Consider the following definition in Lance's book on Hilbert $C^*$-modules. Let $A$ be a $C^*$-algebra.
How is $\alpha(a_i) x_i$ defined? Is this a function evalutation? I.e. does one evaluate $\alpha(a_i) : E \to E$ in $x_i \in E$? Or is this some sort of multiplication? How does one multiply elements in $E$ and in $\mathcal{L}(E)$ which is the set of adjointable maps $E \to E$.
Each $\alpha(a_i)$ is a linear map from $E$ to itself, and $\alpha(a_i)x_i$ should be thought of as evaluation of the function $\alpha(a_i)$ at $x_i$. The notation stems from linear algebra, where $\alpha(a_i)$ is treated like a matrix, $x_i$ is treated like a vector, and $\alpha(a_i)x_i$ is thought of as the product of the "matrix" $\alpha(a_i)$ and the "vector" $x_i$.