Every real valued function $f$ admits a well-known decomposition into its positive and negative parts: $f = f_+ - f_-$ . There is an analogous result for self-adjoint elements in a $C^*$ -algebra that we will now describe.

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Theorem - Let $\mathcal{A}$ be a $C^*$ -algebra and $a \in \mathcal{A}$ a self-adjoint element. Then there are unique positive elements $a_+$ and $a_-$ in $\mathcal{A}$ such that:

• $a= a_+ - a_-$
• $a_+a_- = a_-a_+ = 0$
• Both $a_+$ and $a_-$ belong to $C^*$ -subalgebra generated by $a$ .
• $\|a\| = \max\{\|a_+\|, \|a_-\|\}$

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Remark - As a particular case, the result provides a decomposition of each self-adjoint 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 fix some notation first:

• $\sigma(a)$ denotes the spectrum of $a \in \mathcal{A}$ .
• $C^*[a]$ denotes the $C^*$ -subalgebra generated by $a$ .
• $C_0 \big(\sigma(a)\setminus \{0\}\big)$ denotes the algebra of continuous functions in $\sigma(a)\setminus \{0\}$ that vanish at infinity.

Let $f, f_+, f_- \in C_0\big(\sigma(a)\setminus \{0\}\big)$ be the functions defined by

andandare both positive

(1)

The continuous functional calculus gives an isomorphism $C^*[a] \cong C_0\big(\sigma(a)\setminus \{0\}\big)$ 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 observations made in (1) it is now clear that

• $a_+$ and $a_-$ are both positive elements.
• $a = a_+ - a_-$
• $a_+a_- = a_-a_+ = 0$
• Both $a_+$ and $a_-$ belong to $C^*[a]$ .

From the fact the every $C^*$ -isomorphism is isometric (see this entry) and $\|f\| = \max\{\|f_+\|, \|f_-\|\}$ it follows that $\|a\| = \max\{\|a_+\|, \|a_-\|\}$ .

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 $f_+f_- = 0$ ). $\square$