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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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    @Jonas: thanks!2012-07-03

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$A$ is made a right module over $A$ in the usual way. It is a Hilbert $A$-module with the inner product $\langle a,b\rangle = a^*b$.

If $X$ and $Y$ are Hilbert $A$-modules, then $X\oplus Y$ is a Hilbert $A$-module with componentwise operations and $\langle (x_1,y_1),(x_2,y_2)\rangle=\langle x_1,x_2\rangle_X + \langle y_1,y_2\rangle_Y$. (This generalizes immediately to finite direct sums, and infinite direct sums also exist.)

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