Can you give an examples of non commutative non C*algebras?

Question

Are there examples apart from $B(X)$ where $X$ is not a Hilbert space and not finite dimensional. Do they have a characterization or representation?

Answer 1

Let $A$ be the algebra of all $2\times2$ matrices over $\mathbb{R}$ (or $\mathbb{C}$) of the form $$\begin{bmatrix} a & b \\ 0 & c \end{bmatrix} $$ Then $A$ is a Banach algebra, which is noncommutative, and is not a $C^*$-algebra.

Answer 2

If a complex finite dimensional algebra is a C^* algebra, then it is isomorphic to a direct product of matrix algebras. So any algebra which is not of this form gives you an example.