### Linear isometries of finite codimensions on Banach algebras of holomorphic functions.

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We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T{\left(1\right)}^{-1}T$ is an isometrical group isomorphism. In particular, $T{\left(1\right)}^{-1}T$ extends to an isometrical real algebra isomorphism from A onto B.

We give a complete description of the structure of surjective isometries between the unitary groups of unital C*-algebras. While any surjective isometry between the unitary groups of von Neumann algebras can be extended to a real-linear Jordan *-isomorphism between the relevant von Neumann algebras, this is not the case for general unital C*-algebras. We show that the unitary groups of two C*-algebras are isomorphic as metric groups if and only if the C*-algebras are isomorphic in the sense that...

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = $\overline{{f}^{\circ}\overline{\varphi}}$ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = $({\lambda}_{1}{z}_{{i}_{1}},...,{\lambda}_{n}{z}_{{i}_{n}})$ for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from...

We describe the general form of isometries between uniformly closed function algebras on locally compact Hausdorff spaces in a continuation of the study by Miura. We can actually obtain the form on the Shilov boundary, rather than just on the Choquet boundary. We also give an example showing that the form cannot be extended to the whole maximal ideal space.

We describe all surjective isometries between open subgroups of the groups of invertible elements in unital C*-algebras. As a consequence the two C*-algebras are Jordan *-isomorphic if and only if the groups of invertible elements in those C*-algebras are isometric as metric spaces.

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $$\widehat{T\left(a\right)}\left(y\right)=\left\{\begin{array}{c}\widehat{T\left(e\right)}\left(y\right)\widehat{a}\left(\phi \left(y\right)\right)y\in K\\ \widehat{T\left(e\right)}\left(y\right)\overline{\widehat{a}\left(\phi \left(y\right)\right)}y\in {M}_{\mathcal{B}}\setminus K\end{array}\right.$$ for all a ∈ A, where e is unit element of A. If, in addition, $$\widehat{T\left(e\right)}=1$$ and $$\widehat{T\left(ie\right)}=i$$ on M B, then T is an algebra isomorphism.

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