С* algebras and projective limits

Question

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic limit for the $*$-algebra $\{a \in \prod_i A_i| a_i = \varphi_{ij}(a_j), i \leqslant j\}$? I know that we can topologize projective limits and get a pro-$C^*$-algebra. But I am interested in projective limits in the category of exactly $C^*$-algebras. Thanks!