Draft:Partition of unity (*-algebra)

In mathematics, a partition of unity in a unital *-algebra $A$ is a set $\{p_{1},\dots ,p_{N}\}\subset A$ of projections $p_{i}=p_{i}^{*}=p_{i}^{2}$ that sum to the identity.:

$$\sum _{i=1}^{N}p_{i}=I.$$

In the case of $\mathrm {C} ^{*}$-algebras, it can be shown that the entries are pairwise-orthogonal:

$$p_{i}p_{j}=\delta _{i,j}p_{i}\qquad (p_{i},\,p_{j}\in R).$$

Examples

If $a$ is a normal element of a unital $\mathrm {C} ^{*}$-algebra $A$, and has finite spectrum $\sigma (a)=\{\lambda _{1},\dots ,\lambda _{N}\}$, then the projections in the spectral decomposition:

$$a=\sum _{i=1}^{N}\lambda _{i}\,P_{i},$$

form a partition of unity.

In the field of compact quantum groups, the rows and columns of the fundamental representation $u\in M_{N}(C)$ of a quantum permutation group $(C,u)$ form partitions of unity

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