Proposition 5

Let A be a unital Banach algebra and let B be a full subalgebra of A. Then B is a full subalgebra of A.

In fact, let $x$ be an element of $\overline{B}$ invertible in A, and $(x_n)$ a sequence of elements of B tending to $x$. Then, for $n$ sufficiently large, $x_n$ is invertible in A and $x^{-1}_n$ tends to $x^{-1}$. Since the subalgebra B is full, we have $x^{-1}_n\in B$, whence $x^{-1}\in \overline{B}$.

Let A be a unital Banach algebra and let $(y_i)_{i\in I}$ be a family of elements of A. Let B be the full subalgebra of A generated by the elements $y_i$. Then $\overline{B}$ is the smallest closed full subalgebra of A containing the $y_i$. It is called the closed full subalgebra generated by the elements $y_i$.