# 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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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 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

 $\displaystyle f(t):=t\qquad\qquad f_{+}(t):=\begin{cases}t,&if\;\;t\geq 0\\ 0,&if\;\;t\leq 0\end{cases}\qquad\qquad f_{-}(t):=\begin{cases}0,&if\;\;t% \geq 0\\ -t,&if\;\;t\leq 0\end{cases}$

Since $a$ is , $\sigma(a)\subseteq\mathbb{R}$, so the above functions are well defined. It is clear that

 $\displaystyle f=f_{+}-f_{-}\;\;\;\text{and}\;\;\;f_{+}f_{-}=f_{-}f_{+}=0\;\;\;% \text{and}\;\;\;f_{+},f_{-}\;\text{are both positive}$ (1)

The continuous functional calculus gives an isomorphism $C^{*}[a]\cong C_{0}\big{(}\sigma(a)\setminus\{0\}\big{)}$ 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 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 $\|f\|=\max\{\|f_{+}\|,\|f_{-}\|\}$ 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 $f_{+}f_{-}=0$). $\square$

Title decomposition of self-adjoint elements in positive and negative parts DecompositionOfSelfadjointElementsInPositiveAndNegativeParts 2013-03-22 17:51:49 2013-03-22 17:51:49 asteroid (17536) asteroid (17536) 12 asteroid (17536) Theorem msc 47C15 msc 47B25 msc 47A60 msc 46L05 CAlgebra