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Let $A$ be a Banach algebra. Is there any relation ship between two-sided closed ideals of $A$ and two-sided closed ideals of the operator algebra $\mathscr B(A)$? Is there any characterization for ideals of $\mathscr B(A)$?

Tomasz Kania
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Zeinab
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    Just a simple observation. If $A = \mathbb{C}^n$, then $B(A) \cong M_n(\mathbb{C})$. In this case, $A$ has $2^n$ distinct ideals: one for each subset of ${1,\ldots,n}$. On the other hand, $B(A)$ is simple and has only $2$ ideals. – Mike F Jul 02 '15 at 07:07
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    As for your second question, the ideal structure of $\mathscr B(A)$ is usually a complicated issue. See p. 2 in http://arxiv.org/pdf/1112.4800v1.pdf

    Actually there are very few Banach algebras for which the ideal structure of $\mathscr B(A)$ is completely understood.

     – Tomasz Kania Jul 02 '15 at 22:33

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