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Let $A$ be a $C^*$ Algebra. Let $J$ be a closed ideal in A . Let $B$ be $C^*$-sub-algebra of $A$. Prove $B+J$ is complete space (i.e. every cauchy sequence in $B+J$ converges to an element of $B+J$).

Denote by $x_n + y_n$ a Cauchy sequence in $B+J$ ($x_n \in B$, $y_n \in J$). We need to show $x_n$ is Cauchy sequence in $B$ and $y_n$ is Cauchy sequence in $J$. It seems we need to use the fact $J$ is ideal, But how ?

Any hints ?

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    In general $\{x_n\}$ and $\{y_n\}$ need not be Cauchy. Let $A = B = J = \mathbb{C}$ and take $x_n = n$, $y_n = -n$. Then $\{x_n + y_n\}$ is Cauchy but neither $\{x_n\}$ nor $\{y_n\}$ is. So that can't be the right approach.2012-05-24

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