Suppose that $(A,\parallel\cdot\parallel)$ is a normed algebra (with or without an identity) and $B\subseteq A$ is a subalgebra of $A$. It is said in my textbook that $B$, together with the induced norm, is also a normed algebra.
However, it is assumed that the norm of the identity in a unital normed algebra is $1$. If $B$ happens to contain an identity $\mathbf{1}_B$, then in order for $B$ to become a normed algebra, we need to show that $\parallel\mathbf{1}_B\parallel=1$. Since $\mathbf{1}_B$ is not necessarily the identity in $A$, it is not guaranteed that $\parallel\mathbf{1}_B\parallel=1$. So I think $B$ may not also be a normed algebra.
Any comments on this issue?
This is a subtle point, one that the author barely mentions, but a unital normed algebra is more than just a normed algebra with a unit. A unital normed algebra is a normed algebra $(A,\|\cdot\|)$ with a unit $1_A\in A$ such that $\|1_A\|=1$. While a subalgebra of a normed algebra is itself a normed algebra, it is not true that a unital subalgebra of a unital normed algebra is itself a unital normed algebra
Indeed, suppose $(A,\|\cdot\|)$ is a unital normed algebra containing an idempotent $e$ with $\|e\|>1$, and let $B=eAe$. Then $B$ is a subalgebra of $A$, with unit $e$, but it is not a normed algebra with the norm induced by $A$, as the unit of $B$ does not have norm one.