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Semisimplicity manifesting as categorical smallness

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Alexandru Chirvasitu

For a compact group G, the functor from unital Banach algebras with contractive morphisms to metric spaces with 1-Lipschitz maps sending a Banach algebra A to the space of G-representations in A preserves filtered colimits. Along with this, we prove a number of analogues: one can substitute unitary representations in C*-algebras, as well as semisimple finite-dimensional Banach algebras (or finite-dimensional C*-algebras) for G.

These all mimic results on the metric-enriched finite generation/presentability of finite-dimensional Banach spaces due to Adámek and Rosický. We also give an alternative proof of that finite presentability result, along with parallel results on functors represented by compact metric, metric convex, or metric absolutely convex spaces.

Keywords:
compact group; Haar measure; Banach space; Banach algebra; semisimple; C*-algebra; semiprojective; filtered colimit; representation; diagonal; averaging; finitely presentable; finitely generated; enriched; small object; metric space; non-expansive map; Lipschitz; convex space; absolutely convex space; monad

2020 MSC:
22C05; 46H05; 46H15; 16K99; 18A30; 18A35; 18D20; 46B04; 32K05; 52A21; 46L05; 47B48; 47B01; 51F30; 46A19; 54E40; 54E45; 46A55; 18C15; 18C20

*Theory and Applications of Categories,*
Vol. 41, 2024,
No. 14, pp 470-492.

Published 2024-05-07.

http://www.tac.mta.ca/tac/volumes/41/14/41-14.pdf

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