What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?

Question

$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals to the sum of the minimal left ideals) or $\{0\}$ if there are none minimal right ideals.

Let $\mathcal{B} = M_{2}(\mathcal{A}):=\{\left( \begin{array}{cc} a & c \\ d & b \\ \end{array} \right) : a,b,c,d \in \mathcal{A}$ }. Then $\mathcal{B}$ be a unital semisimple Banach algebra. What is the socle of $\mathcal{B}$? Is it true that $\soc(\mathcal{B}) =\soc(\mathcal{A}) \oplus \soc(\mathcal{A})$?

Answer 1

We have $soc(\mathcal B) =M_2(soc(\mathcal A))$.

For $I$ a minimal right ideal of $\mathcal A$, $\{\begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} \in M_2(\mathcal A) \mid a,b\in I \}$ is a right ideal of $\mathcal B$. It is not hard to check that this is minimal. The same is true of $\{\begin{pmatrix} 0 & 0 \\ a & b \end{pmatrix} \in M_2(\mathcal A) \mid a,b\in I \}$. The sum of these ideals is $M_2(I)$. Applied to all the minimal right ideals, we see $M_2( soc(\mathcal A)) \subseteq soc(\mathcal B)$.

Conversely, let $I$ be a minimal right ideal of $\mathcal B$. We obtain four right ideals of $\mathcal A$ from looking at the upper-left, upper-right, lower-left, or lower-right entries of the elements of $I$. Some of these may be zero but not all of them or else $I$ is zero. Any of these ideals that is not zero is minimal (for any smaller ideal $J$, we could consider the subset of $I$ consisting of elements with all entries in the same row contained in $J$). So all are contained in $soc(\mathcal A)$ and thus $I \subseteq M_2(soc(\mathcal A))$ so $soc(\mathcal B) \subseteq M_2(soc(\mathcal A))$.