special elements in a ${C}^{*}$algebra and their spectral properties
Definition  Suppose $\mathcal{A}$ is a ${C}^{*}$algebra^{} (http://planetmath.org/CAlgebra). An element $x\in \mathcal{A}$ is said to be:

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normal if ${x}^{*}x=x{x}^{*}$

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selfadjoint^{} if ${x}^{*}=x$

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unitary^{} if $\mathcal{A}$ has an identity element^{} $e$ and ${x}^{*}x=x{x}^{*}=e$

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positive^{} if $x={y}^{*}y$ for some element $y\in \mathcal{A}$

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projection if ${x}^{*}=x$ and ${x}^{2}=x$

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partial isometry if ${x}^{*}x$ and $x{x}^{*}$ are both projections
0.0.1 Properties of the special elements in terms of their spectrum
In the following $\sigma (x)$ denotes the spectrum of an element $x$ and ${R}_{\sigma}(x)$ its spectral radius.
Theorem 1  Suppose $\mathcal{A}$ is a ${C}^{*}$algebra and $x\in \mathcal{A}$. If $x$ is normal then $\parallel x\parallel ={R}_{\sigma}(x)$
Theorem 2  Suppose $\mathcal{A}$ is a ${C}^{*}$algebra and $x\in \mathcal{A}$.

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If $x$ is selfadjoint, then $\sigma (x)\subset \mathbb{R}$.

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If $x$ is unitary, then $\sigma (x)\subset \partial D$, where $D\subset \u2102$ is the unit disk.

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If $x$ is positive, then $\sigma (x)\subset {\mathbb{R}}^{+}$

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If $x$ is a projection, then $\sigma (x)\subset \{0,1\}$
Theorem 3  Suppose $\mathcal{A}$ is a commutative^{} ${C}^{*}$algebra and $x\in \mathcal{A}$. Then

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$x$ is selfadjoint if and only if $\sigma (x)\subset \mathbb{R}$.

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$x$ is unitary if and only if $\sigma (x)\subset \partial D$, where $D\subset \u2102$ is the unit disk.

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$x$ is positive if and only if $\sigma (x)\subset {\mathbb{R}}^{+}$

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$x$ is a projection if and only if $\sigma (x)\subset \{0,1\}$
Theorem 4  Suppose $\mathcal{A}$ is a ${C}^{*}$algebra and $x$ is normal in $\mathcal{A}$. Then

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$x$ is selfadjoint if and only if $\sigma (x)\subset \mathbb{R}$.

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$x$ is unitary if and only if $\sigma (x)\subset \partial D$, where $D\subset \u2102$ is the unit disk.

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$x$ is positive if and only if $\sigma (x)\subset {\mathbb{R}}^{+}$

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$x$ is a projection if and only if $\sigma (x)\subset \{0,1\}$
Title  special elements in a ${C}^{*}$algebra and their spectral properties 

Canonical name  SpecialElementsInACalgebraAndTheirSpectralProperties 
Date of creation  20130322 17:28:36 
Last modified on  20130322 17:28:36 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  11 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46L05 
Defines  normal elements and spectral radius 
Defines  spectrum of selfadjoint elements 
Defines  spectrum of unitary elements 
Defines  spectrum of projections 
Defines  spectrum of positive elements 