Some Remarks On Essentially Normal Submodules

TL;DR

This paper explores the properties and relations of essentially normal projections in Hilbert spaces, connecting them to Arveson's conjecture and K-homology, with implications for operator theory and polynomial ideals.

Contribution

It introduces two notions of span for essentially normal projections, proves a representation theorem for two projections, and relates these to Arveson's conjecture and K-homology.

Findings

Representation theorem for two projections.

Relation of span notions to Arveson's conjecture.

Connection between projections' relative position and K-homology elements.

Abstract

Given a *-homomorphism $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$ for a compact metric space $M$, a projection $P$ onto a subspace $\mathcal{P}$ in $\mathcal{H}$ is said to be essentially normal relative to $\sigma$ if $[\sigma(\varphi),P]\in \mathcal{K}$ for $\varphi\in C(M)$, where $\mathcal{K}$ is the ideal of compact operators on $\mathcal{H}$. In this note we consider two notions of span for essentially normal projections $P$ and $Q$, and investigate when they are also essentially normal. First, we show the representation theorem for two projections, and relate these results to Arveson's conjecture for the closure of homogenous polynomial ideals on the Drury-Arveson space. Finally, we consider the relation between the relative position of two essentially normal projections and the $K$ homology elements defined for them.