$Is the $\mathbb{C}$-algebra $Fun(X,\mathbb{C})$ semi-simple?$

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$Is the $\mathbb{C}$-algebra $Fun(X,\mathbb{C})$ semi-simple?$

Let $X$ be a finite set and consider the $\mathbb{C}$-algebra $Fun(X,\mathbb{C})$ of functions from $X$ to the complex numbers, with the obvious definitions of pointwise addition, multiplication and scalar multiplication. Is it semi-simple? In particular, how would we express it as a direct product of semi-simple algebras?

Algebra

Jbuck

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It is semi-simple, it is actually the algebra \[ \mathbb{C}^X = \bigoplus_{x \in X} \mathbb{C} \] with component-wise operations.

For every $x \in X$, let $\chi_x \colon X \rightarrow \mathbb{C}$ be defined as $\chi_x(x) = 1$ and $\chi_x(y) = 0$ for $y \neq x \in X$. Any function $f \colon X \rightarrow \mathbb{C}$ can be written as \[ f = \sum_{x \in X} f(x) \chi_x \] so this is a linear basis of $Fun(X, \mathbb{C})$ as a vector space, and it's clear that operations work component-wise.

Alessandro Iraci

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$Is the $\mathbb{C}$-algebra $Fun(X,\mathbb{C})$ semi-simple?$

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