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arXiv · 2609.25931

Group Actions, Fixed Points and Orbit Quotients in Lipschitz Spaces

Arindam Mandal
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Abstract

We study Lipschitz mappings associated with group actions by similarities, extending the orbit-quotient approach for point-preserving isometric actions. Normalizing by the similarity ratios we obtain isometric representations whose fixed points are character-equivariant Lipschitz mappings. The corresponding quotients by closed spans of orbit differences provide canonical preduals and norm-preserving linearizations, without an amenability assumption. Applications include positively homogeneous, linear, and bilinear mappings. In particular, we prove that the space of positively homogeneous Lipschitz functions form a $1$-complemented subspace of the Lipschitz space (strenghthening the earlier existed result) and interpret existing projections onto linear and bilinear mappings as invariant-mean averages. For amenable groups, the same approach yields isometric realizations of the quotient spaces in the biduals of the original linearization spaces.

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Arindam Mandal. 2026-09-22. Group Actions, Fixed Points and Orbit Quotients in Lipschitz Spaces. https://arxiv.org/abs/2609.25931

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