I am trying to show that the $K$-theory groups of the following $C^*$-algebra $A$ vanish:
Let $\mathcal{H}$ be a separable Hilbert space. Now consider the subalgebra of $\mathcal{B}(\mathcal{H}\oplus\mathcal{H})$ given by those matrices
\begin{pmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{pmatrix}
where $T_{11}, T_{12}$ and $T_{21}$ are compact and $T_{22}$ is arbitrary but bounded. Schematically this is the algebra
$A = \begin{pmatrix} \mathcal{K} & \mathcal{K} \\ \mathcal{K} & \mathcal{B} \end{pmatrix} $
where $\mathcal{K}$ denotes the compact operators of $\mathcal{H}$.
As is said I want to show $K_0(A) = 0 = K_1(A)$.
Unfortunately I do not really know what tools to use here. Any suggestions are appreciated.