1 (2020), 74–82 (in Russian)

[22] E.V. Lipacheva, “Extensions of semigroups and morphisms of semigroup $C^*$-algebras”, Siberian Math. J., 62:1 (2021), 66–76

[23] R. Exel, Partial dynamical systems, Fell bundles and applications, Math. Surv. Monogr., 224, Amer. Math. Soc., Providence, RI, 2017

[24] E.S. Lyapin, Semigroups, Amer. Math. Soc., Providence, R.I., 1974

[25] A.Ya Helemskii, Banach and locally convex algebras, Clarendon Press, Oxford, 1993