# special elements in a $C^{*}$-algebra and their spectral properties

Suppose $\mathcal{A}$ is a $C^{*}$-algebra (http://planetmath.org/CAlgebra). An element $x\in\mathcal{A}$ is said to be:

• normal if $x^{*}x=xx^{*}$

• if $x^{*}=x$

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

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

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

• partial isometry if $x^{*}x$ and $xx^{*}$ 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 $\|x\|=R_{\sigma}(x)$

Theorem 2 - Suppose $\mathcal{A}$ is a $C^{*}$-algebra and $x\in\mathcal{A}$.

• If $x$ is self-adjoint, then $\sigma(x)\subset\mathbb{R}$.

• If $x$ is unitary, then $\sigma(x)\subset\partial D$, where $D\subset\mathbb{C}$ is the unit disk.

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

• 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

• $x$ is self-adjoint if and only if $\sigma(x)\subset\mathbb{R}$.

• $x$ is unitary if and only if $\sigma(x)\subset\partial D$, where $D\subset\mathbb{C}$ is the unit disk.

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

• $x$ 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

• $x$ is self-adjoint if and only if $\sigma(x)\subset\mathbb{R}$.

• $x$ is unitary if and only if $\sigma(x)\subset\partial D$, where $D\subset\mathbb{C}$ is the unit disk.

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

• $x$ is a projection if and only if $\sigma(x)\subset\{0,1\}$

Title special elements in a $C^{*}$-algebra and their spectral properties SpecialElementsInACalgebraAndTheirSpectralProperties 2013-03-22 17:28:36 2013-03-22 17:28:36 asteroid (17536) asteroid (17536) 11 asteroid (17536) Definition msc 46L05 normal elements and spectral radius spectrum of self-adjoint elements spectrum of unitary elements spectrum of projections spectrum of positive elements