Some results on automatic continuity of group representations and morphisms (Q2839814)

This paper gives several criteria for the norm continuity of representations of locally compact groups, or more generally of their bounded homomorphisms to Banach algebras. Norm continuity is a very strong property which in many cases implies that the representation is finite-dimensional. The aim of the article is to link it to other requirements, seemingly weak.NEWLINENEWLINEA typical statement is as follows (Corollary 2.9).NEWLINENEWLINELet \(G\) be a locally compact group and \(\theta\) a norm bounded homomorphism of \(G\) into a unital Banach algebra \(A\). The following facts are equivalent: (1) \(\theta\) is norm continuous; (2) \(\theta\) is continuous when \(A\) is endowed with its weak topology (i.e., \(\omega\circ\theta\) is continuous for all \(\theta\in A'\)).NEWLINENEWLINEIf moreover \(G\) is first countable, continuity of pure states suffices in (2). Pure states are defined in the non-\(C^*\) case as the extreme points of the space \(S(A)=\{\omega\in A': \|\omega\|=\omega(1)=1\}\).NEWLINENEWLINEThe core of the proofs relies on the following theorem of \textit{J.~Esterle} [J. Contemp. Math. Anal., Armen. Acad. Sci. 38, No. 5, 9--19 (2003); translation from Izv. Nats. Akad. Nauk Armen., Mat. 38, No. 5, 11--22 (2003; Zbl 1160.22301)].NEWLINENEWLINELet \(G\) be an abelian locally compact group and \(\theta\) a locally bounded homomorphism of \(G\) into a unital abelian Banach algebra \(A\). The following facts are equivalent: (1) \(\theta\) is norm continuous; (2) \(\lim_{g\to e} \rho(\theta(g)-1)=0\), where \(\rho\) denotes the spectral radius in \(A\).NEWLINENEWLINEIn the last section, the authors prove a similar result where the norm continuity is deduced from the measurability of \(\theta\) as a map to \(A\) endowed with the norm topology. This part relies on a recent result of the reviewer on automatic continuity [Stud. Math. 210, No. 3, 197--208 (2012; Zbl 1290.22002)], cited in the paper by a preprint version on ArXiv.org.NEWLINENEWLINENote: In the article, the authors write just `continuous representation' instead of `norm continuous'.