Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Abstract

Let V be a standard subspace in the complex Hilbert space H and G be a finite dimensional Lie group of unitary and antiunitary operators on H containing the modular group (∆ it V ) t∈R of V and the corresponding modular conjugation JV . We study the semigroupand determine its Lie wedge L(SV ) = {x ∈ g : exp(R+x) ⊆ SV }, i.e., the generators of its oneparameter subsemigroups in the Lie algebra g of G. The semigroup SV is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form G exp(iC), where C ⊆ g is an Ad(G)-invariant closed convex cone.Our main results assert that the Lie wedge L(SV ) spans a 3-graded Lie subalgebra in which it can be described explicitly in terms of the involution τ of g induced by JV , the generator h ∈ g τ of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup SV itself. MSC 2010: Primary 22E45; Secondary 81R05, 81T05.