Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1056.46032
Araujo, Jesús; Jarosz, Krzysztof
Automatic continuity of biseparating maps.
(English)
[J] Stud. Math. 155, No. 3, 231-239 (2003). ISSN 0039-3223; ISSN 1730-6337/e

Let $X$, $Y$ be realcompact spaces, $E$ and $F$ normed spaces, $C(X,E)$ the set of all continuous $E$-valued functions on $X$ and $C_b(X,E)$ the space of all bounded continuous functions from $X$ into $E$. A linear map $T: C(X,E) \to C(Y,F)$ is called separating if $\Vert f(x)\Vert \ \Vert g(x)\Vert =0$ for every $x\in E$ implies that $\Vert (Tf)(y)\Vert \ \Vert (Tg)(y)\Vert =0$ for every $y\in Y$, and biseparating if $T^{-1}$ exists and is separating as well. One result of the paper is that every linear biseparating map $T:C_b(X,E)\to C_b(Y,F)$ is continuous provided that $Y$ has no isolated points. Another result tells us that if additionally $E$ and $F$ are infinite-dimensional, then a linear biseparating map $T: C(X,E)\to C(Y,F)$ is continuous if the interior of the set of $P$-points of $Y$ is empty. This is the best possible result.
[Raymond Mortini (Metz)]
MSC 2000:
*46E40 Spaces of vector-valued functions
47B33 Composition operators
46H40 Automatic continuity
47B38 Operators on function spaces
46E25 Rings and algebras of functions with smoothness properties

Keywords: automatic continuity of biseparating maps; realcompact spaces; P-points

Highlights
Master Server