Group extensions and cohomology for locally compact groups. IV

TLDR

The paper applies cohomology groups constructed in [14] to a variety of problems in analysis. The authors employ these cohomology groups to analyze diverse analytical problems. The authors show that cohomology classes admit direct integral decompositions, providing a new proof for unitary representations, a Frobenius reciprocity theorem for induced modules, splitting theorems for direct integrals of tori, implementation theorems for automorphism groups of von Neumann algebras, agreement of the splitting group topology on two‑dimensional cohomology groups with other natural topologies, and resolve several open questions about splitting groups.

Abstract

In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defined topologies and we find conditions under which this topology is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally we resolve several questions left open concerning splitting groups in a previous paper [13].