Is there a name for the following property of a $C^{*}$ algebra $A$?

$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$

For any such $C^{*}$ algebra, after fixing an isomorphisms between the two algebras, one can consider the following functional equation

$$T(a\otimes b)=T(a) \otimes T(b), \;\;\; T(a^{*})=(T(a))^{*}$$ where $T$ is a linear operator on $A$.

Does this imply that $T$ is a bounded operator? Is there a non scalar example of such $T$ for $A=\mathcal{K}$, the algebra of compact operators?