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I am having trouble with this proof: Let $X$ be a Hilbert $A$-module and let $A$ be a $C^*$-algebra then the direct sum $A\oplus X$ is also a Hilbert $A$-module.

Useful information:

$X$ be a Hilbert $A$-module if $X$ is an inner-product $A$-module which is complete with respect to the norm $\| x \| = \| \langle x, x \rangle \| ^{1/2}$.

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    I'm confused: so the obvious candidate "inner product" $\langle (a,x),(b,y) \rangle_{A \oplus X} = a^\ast b + \langle x,y \rangle_X$ where $\langle \cdot,\cdot \rangle_X \colon X \times X \to A$ is the "inner product" on $X$ doesn't work? Why? I don't see what goes wrong, as all the axioms [here](http://en.wikipedia.org/wiki/Hilbert_C*-module) seem to be satisfied2012-07-03
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    @t.b.: Nothing goes wrong, and that is the usual definition. Peter: Where are you having trouble?2012-07-03
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    @Jonas: thanks!2012-07-03

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