the closure of union of $C^*$-algebras

Question

Suppose $(A_n)$ is a sequence of $C^*$-algebras,let $B$ be the closure of $\cup A_n$,is $B$ also a $C^*$-algebra?I know the fact that if $A_n$ is increasing,the conclusion is true.(I can show that it is a Banach $*$ algebra,how to show that it satisfies the $C^*$ equation?)

Answer 1

An arbitrary union of subspaces is not in general a subspace. For instance, in $\mathbb C\oplus\mathbb C$ take $A_1=\mathbb C\,(1,0)$, $A_2=A_3=\cdots=\mathbb C\,(0,1)$. Then $\bigcup_nA_n$ does not contain $(1,1)=(1,0)+(0,1)$.