decomposition of self-adjoint elements in positive and negative parts

decomposition

Every real valued function $f$ admits a well-known decomposition into its and parts: $f=f_{+}-f_{-}$. There is an analogous result for self-adjoint elements in a $C^{*}$-algebra (http://planetmath.org/CAlgebra) that we will now describe.

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

• $a=a_{+}-a_{-}$

• $a_{+}{a}_{-}=a_{-}{a}_{+}=0$

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

• $\parallel a\parallel =\max \{\parallel a_{+}\parallel ,\parallel a_{-}\parallel \}$

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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 some notation first:

• $\sigma (a)$ denotes the spectrum of $a\in 𝒜$.

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

• $C_0\left(\sigma (a)\setminus \{0\}\right)$ 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

Since $a$ is , $\sigma (a)\subseteq ℝ$, 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

• $a_{+}$ 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 (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)) and $\parallel f\parallel =\max \{\parallel f_{+}\parallel ,\parallel f_{-}\parallel \}$ it follows that $\parallel a\parallel =\max \{\parallel a_{+}\parallel ,\parallel a_{-}\parallel \}$.

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 $f_{+}{f}_{-}=0$). $\mathrm{\square }$