\documentclass{article} \usepackage{amsfonts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2606} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %TCIDATA{Created=Friday, February 26, 2021 18:51:41} %TCIDATA{LastRevised=Friday, February 26, 2021 18:53:28} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=40 LaTeX article.cst} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} Let $X$ be a compact Hausdorff space, and let $\{A_{x}:x\in X\}$ and $% \{B_{x}:x\in X\}$ be collections of Banach algebras such that each $A_{x}$ is a $B_{x}$-bimodule. Let $\mathfrak{A}\subset \prod \{A_{x}:x\in X\}$ and $% \mathfrak{B}\subset \prod \{B_{x}:x\in X\}$ be algebras of functions such that $\mathfrak{A}$ is also a $\mathfrak{B}$-module. For certain choices of $% \mathfrak{A}$ and $\mathfrak{B}$ we use the theory of bundles of Banach spaces as a tool to investigate the module amenability of $\mathfrak{A}$ as a $\mathfrak{B}$-module when each $A_{x}$ is amenable as a $B_{x}$-module.% \newline \end{document}