In this paper, we first give a description of a surjective unit-preserving real-linear uniform isometry $ T : A longrightarrow B$,  where $ A $ and $ B $ are complex function spaces on compact Hausdorff spaces $ X $ and $ Y $, respectively, whenever ${rm ER}left (A, Xright ) = {rm Ch}left (A, Xright )$ and ${rm ER}left (B, Yright ) = {rm Ch}left (B, Yright )$. Next, we give a description of $ T...

We give a sufficient condition on complex Banach space for the homogeneous extension of every surjective isometry from unit sphere it onto another one to be real-linear. The is satisfied by uniform algebras.

We prove that a surjective isometry between the unit spheres of two uniform algebras is extended to real-linear algebras. It provides first positive solution Tingley's problem on Banach space, without being Hilbert consisting analytic functions.

in this paper, we introduce the concepts of $2$-isometry, collinearity, $2$%-lipschitz mapping in $2$-fuzzy $2$-normed linear spaces. also, we give anew generalization of the mazur-ulam theorem when $x$ is a $2$-fuzzy $2$%-normed linear space or $im (x)$ is a fuzzy $2$-normed linear space, thatis, the mazur-ulam theorem holds, when the $2$-isometry mapped to a $2$%-fuzzy $2$-normed linear space...

Every ε-isometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces.

In this paper, we introduce the concepts of $2$-isometry, collinearity, $2$%-Lipschitz mapping in $2$-fuzzy $2$-normed linear spaces. Also, we give anew generalization of the Mazur-Ulam theorem when $X$ is a $2$-fuzzy $2$%-normed linear space or $Im (X)$ is a fuzzy $2$-normed linear space, thatis, the Mazur-Ulam theorem holds, when the $2$-isometry mapped to a $2$%-fuzzy $2$-normed linear space...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

Let X and Y be real normed spaces f : ? a surjective mapping. Then satisfies { ? ( x ) + y , ? } = ? if only is phase equivalent to linear isometry, that is, ? U where isometry 1 . This Wigner's type result for spaces.

It is shown that for many finite dimensional normed vector spaces V over C, a linear projection P : V → V will have nice structure if P + λ(I − P ) is an isometry for some complex unit not equal to one. From these results, one can readily determine the structure of bicircular projections, i.e., those linear projections P such that P + μ(I − P ) is a an isometry for every complex unit μ. The key...

We show that if T is an isometry (as metric spaces) from an open subgroup of the invertible group A of a unital Banach algebra A onto an open subgroup of the invertible group B of a unital Banach algebra B, then T is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear...