Let C be a compact
convex subset of a locally convex topological vector space X. Anzai and
Ishikawa recently proved that if T1,⋯,Tn is a finite commutative family of
continuous affine self-mappings of C, then F(∑i=1nλiTi) =⋂i=1nF(Ti)
for every λi such that 0 < λi< 1 and ∑i=1nλi= 1, where F(T) denotes
the fixed point set of T. It is natural to question whether the conclusion
of their theorem is dependent on the topological properties of X, C and
Ti—in this case, the linear topology, the compactness and the continuity. We
shall see that this is not; the theorem can be formulated in an algebraic
context.
