engleski

Approximately orthogonality preserving mappings on C*-modules

We study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product $C^*$-modules. In particular, if $V$ and $W$ are inner product $C^*$-modules over the $C^*$-algebra $\A, $ any $\A$-linear scalar multiple of an isometry is an $\A$-linear orthogonality preserving mapping. The converse does not hold in general, but it holds if $\A$ contains $\K(\H)$ (the $C^*$-algebra of all compact operators on a Hilbert space $\H$). Furthermore, we give the estimate of $\Vert \langle Tx, Ty \rangle - \Vert T \Vert^2 \langle x, y \rangle \Vert$ for an $\A$-linear approximately orthogonality preserving mapping $T : V \to W$ when $V$ and $W$ are inner product $C^*$-modules over a $C^*$-algebra containing $\K(\H)$. In the case $\A=\K(\H)$ and $V, $ $W$ are Hilbert, we also prove that an $\A$-linear approximately orthogonality preserving mapping can be approximated by an $\A$-linear orthogonality preserving mapping.

Hilbert C*-module ; approximate orthogonality ; orthogonality preserving mapping ; stability

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