Almost commuting matrices, cohomology, and dimension

Dominic Enders; Tatiana Shulman

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arXiv · 1902.10451

Almost commuting matrices, cohomology, and dimension

Dominic Enders·Tatiana Shulman

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Abstract

We investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative $C^*$-algebras $C(X)$ are matricially semiprojective. Extending the works of Davidson, Eilers-Loring-Pedersen, Lin and Voiculescu on almost commuting matrices, we identify the precise dimensional and cohomological restrictions for finite-dimensional spaces $X$ and thus obtain a complete characterization: $C(X)$ is matricially semiprojective if and only if $\dim(X)\leq 2$ and $H^2(X;\mathbb{Q})=0$. We give several applications to lifting problems for commutative $C^*$-algebras, in particular to liftings from the Calkin algebra and to $l$-closed $C^*$-algebras in the sense of Blackadar.

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Dominic Enders, Tatiana Shulman. 2019-02-27. Almost commuting matrices, cohomology, and dimension. https://arxiv.org/abs/1902.10451

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