ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES*

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doi:10.1007/s10473-023-0603-8

Abstract: In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry $f:S_{r}(X)\rightarrow S_{r}(X)$ , where $X$ is a finite-dimensional non-Archimedean normed space and $S_{r}(X)$ is a sphere with radius $r\in \|X\|$ , is surjective if and only if $\mathbb{K}$ is spherically complete and $k$ is finite. Moreover, we prove that if $X$ and $Y$ are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with $|2|=1$ , any phase-isometry $f:X\rightarrow Y$ is phase equivalent to an isometric operator.

Key words: non-Archimedean normed spaces, isometry extension, Wigner's theorem

CLC Number:

46B03

Ruidong WANG, Wenting YAO. ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES^*[J].Acta mathematica scientia,Series B, 2023, 43(6): 2377-2386.

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