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Let $A$ be a Banach algebra and $M(A)$ be its multiplier Banach algebra. Is there any correspondence between closed two sided idaels of $A$ and closed two sided idaels of $M(A)$?

Can we see that every closed two sided ideal of $A$ is a closed two sided ideal of $M(A)$?

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    $\begingroup$ The answer to your question is yes when $A$ is closed in its multiplier algebra and usually no if $A$ is not closed in its multiplier algebra. $\endgroup$
    – Yemon Choi
    Commented Oct 9, 2016 at 2:52
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    $\begingroup$ Clearly not every ideal in M(A) will be an ideal in A; while if J is an ideal in M(A) then clearly A\cap J is an ideal in A. I am therefore not sure you are hoping to find as a "correspondence" between the set of ideals in A and the set of ideals in M(A) $\endgroup$
    – Yemon Choi
    Commented Oct 9, 2016 at 2:54
  • $\begingroup$ Thanks. Can you recommend me a book or a paper aboat your first answer?(The answer to your question is yes when A is closed in its multiplier algebra and usually no if A is not closed in its multiplier algebra) $\endgroup$
    – Albert harold
    Commented Oct 9, 2016 at 6:59
  • $\begingroup$ Provided that you know the definition of the multiplier algebra, you should not need a book. My first comment just relates to your second question, not the first question $\endgroup$
    – Yemon Choi
    Commented Oct 9, 2016 at 11:39

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