TITLE: Non-$C^{*}$ Banach algebras?
QUESTION [7 upvotes]: It suddenly occurred to me almost every Banach algebra I know is actually a $C^{*}$ algebra. Several kinds of function algebras are definitely $C^{*}$ algebras. So is the matrix algebra. Although one gets a non-$C^*$ algebra by focusing on the upper triangular matrices, the norm still satisfies the $C^*$ identity.
The algebra of operators on a general banach space is not $C^*$, but at least for me this is a too abstract class that do not provide much intuition.
Thus I wonder whether someone has some good examples of banach algebras that fail the $C^*$ identity but are on the other hand elementary enough to provide intuition and direct computation, like the function algebras.
Thanks!

REPLY [5 votes]: I must say I find it odd (and a bit worrisome) that one can find textbooks whereby you learn the definition of a Banach algebra (and even a Banach *-algebra) without seeing examples that are not $C^*$-algebras. There is more to life than $B(H)$...
Anyway, some commutative examples which are naturally algebras of functions. In every case the involution is just conjugation of functions.
1) For $G$ a locally compact abelian group (think ${\mathbb Z}^k$ or ${\mathbb T}^k$ or ${\mathbb R}^k$) with dual group $\Gamma$, take
$$A(G) = \{ f\in C_0(G) \mid \widehat{f} \in \ell^1(\Gamma) \} $$
the so-called Fourier algebra of $G$. (One can define $A(G)$ for arbitrary locally compact groups but the definition is more technical.)
2) Algebras of Lipschitz/H\"older functions. Take your favourite compact metric space $(X,d)$, take some $0<\alpha<1$, and define
$$ L_\alpha(f) = \sup_{x,y\in K; x\neq y} \frac{ \vert f(x)-f(y) \vert }{d(x,y)^\alpha} $$
then take
$$ {\rm Lip}_\alpha(X,d) = \{ f: X\to {\mathbb C} \mid L_\alpha(f)<\infty \} $$
equipped with the norm $\Vert f \Vert_\alpha := \Vert f\Vert_{\infty} + L_\alpha(f)$.
3) The algebra $C^k[0,1]$ of $k$-times continuously differentiable functions on $[0,1]$ (for $k\geq 1$), equipped with the natural norm built out of the sup-norms of the derivatives.
If you are willing to consider Banach algebras without involution then there are ${\rm many}^{\rm many}$ more examples.