1. A complex Banach algebra A is a compact (weakly compact) algebra if its left and right regular representations consist of compact (weakly compact) operators. Let E be any subset of A and denote by Ei and Er the left and right annihilators of E. A is an annihilator algebra if A¡= (0) —Ar, Ir^{fS) for each proper closed left ideal / and Ji t¿ (0) for each proper closed right ideal /. In [6, Th...

We investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We give the operator that is semi-compact, but it is not bweakly. W...

It is known that the classical Hilbert–Schmidt theorem can be generalized to the case of compact operators in Hilbert A-modules H∗ A over a W ∗-algebra of finite type, i.e. compact operators in H∗ A under slight restrictions can be diagonalized over A. We show that if B is a weakly dense C∗-subalgebra of real rank zero in A with some additional property then the natural extension of a compact o...

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.

Introduction. We let G denote an infinite compact group, and G its dual. We use the notation of our book [3, Chapters 7 and 8]. Recall that -4(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and J2f°°(G) denotes its dual space under the pairing ( / , # ) , ( / G A (G), 0 Ç if°°(G)). Further, note if°°(G) is identified with the G*-algebra of bounded operators on L(...

Using the Schreier families (Sξ)ξ<ω1 , we define subclasses of weakly compact and Rosenthal operators between two Banach spaces. These subclasses give rise to ordinal ranks defined on each ideal. We prove several results concerning the analytic properties of this rank and give examples of spaces on which the ranks are bounded and unbounded.