Ideals and hereditary subalgebras in operator algebras

Quasitriangular operator algebras

“…This is an extension of a result of Voiculescu which asserts that a separable C*-algebra is QD iff it forms a QD family of operators in some concrete Hilbert space. Since the "third isomorphism theorem" fails in its most general form for operator algebras [2], the QT version of this result for operator algebras satisfying an extra condition (here called essentially embedded, requiring that the quotient map onto the Calkin algebra induces a complete isometry on the given operator algebra; see the next section for details and examples).…”