Let $A$ and $B$ be $C^*$-Algebras and $E$ a Hilbert-$A$-module and $F$ a Hilbert-$B$-module and $\pi: A \rightarrow L(F)$ a $*$-homomorphism where $L(F)$ denotes the $C^*$-algebra of adjointable operators on $F$. Then $F$ is a left $A$ module by $a \cdot f= \pi(a)(f)$. So we can take the algebraic tensor product $E \odot_A F$ over $A$ and definine a sesquilinearform by setting $$ \langle e_1 \otimes f_1, e_2 \otimes f_2\rangle := \langle f_1,\pi(\langle e_1,e_2\rangle)(f_2)\ \rangle$$ which is supposed to be positive. This is obviously true for elementary tensors, but what about sums? Most authors just reference a book by E.C. Lance but I won't be able to access it in the next couple days. The version on google books omits the part I need. Could somebody outline the argument for me? Thank you.
Hilbert-$C^*$-Modules and positivity of inner products
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functional-analysis
operator-algebras
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2[Here](https://i.stack.imgur.com/ZiRu6.png) is the relevant calculation from the book you have mentioned. Are the statements about positivity of $X$ with $(X)_{ij}=\langle x_i,x_j\rangle$ or complete positivity of $\phi$ known to you? I can send you a copy of the book if you wish. – 2017-01-14
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0I am not an expert, so I would really appreciate it, if you could send it to me. Can you do that via stack excchange or do you need an email adress? – 2017-01-14