Groups of Automorphisms Associated to C₀-Representations of ℝⁿ

A representation W of ℝn with values in the group of automorphisms (i.e. linear multiplicative bijections) of a Banach algebra B has many interesting features due to the richness of the algebraic structure which comes into play. We have no intention to present the general theory of such representations (elements of this theory may be found in [Br], [BR], [Cm], [Pd]) but rather to develop a very special aspect in view of later applications in spectral and scattering theory. More precisely, the algebras which will appear in this and in the next chapter are of the form B = B(F), where F is a Banach space equipped with a C ₀-representation W of ℝ nx, while the automorphism W(x) of B is given by W(x) [S] = W(-x)SW(x). Unless the generator of W is bounded in F, the family {W(x)} _{x ∈ ℝn} does not form a C ₀-representation of ℝ ⁿ in the Banach space B but only a C _w-representation (cf. Definition 3.2.6). Nevertheless, by using its continuity in the strong operator topology, one can develop for W the theory obtained in Chapter 3 for C ₀-Groups; the only difference lies in the fact that the domain of the generator of W is not norm dense in B but only strongly dense. Of course, W induces a C ₀-group in the subspace B _u of B consisting of all S ∈ B such that x ↦ W(x)[S] is norm-continuous, and this observation will be frequently used.