Let $A$ be a C*-algebra and $\phi_{i}: A\rightarrow M_{k(i)}(\mathbb{C})$ (the $M_{k(i)}(\mathbb{C})$ denotes the $k(i)\times k(i)$ complex matrices) be c.c.p maps which are asymptotically multiplicative (i.e., $||\phi_{n}(ab)-\phi_{n}(a)\phi_{n}(b)||\rightarrow 0$ for all $a, b\in A$) and asymptotically isometric (i.e., $||a||=limi_{n\rightarrow\infty}||\phi_{n}(a)||$ for all $a\in A$).
My question:
Can we prove that $\oplus\phi_{i}: A\rightarrow \prod M_{k(i)}(\mathbb{C})$ is a *-homomorphism? (Here, the $\prod M_{k(i)}(\mathbb{C})$ denote the direct products of $M_{k(i)}(\mathbb{C})$.)
Certainly not, if any of the $\phi_i$ is not a homomorphism. If $\phi_k(ab)\ne\phi_k(a)\phi_k(b)$ for certain $a,b\in A$, then $$\bigoplus\phi_n(ab)\ne\left(\bigoplus\phi_n(a)\right)\left(\bigoplus\phi_n(b)\right), $$ as they differ in the $k^{\rm th}$ coordinate.