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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Robert C. Eslinger
Journal: Proc. Amer. Math. Soc. 55 (1976), 203-208
MathSciNet review: 0399338
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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 $ \vert\vert x\vert\vert < r$ then $ x \in \operatorname{dom} ({A^{ - 1}})$and $ \vert\vert{A^{ - 1}}(x)\vert\vert \leqslant M\vert\vert x\vert\vert$, (ii) $ cM < 1$, and (iii) if $ \vert\vert{x_i}\vert\vert,\vert\vert{y_i}\vert\vert < r(i = 1,2)$ then $ ({x_i},{y_j}) \in \operatorname{dom} (P)(i,j = 1,2)$,

$\displaystyle \vert\vert P({x_1},{y_1}) - P({x_2},{y_2}) - A({y_1} - {y_2})\vert\vert \leqslant c\vert\vert{y_1} - {y_2}\vert\vert,$


$\displaystyle \vert\vert P({x_1},{y_1}) - P({x_2},{y_1}) - A({x_1} - {x_2})\vert\vert \leqslant c\vert\vert{x_1} - {x_2}\vert\vert.$

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Keywords: Topological semigroup, Banach manifold, implicit function theorem
Article copyright: © Copyright 1976 American Mathematical Society