On Restricted Perturbations in Inverse Images and a Description of Normalizer Algebras in C*-Algebras

Abstract

Let E and F be Banach spaces and T a bounded linear map from E into F. Using a certain perturbation function ƒ(·, T) we find a useful sufficient criterion that T maps the closed unit ball onto a closed set. Applying this to the quotient map from a C*-algebra A onto its quotient Banach space A/(L + R) by the sum L + R of closed left and right ideals L and R of A we obtain that the closed unit ball of A maps onto the closed unit ball of A/(L + R). This gives an independent description of the images of the right normalizer algebra Nr(D) and the normalizer algebra N(D) = Nl(D) ∩ Nr(D) of a hereditary C*-subalgebra D of A by the quotient map from A onto A/(cl(AD) + cl(DA)). We use this to prove a necessary and sufficient criterion for a C*-algebra B to be a C*-quotient-algebra of a C*-subalgebra of a C*-algebra A. The latter criterion will be applied in the situation where A equals the CAR-algebra in some forthcoming papers (cf. Section 6 for more details).