The strongly quasi-local coarse Novikov conjecture and Banach spaces with Property (H)

TL;DR

This paper introduces a strongly quasi-local version of the coarse Novikov conjecture and proves its validity for metric spaces coarsely embedded into Banach spaces with Property (H), expanding the understanding of $K$-theory in coarse geometry.

Contribution

It formulates a new strongly quasi-local coarse Novikov conjecture and proves it for spaces with bounded geometry embedded into Banach spaces with Property (H).

Findings

Conjecture holds for spaces with bounded geometry coarsely embedded into Property (H) Banach spaces.

Generalizes strong quasi-locality to proper metric spaces.

Provides a quasi-local perspective on $K$-homology.

Abstract

In this paper, we introduce a strongly quasi-local version of the coarse Novikov conjecture, which states that certain assembly map from the coarse $K$-homology of a metric space to the $K$-theory of its strongly quasi-local algebra is injective. We prove that the conjecture holds for metric spaces with bounded geometry which can be coarsely embedded into Banach spaces with Property (H), introduced by Kasparov and Yu. Besides, we also generalise the notion of strong quasi-locality to proper metric spaces and provide a (strongly) quasi-local picture for $K$-homology.