Finite dimensional semigroups of unitary endomorphisms of standard subspaces

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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

(D e l t a_{V}^{i t})_{t i n R}

of

V

and the corresponding modular conjugation~

J_{V}

. We study the semigroup [ S_V = { gin G cap U(H) : gV subseteq V} ] and determine its Lie wedge

L (S_{V}) = x i n L (G) : e x p (R_{+} x) s u b s e t e q S_{V}

, i.e., the generators of its one-parameter subsemigroups in the Lie algebra

L (G)

of~

G

. The semigroup

S_{V}

is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form

G e x p (i C)

, where

C s u b s e t e q L (G)

is an

A d (G)

-invariant closed convex cone. Our main results assert that the Lie wedge

L (S_{V})

spans a

3

-graded Lie subalgebra in which it can be described explicitly in terms of the involution

a u

of

L (G)

induced by

J_{V}

, the generator

h i n L (G)^{a} u

of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup

S_{V}

itself

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