The symmetry action of a von Neumann algebra and its associated involutive L-algebra

Abstract

Involutive L-algebras are introduced as a class of L-algebras X which embed into a factor group G of their structure group, so that X generates G and coincides with the set of involutions of G. A particular case exists for every group generated by involutions. In previous work it was shown that the projection lattice of a von Neumann algebra is an L-algebra which is determined, up to isomorphism, by the structure group of this L-algebra. Extending this result, an involutive L-algebra is associated to any von Neumann algebra as a complete invariant. In particular, it is proved that involutive L-algebras admit a self-action by involutive automorphisms which canonically extends to a self-action of the structure group. Several examples are considered, including those which give rise to non-degenerate involutive solutions to the set-theoretic Yang–Baxter equation.