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[parent] special elements in a $C^*$-algebra and their spectral properties (Definition)

Definition - Suppose $ \mathcal{A}$ is a $ C^*$-algebra. An element $ x \in \mathcal{A}$ is said to be:

  • normal if $ x^*x = xx^*$
  • self-adjoint if $ x^* = x$
  • unitary if $ \mathcal{A}$ has an identity element $ e$ and $ x^*x = xx^* = e$
  • positive 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

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 $ \Vert x\Vert = 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\}$



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Also defines:  normal elements and spectral radius, spectrum of self-adjoint elements, spectrum of unitary elements, spectrum of projections, spectrum of positive elements

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Cross-references: commutative, unit disk, spectral radius, spectrum, identity element
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This is version 8 of special elements in a $C^*$-algebra and their spectral properties, born on 2007-08-14, modified 2007-08-14.
Object id is 9862, canonical name is SpecialElementsInACAlgebraAndTheirSpectralProperties.
Accessed 1193 times total.

Classification:
AMS MSC: 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
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