A note on liftings of hermitian elements and unitaries

Extract

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u⁻¹∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.