$^*$-homomorphism $\pi_1 : M_2(A)\to B(K_1)$ is induced by $\pi : A\to B(K)$

Question

Let $A$ be a unital $C^*$-algebra and let $\pi_1:M_2(A)\to B(K_1)$ be a $^*$-homomorphism.

I want to know if $K_1$ may be decomposed as $K_1=K\oplus K$ in such a way that the $^*$-homomorphism $\pi_1:M_2(A)\to B(K\oplus K)$ has the form $$\pi_1\begin{bmatrix}a & b \\ c&d \end{bmatrix}=\begin{bmatrix}\pi(a) & \pi(b) \\ \pi(c)&\pi(d) \end{bmatrix},$$ in other words $\pi_1$ is just $\pi^{(2)}$, where $\pi: A\to B(K)$ is a $^*$-homomorphism.

Thank you!

Answer 1

Yes. Define $K=\pi_1(E_{11})K_1$. Define $W:K\oplus K\to K_1$ by $$ W(\xi,\eta)=\pi_1(E_{11})\xi+\pi_1(E_{21})\eta. $$ Then check that $W$ is a unitary. Define $\pi:A\to B(K)$ by $$ \pi(a)\eta=\pi_1\left(\begin{bmatrix} a&0\\0&0\end{bmatrix}\right)\eta,\ \ \ \ \eta\in K. $$ It is straightforward to check that $\pi$ is a representation.

Finally, prove that $$ \pi_1(x)W=W\pi^{(2)}(x) $$ for all $x\in M_2(A)$.