Let $A$ be an unital Banach *-algebra, let $Lg_{n}(A)$ be the set of n-tuple $(a_{1},..., a_{n})\in A^{n}$ such that there exists $(b_{1},..., b_{n})\in A^{n}$ with $\sum_{i=1}^{n}b_{i}a_{i}=1$. Then define topological rank of $A$ to $n$ if there is a smallest $n$ such that $Lg_{n}(A)$ is dense in $A^{n}$.
I know it is a dumb question, but what confuses me is, for $n=2$, $(a,1-a)$ belongs to $Lg_{2}(A)$ for all $a\in A$. That means $A^{2}\subset A^{2}$, which is dense. This implies all Banach *-algebras have stable rank 2, which is false. What mistakes have I make?