Title: Variants of Invariant Means of Amenability
by Ajit Iqbal Singh (Indian National Science Academy) as part of Topological Groups
Lecture held in Elysium.
It all started, like many other amazing theories, in nineteen twenty-nine,
With John von Neumann, the greatest of the great.
The question of existence of a finitely additive measure on a group, a mean of a kind,
That is invariant, under any translation, neither gaining nor losing any weight.
Mahlon M. Day, in his zest and jest, giving double importance to semigroups, too,
Took up the study of conditions and properties, and named it amenability.
Erling Folner followed it up, more like a combinatorial maze to go through,
Whether or not translated set meets the original in a sizeable proportionality.
How could functional analysts sit quiet, who measure anything by their own norms,
Lo and behold, it kept coming back to the same concept over and over again.
Group algebras were just as good or bad, approximate conditions did no harms,
With the second duals of lofty Richard Arens, it became deeper, but still a fun-game.
Ever since, with the whole alphabet names, reputed experts or budding and slick,
Considering several set-ups and numerous variants of the invariance.
Actions on Manifolds or operators, dynamical systems nimble or quick,
We will have a look at some old and some new, closely or just from the fence.