This work introduces the concept of an M-complete approximate identity (M-cai) for a given operator subspace X of an operator space Y. M-cai's generalize central approximate identities in ideals in C *-algebras, for it is proved that if X admits an M-cai in Y , then X is a complete M-ideal in Y. It is proved, using " special " M-cai's, that if J is a nuclear ideal in a C *-algebra A, then J is ...

It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for r...

We give suucient conditions to approximate the \nonessential" spectrum of a bounded operator L acting on a Banach space B by part of the spectra of a sequence of compact (or nite rank) operators L j = (Id ? j)L(Id ? j), where Id ? j is a suitable family of uniformly bounded operators which approach the identity. (By nonessential spectrum we mean here all the spectrum outside of the disc of radi...

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the larger algebra. Based on this result, we study in detail the existence and properties of representations of the various centralizer algebras of A which are comp...

We prove that a homogeneous Banach space B on the unit circle T can be embedded as a closed subspace of a dual space ΞB contained in the space of bounded Borel measures on T in such a way that the map B → ΞB defines a bijective correspondence between the class of homogeneous Banach spaces on T and the class of prehomogeneous Banach spaces on T. We apply our results to show that the algebra of a...

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the larger algebra. Based on this result, we study in detail the existence and properties of representations of the various centralizer algebras of A which are comp...