Definition:Positive Element of C*-Algebra

Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.

Let $x \in A$ be Hermitian.

Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.

We say that $x$ is positive if and only if:

$\map {\sigma_A} x \subseteq \hointr 0 \infty$

Sources

• 1990: Gerard J. Murphy: C*-Algebras and Operator Theory ... (previous) ... (next): $2.2$: Positive Elements of $C^\ast$-Algebras