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
Canonical name	ExampleOfBanachAlgebraWhichIsNotACalgebraForAnyInvolution
Date of creation	2013-03-22 17:25:54
Last modified on	2013-03-22 17:25:54
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	5
Author	asteroid (17536)
Entry type	Example
Classification	msc 46L05
Defines	finite dimensional $C^{*}$ -algebras are semi-simple

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