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Let $\left({\ell }^{2},{||.‖}_{2}\right)$ with the coordinatewise product.Prove that $\left({\ell }^{2},{||.‖}_{2}\right)$ is a commutative and semisimple Banach algebra
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timmeraared
The space ${\ell }^{2}$ is not isometrically isomorphic to any ${C}^{\ast }$-algebra. It is reflexive, and any reflexive ${C}^{\ast }$-algebra is finite-dimensional.
Clearly ${\ell }^{2}$ is a commutative Banach algebra. To prove that ${\ell }^{2}$ is semi-simple, we must show that the character space $\mathrm{\Gamma }$ of ${\ell }^{2}$ separates the points of ${\ell }^{2}$. This is easy enough: For every $k\in \mathbb{N}$ the map ${\delta }_{k}:{\ell }^{2}\to \mathbb{C}$ is linear, multiplicative and non-zero. Moreover, ${\delta }_{k}\left(f\right)=0$ for all $k\in \mathbb{N}$ implies $f=0$.