# example of Banach algebra which is not a $C^{*}$-algebra for any involution

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

Claim - $\mathcal{A}$ is not a $C^{*}$-algebra (http://planetmath.org/CAlgebra) for any involution $*$.

To prove the above claim we will give a proof of a more general fact about finite dimensional $C^{*}$-algebras, which clearly shows the for a Banach algebra to be a $C^{*}$-algebra for some involution.

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

Proof : Let $\mathcal{B}$ be a finite dimensional $C^{*}$-algebra. Let $a$ 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 $(a^{*}a)^{n}=0$. Then, by the $C^{*}$ condition and the fact that $a^{*}a$ is selfadjoint,

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

so $a=0$ 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 $C^{*}$-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 taken over the other diagonal of the matrix).

Title example of Banach algebra which is not a $C^{*}$-algebra for any involution ExampleOfBanachAlgebraWhichIsNotACalgebraForAnyInvolution 2013-03-22 17:25:54 2013-03-22 17:25:54 asteroid (17536) asteroid (17536) 5 asteroid (17536) Example msc 46L05 finite dimensional $C^{*}$-algebras are semi-simple