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

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 ∞-category of noncommutative CW-spectra, which we denoted NSp. Let M denote the full spectrally enriched subcategory of NSp whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that NSp is equivalent to the ∞-category of spectral presheaves on M. In this paper we investigate the structure of M, and derive some consequences regarding the structure of NSp. To begin with, we introduce a rank filtration of M. We show that the mapping spectra of M map naturally to the connective K-theory spectrum ku, and that the rank filtration of 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 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 M.