Norm decreasing homomorphisms of measure algebras
Roger Rigelhof · Transactions of the American Mathematical Society · 1969
Introduction.Let G be a locally compact group (=locally compact Hausdorff topological group).By the measure algebra of G we mean the Banach *-algebra M(G) of bounded regular Borel measures on G.It is well known that M(G) is the dual of C0(G) the Banach space of continuous complex-valued functions on G which "vanish at infinity".The group algebra L^G) is the *-subalgebra of M(G) consisting of all measures that are absolutely continuous with respect to the Haar measure on G. Alternatively LX{G) can be defined as the Banach *-algebra of (equivalence classes of) Haar summable functions on G.Let F and G be locally compact groups, a a bicontinuous isomorphism of F onto G, and y a character on F. For p. in M(F) and/in C0(G) let Tp(f)=p(y(f ° a)).Then the mapping pv-^Tp is an isometric *-isomorphism of M(F) onto M(G) (Lemma 2).In §3 we show that every norm decreasing isomorphism of M(F) onto M(G) is of the above form and consequently is an isometric *-isomorphism (Theorem 1).A number of other results follow from this, in particular Wendel's theorem on norm decreasing isomorphisms of group algebras.Theorem 1 generalizes a result of B. E. Johnson [7] on isometric isomorphisms of measure algebras.In [7] Johnson used norm properties of measures in L}{F) to show that each isometric isomorphism maps F^F) onto L}(G) and consequently by applying Wendel's result it follows that each isometric isomorphism is of the form described above.Our generalization of Johnson's result has the advantage that Wendel's theorem is a consequence.In §4 we use the results of §3 to prove a similar structure theorem for norm decreasing homomorphisms T of A/(F) onto M(G) such that T(p * L\F)) = {0} implies Tp = 0. (In this case the bicontinuous isomorphism of Fonto G mentioned above becomes a continuous and open homomorphism of Fonto G.)The final section of the paper is concerned with bipositive homomorphisms of M(F) onto M(G).Here we show that the hypothesis of norm decreasing may be dropped provided that the homomorphism maps the positive cone onto the positive cone.