Banach algebras generated by idempotents
The goal of this chapter is to provide a possible tool to study the invertibility of the local cosets which arise after localization of convolution type operators. The basic observation is that the local algebras in some cases are generated by a finite number of elements p which are idempotent in the sense that p 2=p. Under some additional conditions, it turns out that such algebras possess matrix-valued symbols. Thus, one can associate with every element of the algebra a matrix-valued function such that the element is invertible if and only if the associated function is invertible at every point. In this way, one gets an effective criterion for the invertibility of elements in local algebras.
KeywordsIdentity Element Banach Algebra Singular Integral Operator Algebra Homomorphism Local Algebra
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