PlanetMath: example of Banach algebra which is not a $C^*$-algebra for any involution

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example of Banach algebra which is not a $C^*$

-algebra for any involution

(Example)

Consider the Banach algebra $\mathcal{A}= \left\{ \begin{bmatrix} \lambda I_n & A \ 0 & \lambda I_n \end{... ...ix} :\; \lambda \in \mathbb{C},\;\; A \in Mat_{n \times n}(\mathbb{C}) \right\}$ with the usual matrix operations and matrix norm, where denotes the identity matrix in $Mat_{n \times n}(\mathbb{C})$ .

Claim - $\mathcal{A}$ is not a -algebra for any involution .

To prove the above claim we will give a simple proof of a more general fact about finite dimensional -algebras, which clearly shows the algebraic restrictions for a Banach algebra to be a -algebra for some involution.

Theorem - Every finite dimensional -algebra is semi-simple, i.e. its Jacobson radical is $\{0\}$ .

Proof : Let $\mathcal{B}$ be a finite dimensional -algebra. Let be an element of $J(\mathcal{B})$ , the Jacobson radical of $\mathcal{B}$ .

$J(\mathcal{B})$ is an ideal of $\mathcal{B}$ , so $a^*a \in J(\mathcal{B})$ .

The Jacobson radical of a finite dimensional algebra is nilpotent, therefore there exists $n \in \mathbb{N}$ such that . Then, by the condition and the fact that is selfadjoint,

$\displaystyle 0 = \Vert(a^*a)^{2^n}\Vert = \Vert a^*a\Vert^{2^n} = \Vert a\Vert^{2^{n+1}}$

and $J(\mathcal{B})$ is trivial. $\square$

We now prove the above claim.

Proof of the claim: It is easy to see that $\left\{ \begin{bmatrix} 0 & A \ 0 & 0 \end{bmatrix} : \; A \in Mat_{n \times n}(\mathbb{C}) \right\}$ is the only maximal ideal of $\mathcal{A}$ . Therefore the Jacobson radical of $\mathcal{A}$ is not trivial.

By the theorem we conclude that there is no involution that makes $\mathcal{A}$ into a -algebra. $\square$

Remark - It could happen that there were no involutions in $\mathcal{A}$ and so the above claim would be uninteresting. That's not the case here. For example, one can see that $[a_{i,j}] \longrightarrow [\overline{a}_{2n+1-j,2n+1-i}]$ defines an involution in $\mathcal{A}$ (this is just the conjugate transpose taken over the other diagonal of the matrix).