Free and projective generalized multinormed spaces

DOI10.1016/J.JMAA.2022.126660arXiv2109.01786MaRDI QIDQ6376827

Timur Oikhberg, A. Ya Helemskiǐ

Publication date: 4 September 2021

Abstract: The paper investigates free and projective $$-spaces, where $$ is a given normed space. These spaces form a far-reaching generalization of known $p$-multinormed spaces; in particular, if $$, the $$-spaces can be considered as $p$-multinormed spaces, based on arbitrary $sigma$-finite measure spaces $X$ (for "canonical" $p$-multinormed spaces, $X=mathbbN$ with the counting measure). We first describe a "naturally appearing" functor, based on paving $$ with contractively complemented finite dimensional subspaces. This finite dimensionality is essential; it permits us to describe a free $$-space for this functor. As a corollary, we obtain a wide variety of projective $$-spaces. For "nice" $$ (such as the space of simple $p$-integrable functions on a measure space), we obtain a full description of projective $$-spaces.

Mathematics Subject Classification ID

Normed linear spaces and Banach spaces; Banach lattices (46B99) Operator spaces and completely bounded maps (46L07) Projective and injective objects in functional analysis (46M10)

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