Calkin algebra of direct sum and direct sum of calkin algebra

Question

Calkin algebra is the quotient algebra of bounded linear operator module compact operator on Hilbert space. bounded operator on direct sum of Hilbert space is contained in the direct sum of bounded operator, and compact operator on direct sum of Hilbert space is contained in the direct sum of compact operator, so what about the Calkin algebra?

Answer 1

It looks to me that you are asking if $$ \bigoplus_j A_j/\bigoplus_j J_j \simeq \bigoplus A_j/J_j. $$ Let $\gamma:\bigoplus_j A_j/\bigoplus_j J_j \to \bigoplus A_j/J_j$ be given by $$ \gamma(\bigoplus_j a_j+\bigoplus_jJ_j)=\bigoplus_ja_j+J_j. $$ This is well-defined: if $\bigoplus (a_j'-a_j)\in\bigoplus_j J_j$, then $a_j'-a_j\in J_j$ for all $j$. It is trivial to check that is a $*$-homomorphism. It is one-to-one: if $\bigoplus_ja_j+J_j=\bigoplus_j J_j$, then $a_j\in J_j$ for all $j$, so $\bigoplus_j a_j\in \bigoplus_j J_j$. And onto: given $\bigoplus_j a_j+J_j$, this is clearly $\gamma(\bigoplus_j a_j+\bigoplus_j J_j)$.

So $\gamma$ is a $*$-isomorphism.