Forgive me if my question is too elementary however I haven't found this issue discussed anywhere. Suppose that $A$ is arbitrary $C^*$-algebra and consider the multiplication map $A \overline{\otimes} A \to A$ where on the left hand side is some $C^*$-tensor product. Can one say when this map is bounded (as a linear operator)?/how does it depends from the choice of $C^*$-norm on tensor product? For example if $A$ is commutative then the multiplication map is $*$-homomorphism, therefore it is norm decreasing since it acts between $C^*$-algebras. Note that in case of $A$ being commutative the problem with the various choices of tensor product is irrelevant, since every commutative $C^*$-algebra is nuclear.
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1$\begingroup$ Since the map factorises through the minimal tensor product, the question is equivalent to asking whether or not $A\otimes_{min} A\to A$ is bounded. $\endgroup$– Johannes HahnCommented Sep 23, 2016 at 21:41
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9$\begingroup$ This question was asked and answered here: mathoverflow.net/questions/171624/continuity-of-the-product-map/… $\endgroup$– Caleb EckhardtCommented Sep 23, 2016 at 23:53
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