I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that 



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> Maximal Tensor products commute with arbitrary limits. 
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In the same book the proof of this fact is not given.In the literature also i could not find a proof of this. Can someone give a sketch or reference for the proof of the same result?


Thanks in advance



Need a reference of a fact given in B. Blackadar's Operator Algebras
I think it goes like this (I am only dealing with the unital case, I am not exactly sure how this works in the non-unital case).


Let $(A\_{i})\_{i\in I}$ be an inductive system of unital $C^{\ast}$-algebras with unital connecting homomorphisms $f\_{ij}: A\_{i} \to A\_{j}$ and let's denote the limit by $A$. If $B$ is another unital $C^{\ast}$-algebra, we would like to show that the inductive limit of $(A\_{i}\otimes\_{\rm{max}} B)\_{i\in I}$ is isomorphic to $A\otimes\_{\rm{max}} B$. In order to do so, we will show that $A\otimes\_{\rm{max}}B$ has the required universal property, i.e. any compatible family of $\ast$-homomorphisms from $A\_{i}\otimes\_{\rm{max}} B$ to a unital $C^{\ast}$-algebra $C$ gives rise to a $\ast$-homomorphism from $A\otimes\_{\rm{max}} B$. Note now that, by the universal property of the maximal tensor product, a $\ast$-homomorphism $\varphi\_{i}: A\_{i}\otimes\_{\rm{max}} B \to C$ is given by $\varphi\_{i} = \theta\_{i}\cdot \psi\_{i}$, where $\theta\_{i}:A\_{i} \to C$ and $\psi\_{i}: B \to C$ are $\ast$-homomorphisms with commuting ranges. As connecting maps are of the form $f\_{ij}\otimes \rm{Id}$, we can check that $\psi\_{i}=\psi\_{j}$, which we call $\psi$ from now on, and $\theta\_{i} = \theta\_{j}\circ f\_{ij}$. Since $A$ is the limit of $A\_i$'s, we get a $\ast$-homomorphism $\theta:A\to C$. As the union of ranges of $A\_i$'s inside $A$ is dense, this $\ast$-homomorphism has range commuting with the range of $\psi$, so we get a $\ast$-homomorphism $\varphi: A\otimes\_{\rm{max}} B \to C$ given by $\varphi(x\otimes y):= \theta(x)\psi(y)$.


