Browsing by Author "Katsoulis, Elias G."
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ItemCrossed products of operator algebras(2019) Katsoulis, Elias G.; Ramsey, ChristopherWe study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C°-algebras. We make contributions to the study of C°-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature. ItemCrossed products of operator algebras: applications of Takai duality(2018) Katsoulis, Elias G.; Ramsey, ChristopherIn this paper we continue our study of crossed products of approximately unital operator algebras begun with our monograph. The main objectives of the paper is to strengthen ties and establish new connections between our theory of crossed products and the well-established theory of semicrossed products. Using these connections we broaden our understanding for various topics of investigation in both theories, including semisimplicity and the structure of invariant ideals by the dual action. ItemThe Hyperrigidity of Tensor Algebras of C*-Correspondences(2019) Katsoulis, Elias G.; Ramsey, ChristopherGiven a C∗-correspondence X, we give necessary and sufficient conditions for the tensor algebra T+X to be hyperrigid. In the case where X is coming from a topological graph we obtain a complete characterization. ItemThe isomorphism problem for tensor algebras of multivariable dynamical systems(2022) Katsoulis, Elias G.; Ramsey, ChristopherWe resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not. 2014 (2014), 1289–1311 relating to work of Arveson, Acta Math. 118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom. 8 (2014), 771–787. ItemLimit algebras and integer-valued cocycles, revisited(2016) Katsoulis, Elias G.; Ramsey, ChristopherA triangular limit algebra A is isometrically isomorphic to the tensor algebra of a C*-correspondence if and only if its fundamental relation R(A) is a tree admitting a Z+0-valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining C*-correspondence and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho-Hopcroft-Ullman algorithm from computer aided graph theory. ItemThe non-selfadjoint approach to the Hao–Ng isomorphism(2019) Katsoulis, Elias G.; Ramsey, ChristopherIn an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of C∗-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C∗-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of C∗-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid C∗-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.