Pointwise eventually non-expansive action of semi-topological semigroups and fixed points

C*-algebraic approach to fixed point theory generalizes Baggett's theorem to groups with discrete reduced duals

“…When the action is separately continuous, each member of S is continuous. We say that S has a common fixed point in K if there exists a point x in K such that sx = x for all s ∈ S. When E is a normed space, the action of S on K is non-expansive if s(x) − s(y) ≤ x − y for all s ∈ S and x, y ∈ K. There are also other types of action for a semi-topological semigroup (see [16] and [2]).…”

An ultraproduct method via left reversible semigroups to study Bruck's generalized conjecture

“…An action of S on a subset K of a topological space E is a mapping (s, x) → s(x) from S × K into K such that (st)(x) = s(t(x)) for s, t ∈ S, x ∈ K. The action is separately continuous if it is continuous in each variable when the other is kept fixed. We say that S has a common fixed point in K if there exists a point x in K such that sx = x for all s ∈ S. When E is a normed space, the action of S on K is non-expansive if s(x)−s(y) ≤ x−y for all s ∈ S and x, y ∈ K. There are also other types of action for a semi-topological semigroup (see [1] and [5]).…”