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

Title	decomposition of self-adjoint elements in positive and negative parts
Canonical name	DecompositionOfSelfadjointElementsInPositiveAndNegativeParts
Date of creation	2013-03-22 17:51:49
Last modified on	2013-03-22 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