B. Blackadar in his book Operator algebras: Theory of C${}^\ast$-Algebras and Von Neumann Algebras defines a C*-algebra $A$ to be nuclear if for every C*-algebra $B$ the algebraic tensor product $A\odot B$ admits a unique C*-norm. In particular, the minimal (spatial) and maximal tensor products $A\otimes_{{\rm min}} B$ and $A\otimes_{{\rm max}} B$ agree. Okay.

Later on he proves that commutative C*-algebras are nuclear showing that $A\otimes_{{\rm min}} B = A\otimes_{{\rm max}} B$ if $A$ is commutative. But how do we know a priori that this is enough once we have not yet proved that $\|\cdot\|_{{\rm min}}\leqslant \|\cdot\|_\gamma$ for any C*-norm on $A\odot B$? In particular, he uses nuclearity of commutative C*-algebras to infer that $\|\cdot\|_{{\rm min}}$ is indeed minimal.

Can one please clarify this confusion?


I think that Blackadar's goal was to condense most of the theory in a readable number of pages, and a consequence is that logic suffers a little.

If you look at the proof in Takesaki, the fact he uses is not nuclearity of abelian algebras, but rather a slightly weaker condition that he can prove without recourse to nuclearity.


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