Examples of Banach $^*$-algebras where $^*$ is not an isometry I'm reading up on $C^*$-algebras at the moment, and in the Wiki article (for instance) on the topic they note that the condition $||a^*a||=||a||^2$ implies that the $^*$-involution is an isometry.
But what are examples of Banach $^*$-algebras where the $^*$-involution is not an isometry? While there must be a reason why it's noted in particular that the $C^*$ condition implies that $||a^*||=||a||$, I don't know any examples of a Banach $^*$-algebra $B$ where $||a^*||\neq ||a||$ for some $a\in B$.
As a bonus question, what are some examples of Banach $^*$-algebras, where the $^*$-involution is an isometry, but the $C^*$ condition does not hold?
 A: Concerning the bonus question: Consider the Disk algebra $A(\mathbb{D})$ (the continuous functions on $\overline{\mathbb{D}}$ which are holomorphic on
$\mathbb{D}$, endowed with the maximum norm $||\cdot||$) and set
$$
f^\ast(z)=\overline{f(\overline{z})}.
$$
Then $\ast$ is an involution. It is an isometry, but for $f(z)=z+i$ we have
$f^\ast(z)=z-i$, hence $f^\ast (z)f(z)=z^2+1$. Thus $||f^\ast f||= 2$ but
$||f||^2=4$. Also note that $g(z)=z$ is selfadjoint with spectrum
$\sigma(g)=\overline{\mathbb{D}}$. This also shows that we have no $C^\ast$-algebra, as in that case the spectrum of a selfadjoint element has to be real.
Edit: Concerning the underlying question: Consider the matrix algebra $\mathbb{C}^{n\times n}$ endowed with the row-sum norm $||\cdot||_\infty$. This is a Banach algebra, as the row-sum norm is an operator norm. Let $\ast$ be the usual matrix involution. Then $||A^\ast||_\infty = ||A||_1$ (with $||\cdot||_1$ the column-sum norm). It is easy to find matrices with $||A||_1 \not= ||A||_\infty$, for those
$||A^\ast||_\infty \not= ||A||_\infty$.
