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YCor
YCor
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Let$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $soc(\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})$$\soc(\mathcal{B}) =\soc(\mathcal{A}) \oplus \soc(\mathcal{A})$?

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 $soc(\mathcal{B}) =soc(\mathcal{A}) \oplus soc(\mathcal{A})$?

$\DeclareMathOperator\soc{soc}$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 $\soc(\mathcal{B}) =\soc(\mathcal{A}) \oplus \soc(\mathcal{A})$?

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

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 $soc(\mathcal{B}) =soc(\mathcal{A}) \oplus soc(\mathcal{A})$?