2018/12/29 by Naarmann, Simon
#FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1812.11442
Let A be a C*-algebra that is the norm closure A = ∑β∈ α Iβ of an arbitrary sum of C*-ideals Iβ⊆ A. We construct a homological spectral sequence that takes as input the K-theory of \bigcapj ∈ J Ij for all finite nonempty index sets J ⊆ α 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 \Xβ\β∈ α of X, we reshape \mathfrak C^* Xβ as input for the spectral sequence. From the K-theory of \mathfrak C^*X ( \bigcapj ∈ J Xj ) for finite nonempty index sets J ⊆ α, we compute the K-theory of \mathfrak C^* X if α is finite, or of a direct limit C*-ideal of \mathfrak C^* X if α 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.