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Every real valued function  admits a well-known decomposition into its positive and negative parts:
 . (These functions are defined within the entry Lebesgue integral.) There is an analogous result for self-adjoint elements in a  -algebra that we will now describe.
Theorem - Let
 be a  -algebra and
 a self-adjoint element. Then there are unique positive elements  and  in
 such that:
Remark - As a particular case, the result provides a decomposition of each self-adjoint operator  on a Hilbert space as a difference of two positive operators
 such that
 and
 , where
 and
 denote, respectively, the range and kernel of an operator.
Proof:
Let us fix some notation first:
Let
 be the functions defined by
Since  is self-adjoint,
 , so the above functions are well defined. It is clear that
 and and are both positive |
(1) |
The continuous functional calculus gives an isomorphism
![$ C^*[a] \cong C_0\big(\sigma(a)\setminus \{0\}\big)$](http://images.planetmath.org:8080/cache/objects/10341/l2h/img40.png "$ C^*[a] \cong C_0\big(\sigma(a)\setminus \{0\}\big)$") such that the element  corresponds to the function  . Let  and  be the elements corresponding to  and  respectively. From the observations made in (1) it is now clear that
 and  are both positive elements.-
-
- Both
 and  belong to ![$ C^*[a]$](http://images.planetmath.org:8080/cache/objects/10341/l2h/img53.png "$ C^*[a]$") .
From the fact the every  -isomorphism is isometric (see this entry) and
 it follows that
 .
The uniqueness of the decomposition follows from the uniqueness of the decomposition of real valued functions in its positive and negative parts
 (with
 ). 
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![$ C^*[a]$](http://images.planetmath.org:8080/cache/objects/10341/l2h/img28.png "$ C^*[a]$"){kind=small}
