XXXVII Workshop on Geometric Methods in Physics 1-7.07.2018 VII School on Geometry and Physics 25-29.06.2018
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# Eduardo Chiumiento

## Essentially commuting projections

Let $\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-$ be a fixed orthogonal decomposition of a Hilbert space, with both subspaces of infinite dimension, and let $E_+, E_-$ be the projections onto $\mathcal{H}_+$ and $\mathcal{H}_-$. We study the set $\mathcal{P}_{cc}$ of orthogonal projections $P$ in $\mathcal{H}$ which {\it essentially commute} with $E_+$, i.e.
$$[P,E_+]=PE_+-E_+P \ \ \hbox{ is compact.}$$
Using the projection $\pi$ onto the Calkin algebra, one sees that these projections $P\in\mathcal{P}_{cc}$ fall into nine classes.
Indeed, $\pi(P)$ can be represented as a $2 \times 2$ diagonal projection, where the diagonal elements can be $0$, $1$ or a proper projection. We define the {\it discrete classes}, when the diagonal elements are $0$ or $1$, which corresponds to finite rank projections, finite co-rank projections, the restricted Grassmannian of $\mathcal{H}_+$ and the restricted Grassmannian of $\mathcal{H}_-$. The connected components of each of these classes are parametrized by the integers. The five remaining classes are called {\it essential classes}, and they are connected. We show that the Hopf-Rinow Theorem holds in the discrete classes, but not in the essential classes. This is joint work with Esteban Andruchow and María Eugenia Di Iorio.

 Event sponsored by: University of Bialystok

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