Phase-isometries between two $$\ell ^p(\Gamma , H)$$-type spaces

Abstract

Let $$\Gamma ,\Delta $$ be nonempty index sets, and let H, K be inner product spaces. We prove that for $$p\ge 1$$ any surjective phase-isometry between $$\ell ^p(\Gamma ,H)$$ and $$\ell ^p(\Delta , K)$$ is a plus–minus linear isometry. This can be considered as an extension of Wigner’s theorem for real $$\ell ^p(\Gamma , H)$$ -type spaces.