Baruch Solel

Tensor Algebras, Induced Representations, and the Wold Decomposition

Our objective in this sequel to (18) is to develop extensions, to representations of tensor algebras over C � -correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decompo- sition theorem and Beurlings theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

On the Morita Equivalence of Tensor Algebras

Our objective is two fold. First, we want to develop a notion of Morita equivalence for C-correspondences that guarantees that if two C-correspondences E and F are Morita equivalent, then the tensor algebras of E and F, TaOEU and TaOFU, are strongly Morita equivalent in the sense of [8], the Toeplitz algebras, TOEU and TOFU, are strongly Morita equivalent in the sense of Rieffel [32] (see [33] also), and the Cuntz‐Pimsner algebras [28], OOEU and OOFU, are strongly Morita equivalent in the same sense. We were inspired by earlier work of Curto, Williams and the first author in [15] and Combes [14]. These papers investigate conditions implying that the crossed product of two C-dynamical systems are strongly Morita equivalent, given that the coefficient algebras are strongly Morita equivalent. Abadie, Eilers, and Exel [1] were similarly inspired and developed a theory of Morita equivalence for special C-correspondences. Our analysis extends theirs but our arguments are different. (See Example 2.5 and Remarks 3.3 and 3.6 for a comparison of their results and ours.) Second, we want to investigate the converse implication. That is, we study the following question: if TaOEU and TaOFU are strongly Morita equivalent in the sense of [8], when are E and F Morita equivalent in the sense we define? For this converse direction, we focus on the tensor algebras rather than the Toeplitz and Cuntz‐Pimsner algebras because very little can be said in the self-adjoint setting. Here, we were inspired by Arveson’s paper [2] in which it is proved that two ergodic measure-preserving transformations are conjugate if and only if certain non-self-adjoint crossed products associated with the transformations are algebraically isomorphic. (See [3] also.) His analysis was further extended by K. Saito and the first author to cover non-self-adjoint crossed products built from automorphisms of general von Neumann algebras. They showed that non-self-adjoint crossed products built from properly outer automorphisms of von Neumann algebras are isomorphic if and only if the automorphisms are conjugate. See [22] and [23]. Our principal result in the converse direction, Theorem 7.2, asserts that if E and F are two aperiodic correspondences, and if TaOEU is strongly Morita equivalent to TaOFU in the sense of [8], then E and F are Morita equivalent. As we shall see in § 5, the notion of aperiodicity is an exact analogue for correspondences of the concept of ‘free action’ in ergodic theory. Free action was a central hypothesis in Arveson’s

QUANTUM MARKOV PROCESSES (CORRESPONDENCES AND DILATIONS)

We study the structure of quantum Markov Processes from the point of view of product systems and their representations.

You can see the arrows in a quiver operator algebra

We prove that two quiver operator algebras can be isometrically isomorphic only if the quivers ( = directed graphs) are isomorphic. We also show how the graph can be recovered from certain representations of the algebra.

Coordinates for triangular operator algebras

On demontre un theoreme de structure pour des algebres sous diagonales contenant une sous algebre de Cartan. On etudie les isomorphismes pour ces algebres

An Algebraic Characterization of Boundary Representations

We show that boundary representations of an operator algebra may be characterized as those (irreducible) completely contractive representations that determine Hilbert modules that are simultaneously orthogonally projective and orthogonally injective. As a corollary, we conclude that if an operator algebra is an admissable subalgebra of its C * —envelope, in the sense of Arveson, then it has a completely isometric representation such that the associated Hilbert module is simultaneously orthogonally projective and orthogonally injective.

REGULAR DILATIONS OF REPRESENTATIONS OF PRODUCT SYSTEMS

We study completely contractive representations of product systems

The Curvature and Index of Completely Positive Maps

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Isometries of CSL algebras

of correspondences over the semigroup

QUANTUM MARKOV SEMIGROUPS: PRODUCT SYSTEMS AND SUBORDINATION

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