Semi-Cosimplicial Hilbert Spaces with Isometric Coface Operators

TL;DR

This paper introduces semi-cosimplicial Hilbert spaces with isometric coface operators, exploring their structure, cohomology, and connections to non-commutative probability and group representations.

Contribution

It systematically develops semi-cosimplicial Hilbert spaces with isometries, linking them to spreadability, cohomology, and representation theory of symmetric and braid groups.

Findings

Develops a framework for semi-cosimplicial Hilbert spaces with isometric coface operators.

Connects the structure to non-commutative probability and spreadability.

Explores representation theory of infinite symmetric and braid groups within this context.

Abstract

Semi-cosimplicial objects in the category of Hilbert spaces with isometries which are motivated by non-commutative probability theory, in particular by the distributional symmetry of spreadability, are introduced and systematically developed in various directions: partial shifts, cohomology, a related graph, decomposition into labeled subspaces, representation theory of the infinite symmetric and braid groups, extension problems and a toy version of spreadability.