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?

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

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

Example of this situation is $A=C(X)$ where $X$ is the Cantor set or $A=\mathcal{K}$ where $\mathcal{K}$ is the algebra of compact operators on a separable Hilbert space.

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?