Abstract
Starting from an arbitrary endomorphism α of a unital C*-algebra A we construct in a canonical way a bigger algebra B and extend α onto B in such a way that α : B → B possess a unique non-degenerate transfer operator L: B → itB called complete transfer operator. The pair (B, α) is universal with respect to a suitable notion of a covariant representation and in general depends on a choice of an ideal in A.
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References
J. Cuntz and W. Krieger, “A class of C*-algebras and topological Markov chains,” Inventiones Math. 56, 251–268 (1980).
W. L. Paschke, “The crossed product of a C*-algebra by an endomorphism,” Proc. Amer. Math. Soc., No. 1, 113–118 (1980).
P. J. Stacey, “Crossed products of C*-algebras by *-endomorphisms,” J. Austral. Math. Soc. Ser. A 54, 204–212 (1993).
G. J. Murphy, “Crossed products of C*-algebras by endomorphisms,” Integral Equat. Oper. Theory 24, 298–319 (1996).
R. Exel, “Circle actions on C*-algebras, partial automorhisms and generalized Pimsner-Voiculescu exact sequence,” J. Funct. Analysis 122, 361–401 (1994).
R. Exel, “A new look at the crossed-product of a C*-algebra by an endomorphism,” Ergodic Theory Dynam. Systems 23, 1733–1750 (2003).
A. B. Antonevich, V. I. Bakhtin and A. V. Lebedev, “Crossed product of C*-algebra by an endomorphism, coefficient algebras and transfer operators,” Math. Sbornik 202 (9), 1253–1283 (2011).
B. K. Kwaśniewski, “Covariance algebra of a partial dynamical system,” Central European J. Math. 3 (4), 718–765 (2005).
V. I. Bakhtin and A. V. Lebedev, “When a C*-algebra is a coefficient algebra for a given endomorphism,” arXiv: math. OA/0502414 v1 19 Feb 2005.
B. K. Kwaśniewski, “On transfer operators for C*-dynamical systems,” Rocky Mountain J. Math. 42 (3), 919–938 (2012).
B. K. Kwaśniewski and A. V. Lebedev, “Crossed products by endomorphisms and reduction of relations in relative Cuntz-Pimsner algebras,” J. Funct. Anal. 264 (8), 1806–1847 (2013).
B. K. Kwaśniewski and A. V. Lebedev, “Reversible extensions of irreversible dynamical systems: C*-method,” Math. Sbornik 199 (11), 45–74 (2008).
B. K. Kwaśniewski, “C*-algebras associated with reversible extensions of logistic maps,” Math. Sbornik 203 (10), 1448–1489 (2012).
M. Laca, “From endomorphisms to automorphisms and back: dilations and full corners,” J. London Math. Soc. 61, 893–904 (2000).
A. V. Lebedev and A. Odzijewicz, “Extensions of C*-algebras by partial isometries,” Math. Sbornik 195 (7), 37–70 (2004).
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Kwaśniewski, B.K. Extensions of C*-dynamical systems to systems with complete transfer operators. Math Notes 98, 419–428 (2015). https://doi.org/10.1134/S0001434615090072
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DOI: https://doi.org/10.1134/S0001434615090072