Geodesic
On a class of graded ideals of semigroup $C^*$-algebras
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2018), pp. 43-54.

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We present general results about graded $C^*$-algebras and continue the previously initiated research of the $C^*$-algebra generated by the left regular representation of an Abelian semigroup. We study the invariant ideals of this $C^*$-algebra invariant with respect to the representation of a compact group $G$ in the automorphism group of this algebra. We prove that the invariance of the ideal is equivalent to the fact that this ideal is graded $C^*$-algebra, that there is a maximum of all invariant ideals, and it is the commutator ideal. Separately we study a class of graded primitive ideals generated by a single projector.
Keywords: $C^*$-algebra, graded $C^*$-algebra, semigroup, left regular representation, invariant subspace, representation in the automorphism group, invariant ideal, commutator ideal. 
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     title = {On a class of graded ideals of semigroup $C^*$-algebras},
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E. V. Lipacheva. On a class of graded ideals of semigroup $C^*$-algebras. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2018), pp. 43-54. http://geodesic.mathdoc.fr/item/IVM_2018_10_a4/

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

[2] Douglas R. G., “On the $C^*$-algebra of a one-parameter semigroup of isometries”, Acta Math., 128 (1972), 143–152 | DOI | MR

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

[4] Murphy G. J., “Crossed products of $C^*$-algebras by semigroups of automorphisms”, Proc. London Math. Soc., 68:2 (1994), 423–448 | DOI | MR | Zbl

[5] Murphy G. J., “Ordered groups and Toeplitz algebras”, J. Operator Theory, 18:2 (1987), 303–326 | MR | Zbl

[6] Murphy G. J., “Invariant ideals in Toeplitz algebras”, Integr. Equ. Operator Theory, 35:1 (1999), 118–121 | DOI | MR | Zbl

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

[8] Lipacheva E. V., Ovsepyan K. G., “Struktura podalgebr algebry Teplitsa, nepodvizhnykh otnositelno konechnoi gruppy avtomorfizmov”, Izv. vuzov. Matem., 2015, no. 6, 14–23

[9] Lipacheva E. V., Ovsepyan K. G., “Struktura invariantnykh idealov nekotorykh podalgebr algebry Teplitsa”, Izv. NAN Armenii. Matem., 50:2 (2015), 38–52

[10] Lipacheva E. V., Ovsepyan K. G., “Avtomorfizmy nekotorykh podalgebr algebry Teplitsa”, Sib. matem. zhurn., 57:3 (2016), 666–674 | Zbl

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

[12] Rieffel M. A., “Induced representations of $C^*$-algebras”, Advan. in Math., 13 (1974), 176–257 | DOI | MR | Zbl

[13] Stratilǎ S., Voiculescu D., Representations of $AF$-algebras and of the group $U(\infty)$, Lect. Notes in Math., 486, Springer-Verlag, 1975 | DOI | MR

[14] Grigoryan S. A., Salakhutdinov A. F., “$C^*$-algebry, porozhdennye polugruppami s sokrascheniem”, Sib. matem. zhurn., 51:1 (2010), 16–25 | Zbl

[15] Aukhadiev M. A., Grigoryan S. A., Lipacheva E. V., “Operatornyi podkhod k kvantovaniyu polugrupp”, Matem. sb., 205:3 (2014), 15–40 | DOI | Zbl

[16] Grigoryan S. A., Lipacheva E. V., $C^*$-algebry, porozhdennye abelevymi polugruppami, KGEU, Kazan, 2016