decomposition of selfadjoint elements in positive and negative parts
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decomposition
Every real valued function $f$ admits a wellknown decomposition into its and parts: $f={f}_{+}{f}_{}$. There is an analogous result for selfadjoint elements in a ${C}^{*}$algebra (http://planetmath.org/CAlgebra) that we will now describe.
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Theorem  Let $\mathcal{A}$ be a ${C}^{*}$algebra and $a\in \mathcal{A}$ a selfadjoint element. Then there are unique positive elements^{} ${a}_{+}$ and ${a}_{}$ in $\mathcal{A}$ such that:

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$a={a}_{+}{a}_{}$

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${a}_{+}{a}_{}={a}_{}{a}_{+}=0$

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Both ${a}_{+}$ and ${a}_{}$ belong to ${C}^{*}$subalgebra generated by $a$.

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$\parallel a\parallel =\mathrm{max}\{\parallel {a}_{+}\parallel ,\parallel {a}_{}\parallel \}$
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Remark  As a particular case, the result provides a decomposition of each selfadjoint operator $T$ on a Hilbert space^{} as a difference of two positive operators $T={T}_{+}{T}_{}$ such that $\mathrm{Ran}{T}_{}\subseteq \mathrm{Ker}{T}_{+}$ and $\mathrm{Ran}{T}_{+}\subseteq \mathrm{Ker}{T}_{}$, where $\mathrm{Ran}$ and $\mathrm{Ker}$ denote, respectively, the range and kernel of an operator^{}.
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Proof:
Let us some notation first:

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$\sigma (a)$ denotes the spectrum of $a\in \mathcal{A}$.

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${C}^{*}[a]$ denotes the ${C}^{*}$subalgebra generated by $a$.

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${C}_{0}\left(\sigma (a)\setminus \{0\}\right)$ denotes the algebra of continuous functions^{} in $\sigma (a)\setminus \{0\}$ that vanish at infinity.
Let $f,{f}_{+},{f}_{}\in {C}_{0}\left(\sigma (a)\setminus \{0\}\right)$ be the functions defined by
$f(t):=t\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{f}_{+}(t):=\{\begin{array}{cc}t,\hfill & \text{if}t\ge 0\hfill \\ 0,\hfill & \text{if}t\le 0\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{f}_{}(t):=\{\begin{array}{cc}0,\hfill & \text{if}t\ge 0\hfill \\ t,\hfill & \text{if}t\le 0\hfill \end{array}$ 
Since $a$ is , $\sigma (a)\subseteq \mathbb{R}$, so the above functions are well defined. It is clear that
$f={f}_{+}{f}_{}\text{and}{f}_{+}{f}_{}={f}_{}{f}_{+}=0\text{and}{f}_{+},{f}_{}\text{are both positive}$  (1) 
The continuous functional calculus gives an isomorphism^{} ${C}^{*}[a]\cong {C}_{0}\left(\sigma (a)\setminus \{0\}\right)$ such that the element $a$ corresponds to the function $f$. Let ${a}_{+}$ and ${a}_{}$ be the elements corresponding to ${f}_{+}$ and ${f}_{}$ respectively. From the made in (1) it is now clear that

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${a}_{+}$ and ${a}_{}$ are both positive elements.

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$a={a}_{+}{a}_{}$

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${a}_{+}{a}_{}={a}_{}{a}_{+}=0$

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Both ${a}_{+}$ and ${a}_{}$ belong to ${C}^{*}[a]$.
From the fact the every ${C}^{*}$isomorphism is isometric (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)) and $\parallel f\parallel =\mathrm{max}\{\parallel {f}_{+}\parallel ,\parallel {f}_{}\parallel \}$ it follows that $\parallel a\parallel =\mathrm{max}\{\parallel {a}_{+}\parallel ,\parallel {a}_{}\parallel \}$.
The uniqueness of the decomposition follows from the uniqueness of the decomposition of real valued functions in its positive and negative parts $f={f}_{+}{f}_{}$ (with ${f}_{+}{f}_{}=0$). $\mathrm{\square}$
Title  decomposition of selfadjoint elements in positive and negative parts 

Canonical name  DecompositionOfSelfadjointElementsInPositiveAndNegativeParts 
Date of creation  20130322 17:51:49 
Last modified on  20130322 17:51:49 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  12 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 47C15 
Classification  msc 47B25 
Classification  msc 47A60 
Classification  msc 46L05 
Related topic  CAlgebra 