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

Abstract: In a companion paper [ABS1] 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 [ABS1] 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 complexes 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 $\mathcal{M}$.