A Mayer-Vietoris Spectral Sequence for C*-Algebras and Coarse Geometry

Abstract

Let $A$ be a C*-algebra that is the norm closure $A = \overline{\sum_{\beta \in \alpha} I_\beta}$ of an arbitrary sum of C*-ideals $I_\beta \subseteq A$. We construct a homological spectral sequence that takes as input the K-theory of $\bigcap_{j \in J} I_j$ for all finite nonempty index sets $J \subseteq \alpha$ and converges strongly to the K-theory of $A$. For a coarse space $X$, the Roe algebra $\mathfrak C^* X$ encodes large-scale properties. Given a coarsely excisive cover $\{X_\beta\}_{\beta \in \alpha}$ of $X$, we reshape $\mathfrak C^* X_\beta$ as input for the spectral sequence. From the K-theory of $\mathfrak C^*X \big( \bigcap_{j \in J} X_j \big)$ for finite nonempty index sets $J \subseteq \alpha$, we compute the K-theory of $\mathfrak C^* X$ if $\alpha$ is finite, or of a direct limit C*-ideal of $\mathfrak C^* X$ if $\alpha$ is infinite. Analogous spectral sequences exist for the algebra $\mathfrak D^* X$ of pseudocompact finite-propagation operators that contains the Roe algebra as a C*-ideal, and for $\mathfrak Q^* X = \mathfrak D^* X / \mathfrak C^* X$.