Crossed Products of Operator Algebras

Abstract

Chapter 1. Introduction Chapter 2. Preliminaries 2.1. Generalities 2.2. C ˚-correspondences and tensor algebras 2.3. Crossed products of C ˚-algebras Chapter 3. Definitions and Fundamental Results Chapter 4. Maximal C ˚-covers, Iterated Crossed Products and Takai Duality Chapter 5. Crossed Products and the Dirichlet Property Chapter 6. Crossed Products and Semisimplicity 6.1. Crossed products by discrete abelian groups 6.2. Crossed products by compact abelian groups 6.3. More examples of crossed product Dirichlet algebras 6.4. Semicrossed products and semisimplicity Chapter 7. The Crossed Product as the Tensor Algebra of a C ˚correspondence. 7.1. Discrete groups 7.2. The general case of a locally compact group. 7.3. Hilbert C ˚-bimodules Chapter 8. Concluding Remarks and Open Problems Bibliography

Full Cuntz-Krieger dilations via non-commutative boundaries

“…The notions of the C * -envelope, also known as the non-commutative Shilov boundary, and the more delicate non-commutative Choquet boundary are very useful in operator algebras. A good instance of this appears in the recent work of Katsoulis and Ramsey, and Katsoulis on the Hao-Ng isomorphism problem [26,30], where non-self-adjoint algebras and their C * -envelopes play a prominent role.…”

“…Characterizing the C * -envelope of various operator structures was of use and intrigue to many authors, as can be seen for instance in [14,25,30]. In [29], Katsoulis and Kribs improve on the work in [35] and [22,Theorem 5.3] and show that the C * -envelope of a tensor algebra associated to a general C * -correspondence, is the Cuntz-Pimsner-Katsura algebra of the C * -correspondence.…”

Residually finite-dimensional operator algebras

“…First, we mention that the property of an operator algebra being RFD isn't preserved by crossed products with groups. This is an immediate consequence of the so-called Takai duality [28,Theorem 4.4]. We refer the interested reader to [28] for details on these topics.…”

Crossed products of operator systems

“…We show that the reduced and full enveloping crossed products agree when the group G is amenable. We turn to nuclearity detectors in Section §5, allowing us to give counterexamples to the first two problems of the monograph of Katsoulis and Ramsey [23]. Finally, we show in Section §6 that the operator algebra U(W 3,2 ) is hyperrigid in its C * -envelope, which shows that the identity…”

“…Problem 1 asks whether the identity C * env (A ⋊ C * u (A),α G) = C * env (A) ⋊ α G holds for all operator algebra dynamical systems. If the identity above was true for all operator algebraic dynamical systems, then the Hao-Ng isomorphism theorem would have a positive answer for all C * -correspondences and all locally compact groups [23,Chapter 7]. In a similar direction, Problem 2 asks whether the images of C c (G, A) in C * env (A) ⋊ G and C * u (A) ⋊ G are canonically completely isometrically isomorphic.…”