Direct sum of Hilbert modules

Question

Let $E$ and $F$ be Hilbert modules over C*-algebras $A$ and $B$ respectively. Then we define the following $A\oplus B$-valued innerproduct structure $\langle , \rangle \colon E \oplus F \times E \oplus F \to A \oplus B $ by $$\langle (e_1, f_1), (e_2, f_2) \rangle_{A\oplus B} := (\langle, e_1, e_2 \rangle_A, \langle f_1, f_2 \rangle_B),$$ with componentwise actions. I believe all the properties of the inner product follows, but I could not find any standard text to consider such direct sum. I am unable to get my head around the fact why is so, am I missing something?

Answer 1

You can make $E$ into a $C^\ast$-module over $A\oplus B$ by considering the action $e(a,b)=ea$ and the new inner product $\langle e_1\mid e_2\rangle=(\langle e_1,e_2\rangle,0)$. If you do the same thing for $F$, just with everything in the second coordinate, then the usual direct sum of $E$ and $F$ as $C^\ast$-modules over $A\oplus B$ is the $C^\ast$-module you described in your question.