Crossed product of a $C^*$-algebra by an endomorphism, coefficient algebras and transfer operators
Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1253-1283 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper presents a construction of the crossed product of a $C^*$-algebra by an endomorphism generated by partial isometry. Bibliography: 26 titles.
Keywords: $C^*$-algebra, transfer operator, crossed product.
Mots-clés : endomorphism
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A. B. Antonevich; V. I. Bakhtin; A. V. Lebedev. Crossed product of a $C^*$-algebra by an endomorphism, coefficient algebras and transfer operators. Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1253-1283. http://geodesic.mathdoc.fr/item/SM_2011_202_9_a0/

[1] J. Cuntz, “Simple $C^*$-algebra generated by isometries”, Comm. Math. Phys., 57:2 (1977), 173–185 | DOI | MR | Zbl

[2] J. Cuntz, W. Krieger, “A class of $C^*$-algebras and topological Markov chains”, Invent. Math., 56:3 (1980), 251–268 | DOI | MR | Zbl

[3] W. L. Paschke, “The crossed product of a $C^*$-algebra by an endomorphism”, Proc. Amer. Math. Soc., 80:1 (1980), 113–118 | DOI | MR | Zbl

[4] P. J. Stacey, “Crossed products of $C^*$-algebras by $^*$-endomorphisms”, J. Austral. Math. Soc. Ser. A, 54:2 (1993), 204–212 | DOI | MR | Zbl

[5] G. J. Murphy, “Crossed products of $C^*$-algebras by endomorphisms”, Integral Equations Operator Theory, 24:3 (1996), 298–319 | DOI | MR | Zbl

[6] R. Exel, “A new look at the crossed-product of a $C^*$-algebra by an endomorphism”, Ergodic Theory Dynam. Systems, 23:6 (2003), 1733–1750 | DOI | MR | Zbl

[7] B. K. Kwaśniewski, “Covariance algebra of a partial dynamical system”, Cent. Eur. J. Math., 3:4 (2005), 718–765 | DOI | MR | Zbl

[8] R. Exel, “Interactions”, J. Funct. Anal., 244:1 (2007), 26–62 | DOI | MR | Zbl

[9] N. Brownlowe, I. Raeburn, “Exel's crossed product and relative Cuntz–Pimsner algebras”, Math. Proc. Cambridge Philos. Soc., 141:3 (2006), 497–508 | DOI | MR | Zbl

[10] A. V. Lebedev, A. Odzijewicz, “Extensions of $C^*$-algebras by partial isometries”, Sb. Math., 195:7 (2004), 951–982 | DOI | MR | Zbl

[11] V. I. Bakhtin, A. V. Lebedev, When a $C^*$-algebra is a coefficient algebra for a given endomorphism, arXiv: math.OA/0502414

[12] A. B. Antonevich, V. I. Bakhtin, A. V. Lebedev, Crossed product of a $C^*$-algebra by an endomorphism, coefficient algebras and transfer operators, arXiv: math.OA/0502415

[13] D. P. O'Donovan, “Weighted shifts and covariance algebras”, Trans. Amer. Math. Soc., 208 (1975), 1–25 | DOI | MR | Zbl

[14] A. V. Lebedev, On certain $C^*$-methods that are used while investigating algebras associated with automorphisms and endomorphisms, VINITI, No 5351-B87 (Russian), 1987

[15] A. Antonevich, A. Lebedev, Functional differential equations: I. $C^*$-theory, Pitman Monogr. Surveys Pure Appl. Math., 70, Longman, Harlow, 1994 | MR | Zbl

[16] S. Boyd, N. Keswani, I. Raeburn, “Faithful representations of crossed products by endomorphisms”, Proc. Amer. Math. Soc., 118:2 (1993), 427–436 | DOI | MR | Zbl

[17] S. Adji, M. Laca, M. Nilsen, I. Raeburn, “Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups”, Proc. Amer. Math. Soc., 122:4 (1994), 1133–1141 | DOI | MR | Zbl

[18] R. Exel, “Amenability for Fell bundles”, J. Reine Angew. Math., 492 (1997), 41–73 | DOI | MR | Zbl

[19] A. Antonevich, M. Belousov, A. Lebedev, Functional differential equations. II. $C^*$-applications, Pitman Monogr. Surveys Pure Appl. Math., 94, Part 1. Equations with continuous coefficients, Longman, Harlow, 1998 | MR | Zbl

[20] A. Antonevich, M. Belousov, A. Lebedev, Functional differential equations. II. $C^*$-applications, Pitman Monogr. Surveys Pure Appl. Math., 95, Part 2. Equations with discontinuous coefficients and boundary value problems, Longman, Harlow, 1998 | MR | Zbl

[21] R. Exel, “Circle actions on $C^*$-algebras, partial automorphisms, and a generalized Pimsner–Voiculescu exact sequence”, J. Funct. Anal., 122:2 (1994), 361–401 | DOI | MR | Zbl

[22] K. McClanahan, “$K$-theory for partial crossed products by discrete groups”, J. Funct. Anal., 130:1 (1995), 77–117 | DOI | MR | Zbl

[23] A. V. Lebedev, “Topologically free partial actions and faithful representations of crossed products”, Funct. Anal. Appl., 39:3 (2005), 207–214 | DOI | MR | Zbl

[24] P. R. Halmos, L. J. Wallen, “Powers of partial isometries”, J. Math. Mech., 19:8 (1970), 657–663 | DOI | MR | Zbl

[25] B. K. Kwaśniewski, A. V. Lebedev, “Reversible extensions of irreversible dynamical systems: $C^*$-method”, Sb. Math., 199:11 (2008), 1621–1648 | DOI | MR | Zbl

[26] J. Lindiarni, I. Raeburn, “Partial-isometric crossed products by semigroups of endomorphisms”, J. Operator Theory, 52:1 (2004), 61–87 | MR | Zbl