Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

TL;DR

This paper studies the structure of Lie-group-valued cocycles, showing they form analytic manifolds and fiber bundles, leading to new insights into cohomology spaces, morphisms between C*-algebras, and projective representations.

Contribution

It introduces a framework for understanding the manifold structure of cocycles and cohomology spaces in the context of Lie groups and C*-algebras, with applications to morphisms and representations.

Findings

Cocycles form analytic submanifolds of continuous maps.

Cohomology spaces are discrete and have open orbits.

Morphisms between finite-dimensional C*-algebras form analytic manifolds.

Abstract

Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps ${G^n\to U}$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's).