An amenability-like property of finite energy path and loop groups

Abstract

We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B (H π ).Résumé. Nous montrons que les groupes de lacets et de chemins à énergie finie (c.à.d. de classe H 1 de Sobolev) à valeurs dans un groupe de Lie compact et connexe, ainsi que leurs extensions centrales, satisfont une version de la moyennabilité: ils admettent une moyenne invariante à gauche sur l'espace de fonctions bornées uniformément continues par rapport a une métrique invariante à gauche. Chaque représentation unitaire continue, π, d'un tel groupe (que nous disons d'être "moyennable en biais") possède un état sur B (H π ) invariant sous conjugaison.

On invariant means and pre-syndetic subgroups

“…Now, if a topological group is SIN, then amenability and skew-amenability are equivalent properties, and hence Corollary 1.2 follows immediately. Previously the conclusion of Theorem 1.4 was known for normal co-compact subgroups ( [22], Thm. 12).…”

“…While the choice for what equivalent property to call amenability is motivated by its usefulness in many applications (see e.g. the survey [12] for motivation), the properties (2) and (3) also emerge naturally in the context of path and loop groups, see [18] and [22], respectively. We call groups satisfying (2) skew-amenable, and (3), Malliavin-Malliavin amenable, or M-M amenable.…”

“…The above result is a corollary of a (rather) more general result involving a different version of amenability. As we already mentioned above, we call a topological group G skew-amenable [22,16] if it admits a leftinvariant mean on the space LUCB(G) of all left uniformly continuous bounded real-valued functions on G. A function f : G → R is left uniformly continuous if for every ǫ > 0 and x ∈ G, we have |f (x) − f (xv)| < ǫ when v ∈ V and V is a sufficiently small neighbourhood of identity. It is equivalent to the uniform continuity of f in relation with a suitable continuous left-invariant pseudometric.…”

“…Skew-amenability was formally introduced by the first-named author in [22] (adopting the term suggested by the second-named author), but it had already made cameo appearance, nameless, in [10,21]. As we have already mentioned, for locally compact groups skew-amenability is simply one of many equivalent forms of amenability (see for example Thm.…”

“…For this reason, it may well be that the open question by Carey and Grundling [7] asking if the groups C ∞ (X, K) of infinitely smooth maps from a compact manifold X to a compact Lie group K, equipped with the C ∞ -topology, are amenable, should also ask whether those groups are skew-amenable. (See a more detailed discussion in [22]. )…”