Duality and operator algebras II: operator algebras as Banach algebras

Duality and Operator Algebras

We investigate some subtle and interesting phenomena in the du-ality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are always weak *-continuous on dual operator spaces. For example , if X is a subspace of a C *-algebra A, and if a ∈ A satisfies aX ⊂ X and a * X ⊂ X, and if X i...

Banach Algebras of Operator Sequences

During the last decades it turned out to be fruitful to apply certain Banach algebra techniques in the theory of approximation of operators by matrix sequences. Here we discuss the case of operator sequences (acting on infinite dimensional Banach spaces and which do not necessarily converge strongly) and we derive analogous results concerning the stability and Fredholm properties of such sequen...

m at h . O A ] 2 0 A ug 2 00 4 DUALITY AND OPERATOR ALGEBRAS II : OPERATOR ALGEBRAS AS

We answer, by counterexample, several open questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. We also answer a natural question about automatic w*-continuity arising in the preceding pape...

Operator Algebras

Notice that the left-hand side of the third equation is the sum of the left-hand sides of the first two. As a result, no solution to the system exists unless a + b = c. But if a + b = c, then any solution of the first two equations is also a solution of the third; and in any linear system involving more unknowns than equations, solutions, when they exist, are never unique. In the present case, ...

Operator Theory and Operator Algebras