Vladimir G. Pestov

On the problem of amenability of groups of maps.

Amenability of a locally compact group admits numerous equivalent definitions, which are pairwise inequivalent for more general topological groups. The property that is usually called amenability is this: a topological group G is amenable if every continuous action of G on a compact space X admits an invariant regular Borel probability measure.
The open problem motivating this minicourse comes from mathematical physics. If X is a compact manifold and K a compact Lie group, are the groups of currents C^{\infty}(X,K), C^{k}(X,K) and H^k(X,K), with their natural topologies, amenable? [2]
The question appears to be still open for all values of k\geq 0 and all dimensions of X, with the only exception of groups of continuous loops C(\mathbb{S}^1,K) and continuous paths C(\mathbb{I},K) (Malliavin and Malliavin [4]). Recently the speaker has proved [7] amenability of the groups of paths and loops of Sobolev class H^1 (the finite energy paths and loops).
Another source of interest in amenability of groups of maps is the open problem of describing C^\ast-algebras whose unitary groups are amenable in the norm topology [1,5].
We plan on discussing amenability and its diverging variants beyond the locally compact case (the classical definition; a version belonging to Malliavin and Malliavin; skew-amenability [6,3]), the advances in the direction of the above problems and around (like Ozawa’s results [5]), and the techniques used (logarithmic derivative, product integral, Wiener measure…).

[1] Vadim Alekseev, Max Schmidt, Andreas Thom, Amenability for unitary groups of C^\ast-algebras, arXiv preprint arXiv:2305.13181v1 [math.OA], 12 pp.

[2] VA. Carey and H. Grundling, On the problem of the amenability of the gauge group, Lett. Math. Phys. 68 (2004), 113—120.

[3] K. Juschenko, F.M. Schneider, Skew-amenability of topological groups, Comment. Math. Helv. 96 (2021), pp. 805—851.

[4] M.-P. Malliavin and P. Malliavin, Integration on loop group III. Asymptotic Peter—Weyl orthogonality, J. Funct. Analysis 108 (1992), 13—46.

[5] Narutaka Ozawa, Amenability for unitary groups of simple monotracial C^{\ast}-algebras, arXiv preprint arXiv:2307.08267v2, [math.OA], 8 pp.

[6] V. Pestov, An amenability-like property of finite energy path and loop groups, C.R. Math. Acad. Sci. Paris 358 (2020), pp. 1139—1155.

[7] Vladimir G. Pestov, Amenability of finite energy path and loop groups, arXiv preprint arXiv:2307.00403, [math.FA]. 16 pp.