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

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

• normal if $x^{*}x=x{x}^{*}$

• self-adjoint if $x^{*}=x$

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

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

• projection if $x^{*}=x$ and $x^2=x$

• 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 $𝒜$ is a $C^{*}$-algebra and $x\in 𝒜$. If $x$ is normal then $\parallel x\parallel =R_{\sigma }(x)$

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

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

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

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

• If $x$ is a projection, then $\sigma (x)\subset \{0,1\}$

Theorem 3 - Suppose $𝒜$ is a commutative $C^{*}$-algebra and $x\in 𝒜$. Then

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

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

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

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

Theorem 4 - Suppose $𝒜$ is a $C^{*}$-algebra and $x$ is normal in $𝒜$. Then

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

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

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

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