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

Abstract:

Let $S$ be a multiplicative topological semigroup with identity $e$. Suppose $D$ is an open subset containing $e$ and $h$ is a homeomorphism from $D$ onto a Banach space $B$ with $h(e) = 0$. Define the function $P$ by $P(x,y) = h[{h^{ - 1}}(x) \cdot {h^{ - 1}}(y)]$. 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 $B$ into $B$ and positive numbers $r,M$, and $c$ such that (i) if $||x|| < r$ then $x \in \operatorname {dom} ({A^{ - 1}})$and $||{A^{ - 1}}(x)|| \leqslant M||x||$, (ii) $cM < 1$, and (iii) if $||{x_i}||,||{y_i}|| < r(i = 1,2)$ then $({x_i},{y_j}) \in \operatorname {dom} (P)(i,j = 1,2)$, \[ ||P({x_1},{y_1}) - P({x_2},{y_2}) - A({y_1} - {y_2})|| \leqslant c||{y_1} - {y_2}||,\] and \[ ||P({x_1},{y_1}) - P({x_2},{y_1}) - A({x_1} - {x_2})|| \leqslant c||{x_1} - {x_2}||.\]