We provide examples of nonseparable spaces $X$ for which C(X) admits an isometric shift of type I... more We provide examples of nonseparable spaces $X$ for which C(X) admits an isometric shift of type I, which solves in the negative a problem proposed by Gutek {\em et al.} (J. Funct. Anal. {\bf 101} (1991), 97-119). We also give two independent methods for obtaining separable examples. The first one allows us in particular to construct examples with infinitely many nonhomeomorphic components in a subset of the Hilbert space $\ell^2$. The second one applies for instance to sequences adjoined to any n-dimensional compact manifold (for $n \ge 2$) or to the Sierpi\'nski curve. The combination of both techniques lead to different examples involving a convergent sequence adjoined to the Cantor set: one method for the case when the sequence converges to a point in the Cantor set, and the other one for the case when it converges outside.
It is shown that the existence of a biseparating map between a large class of spaces of vector-va... more It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps bet... more For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
It is shown that the existence of a biseparating map between a large class of spaces of vector-va... more It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps bet... more For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
It is shown that the existence of a biseparating map between a large class of spaces of vector-va... more It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps bet... more For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
We provide examples of nonseparable spaces $X$ for which C(X) admits an isometric shift of type I... more We provide examples of nonseparable spaces $X$ for which C(X) admits an isometric shift of type I, which solves in the negative a problem proposed by Gutek {\em et al.} (J. Funct. Anal. {\bf 101} (1991), 97-119). We also give two independent methods for obtaining separable examples. The first one allows us in particular to construct examples with infinitely many nonhomeomorphic components in a subset of the Hilbert space $\ell^2$. The second one applies for instance to sequences adjoined to any n-dimensional compact manifold (for $n \ge 2$) or to the Sierpi\'nski curve. The combination of both techniques lead to different examples involving a convergent sequence adjoined to the Cantor set: one method for the case when the sequence converges to a point in the Cantor set, and the other one for the case when it converges outside.
It is shown that the existence of a biseparating map between a large class of spaces of vector-va... more It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps bet... more For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
It is shown that the existence of a biseparating map between a large class of spaces of vector-va... more It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps bet... more For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
It is shown that the existence of a biseparating map between a large class of spaces of vector-va... more It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
For realcompact spaces X and Y we give a complete description of the linear biseparating maps bet... more For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
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