The heart of the Banach spaces

Sven Ake Wegner

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Abstract

Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beĭlinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel-cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.

Original languageEnglish
JournalJournal of Pure and Applied Algebra
DOIs
Publication statusAccepted/In press - Jul 31 2016

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ASJC Scopus subject areas

  • Algebra and Number Theory

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