Geodesic
A note on liftings of hermitian elements and unitaries
Glasgow mathematical journal, Tome 21 (1980) no. 2, pp. 183-185

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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(ifh)‖ = 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. 
Fong, C. K. A note on liftings of hermitian elements and unitaries. Glasgow mathematical journal, Tome 21 (1980) no. 2, pp. 183-185. doi: 10.1017/S0017089500004341
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[2] 2.Allen, G. D., Ward, J. D. and Legg, D. A., Hermitian liftings in Orlicz sequence spaces. To appear. Google Scholar

[3] 3.Bonsall, F. F. and Duncan, J., Complete normed algebras. Ergebnisse der Mathematik and ihrer Grenzgebiete, Band 80. (Springer-Verlag, 1973). Google Scholar | DOI

[4] 4.Sinclair, A. M., Eigenvalues in the boundary of the numerical range, Pacific J. Math. 35 (1970), 231–234. Google Scholar | DOI

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