Study on preserver problems on Banach alebras

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We give a structure theorem for isometries between the general linear groups of Banach algebras. As an application of it, we deduce the form of isometries between the general linear group of C*-algebras. We get a non-commutative Mazur-Ulam theorem and we describe the form of isometries between the full unitary group of a Hilbert space. We also get a result on the maps between certain commutative Banach algebras which preserve the norm non-symmetrically. We describe the form of the multiplicative isometries between F-algebras of holomorphic functions.