Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology

TL;DR

This paper establishes automatic continuity of derivations on von Neumann algebras, explores the behavior of the first continuous L^2-Betti number under algebraic operations, and derives implications for factors with specific properties.

Contribution

It proves automatic continuity of derivations into affiliated operators and analyzes the quadratic scaling of the first continuous L^2-Betti number.

Findings

Derivations are automatically continuous in specified topologies.

The first continuous L^2-Betti number scales quadratically for corner algebras.

Vanishing of the first continuous L^2-Betti number for certain factors with property (T), non-trivial fundamental group, or property Gamma.

Abstract

We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L^2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L^2-Betti number for II_1 factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.