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We will show that the bounded part of the locally C*-algebra of all adjointable operators on the Hilbert A-module E is isomorphic to the C*-algebra L b(A)(b(E)) of all adjointable operators on the Hilbert b(A)-module b(E).

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In this paper we study the unitary equivalence between Hilbert modules over a locally \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $C^{*}$ \end{document}-algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $C^{*}$ \end{document}-algebra and show that a Hilbert module over a Fr\'{e}chet locally \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $C^{*}$ \end{document}-algebra is countably generated if and only if the locally \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $C^{*}$ \end{document}-algebra of all ``compact'' operators has an approximate unit.

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. [5] Joita , M ., Hilbert Modules Over Locally C -Algebras , Editura Universitatii din Bucuresti , 2006 . [6] Mallios , A ., Topological Algebras: Selected Topics , North Holland , Amsterdam , 1986 . [7] Misiak , A ., n -Inner Product Space

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