On Reflexivity of Certain Hyponormal Operators with Double Commutant Property

Sarason did pioneer work on the reflexivity and purpose of this paper is to discuss the reflexivity of different class of contractions. Among contractions it is now known that C11 contractions with finite defect indices, C.o contractions with unequal defect indices and C1. contractions with at least one finite defect indices are reflexive. More over the characterization of reflexive operators among co contractions and completely non unitary weak contractions with finite defect indices has been reduced to that of S (F), the compression of the shift on H2 ⊖ F H2, F is inner. The present work is mainly focused on the reflexivity of contractions whose characteristic function is constant. This class of operator include many other isometries, co-isometries and their direct sum. We shall also discuss the reflexivity of hyponormal contractions, reflexivity of C1. contractions and weak contractions. It is already known that normal operators isometries, quasinormal and sub-normal operators are reflexive. We partially generalize these results by showing that certain hyponormal operators with double commutant property are reflexive. In addition, reflexivity of operators which are direct sum of a unitary operator and C.o contractions with unequal defect indices,is proved Each of this kind of operator is reflexive and satisfies the double commutant property with some restrictions.

Characterizations of Double Commutant Property on B H

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

A new proof of the Tikuisis–White–Winter theorem

A trace on a {\mathrm{C}^{*}}-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear {\mathrm{C}^{*}}-algebra is hyperfinite, it is easy to see that traces on nuclear {\mathrm{C}^{*}}-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear {\mathrm{C}^{*}}-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of {\mathrm{C}^{*}}-algebras and, in particular, using a version of the Weyl–von Neumann Theorem due to Elliott and Kucerovsky.

Centralizers of the infinite symmetric group

International audience We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces. Nous étudions les algèbres du centralisateur du groupe symétrique infini $S_{\infty}$, passant en revue certaines approches et en introduisant de nouvelles. Notre travail est basé sur la relation du double commutant entre le groupe symétrique fini et les algèbres de partition; dans le cas de $S_{\infty}$, nous obtenons des algèbres du centralisateur contenues dans les algèbres de partition. Compte tenu de la théorie des fonctions symétriques en variables non commutatives, nous considérons les représentations de $S_{\infty}$ qui sont fidèles et contiennent les invariants; c’est-à-dire, les représentations non unitaires sur les espaces de suites.