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Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra. I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...

There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book ...