Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

Abstract

In a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775, 2021) we introduced the stable infty -category of noncommutative CW-spectra, which we denoted mathtt {NSp}. Let {mathcal {M}} denote the full spectrally enriched subcategory of mathtt {NSp} whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that mathtt {NSp} is equivalent to the infty -category of spectral presheaves on {mathcal {M}}. In this paper we investigate the structure of {mathcal {M}}, and derive some consequences regarding the structure of mathtt {NSp}. To begin with, we introduce a rank filtration of {mathcal {M}}. We show that the mapping spectra of {mathcal {M}} map naturally to the connective K-theory spectrum ku, and that the rank filtration of {mathcal {M}} is a lift of the classical rank filtration of ku. We describe the subquotients of the rank filtration in terms of spaces of direct-sum decompositions which also arose in the study of K-theory and of Weiss’s orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of mathtt {NSp} as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the p-localization and the chromatic localization of {mathcal {M}}.