Let $f$ be a faithful state on a $C^\*$-algebra $A$, i.e. $f(a^\*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^\*$-algebras faithful if $T(a^\*a)=0$ implies $a=0$. How to prove that if $f$ is faithful then the map $f \otimes id: A \otimes A \to A$ is faithful? I think that it should be simple however I don't have an idea how to prove this. 



Faithul map and (minimal) tensor product of $C^*$-algebras
See p. 434 in:



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> D. Avitzour, [Free products of C\*-algebras](http://www.ams.org/journals/tran/1982-271-02/S0002-9947-1982-0654842-1/S0002-9947-1982-0654842-1.pdf), *Trans. Amer. Math. Soc.* **271** (1982), 423–435.
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Actually a more general statement holds true; namely tensor products of c.p. faithful maps are faithful.


