Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Abstract

Let V \mathtt {V} be a standard subspace in the complex Hilbert space H \mathcal {H} and G G be a finite dimensional Lie group of unitary and antiunitary operators on H \mathcal {H} containing the modular group ( Δ V i t ) t ∈ R (\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}} of V \mathtt {V} and the corresponding modular conjugation J V J_{\mathtt {V}} . We study the semigroup \[ S V = { g ∈ G ∩ U ⁡ ( H ) : g V ⊆ V } S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] and determine its Lie wedge L ⁡ ( S V ) = { x ∈ g : exp ⁡ ( R + x ) ⊆ S V } \operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\} , i.e., the generators of its one-parameter subsemigroups in the Lie algebra g \mathfrak {g} of G G . The semigroup S V S_{\mathtt {V}} is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form G exp ⁡ ( i C ) G \exp (iC) , where C ⊆ g C \subseteq \mathfrak {g} is an Ad ⁡ ( G ) \operatorname {Ad}(G) -invariant closed convex cone. Our main results assert that the Lie wedge L ⁡ ( S V ) \operatorname {\textbf {L}}(S_{\mathtt {V}}) spans a 3 3 -graded Lie subalgebra in which it can be described explicitly in terms of the involution τ \tau of g \mathfrak {g} induced by J V J_{\mathtt {V}} , the generator h ∈ g τ h \in \mathfrak {g}^\tau of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup S V S_{\mathtt {V}} itself.

Elements in pointed invariant cones in Lie algebras and corresponding affine pairs

“…For an affine pair (x, h) the element x is nilpotent, hence in particular not elliptic if it is not central. As we know from [Ne21], the most important affine pairs are those for which h in an Euler element of g, i.e., ad h is diagonalizable with possible eigenvalues {−1, 0, 1} and the Lie algebra g is generated by h and the cones C U ∩ g ±1 (h) (see [Oeh20a,Oeh20b] for related classification results). Considering representations with discrete kernel then leads to the situation, where the cone C U ⊆ g is pointed, so that g = g U ⋊ Rh, and g U = C U − C U is an ideal containing the pointed generating invariant cone C U .…”

Nets of standard subspaces induced by antiunitary representations of admissible Lie groups I

“…Let τ := L(τ G ) ∈ Aut(g) and let h ∈ g τ be a fixed point of τ . Then J := π(τ G ) and ∆ −it/2π := π(exp(th)), t ∈ R, determine a standard subspace V := Fix(J∆ 1/2 ) of H. The subsemigroup S V := {g ∈ G : π(g)V ⊂ V} encodes the order structure on the G-orbit of V in the set of standard subspaces of H. It has been studied extensively in [Ne19] and [Ne20]. Its Lie wedge L(S V ) := {x ∈ g : exp(R + x) ⊂ S V } can be described in terms of the Lie algebraic data (π, H), τ , and h by…”

“…where g λ (h) := ker(ad(h) − λ id g ) and C π := {x ∈ g : −i∂π(x) ≥ 0} is the positive cone of π (cf. [Ne19,Thm. 4.4]).…”

Nets of standard subspaces on Lie groups

“…(the set of infinitesimal generators of one-parameter subsemigroups of SV) spans the Lie algebra g. Here we build on the previous work [Ne19,Ne19b] of the first author on these semigroups.…”