Existence of inverses and square roots in locally Banach semigroups with identity

Robert C. Eslinger
1976 Proceedings of the American Mathematical Society  
Let S be a multiplicative topological semigroup with identity e. Suppose D is an open subset containing e and A is a homeomorphism from D onto a Banach space B with h(e) = 0. Define the function P by P(x, y) = h[h~l(x) ■ h~l(y)]. A new implicit function theorem is applied to the function P to show the existence of inverses and square roots of elements in a neighborhood of the identity. It is assumed that P satisfies the following condition: There exist a one-one function A from a subset of B
more » ... om a subset of B into B and positive numbers r, M, and c such that (i) if 11*11 < r then x E dom(A ~')and \\A -l(*)ll < M\\x\\, (ii) cM < 1, and (iii) if ||*,.||, ||y,-|| < r (i = 1, 2) then (x"yj) G dom(P) (i,j = 1,2),
doi:10.1090/s0002-9939-1976-0399338-2 fatcat:2ds5icqp7zhdbl54coz4zgg7te