Set of Positive Elements of C*-Algebra is Closed

Theorem

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

Let $A^+$ be the set of positive elements of $A$.

Then $A^+$ is closed.

Proof

Let $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ be the unitization of $A$.

Let ${\mathbf 1}_+$ be the identity element of $A_+$.

Let $\iota : A \to A_+$ be the mapping defined by:

$\map \iota a = \tuple {a, 0}$

for each $a \in A$.

Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $A^+$ such that $a_n \to a$ in $\struct {A, \norm {\, \cdot \,} }$.

From Convergence in Direct Product Norm, $\sequence {\tuple {a_n, 0} }_{n \mathop \in \N}$ converges to $\struct {a, 0}$ in $\struct {A_+, \norm {\, \cdot \,}_\ast}$.

From Convergent Sequence in Normed Vector Space is Bounded, there exists $t \ge 0$ such that:

$\norm {\tuple {a_n, 0} } \le t$

From Element of C*-Algebra is Positive iff Positive in Unitization, $\tuple {a_n, 0}$ is positive for each $n \in \N$.

Hence from Characterization of Positive Element of Unital C*-Algebra, we must have:

$\norm {\tuple {a_n, 0} - t {\mathbf 1}_+} \le t$

Taking $n \to \infty$ we obtain:

$\norm {\tuple {a, 0} - t {\mathbf 1}_+} \le t$

from Norm is Continuous.

Hence by Characterization of Positive Element of Unital C*-Algebra, $\tuple {a, 0}$ is positive in $A_+$.

Hence by Element of C*-Algebra is Positive iff Positive in Unitization, $a \in A^+$.

Hence by the definition of a closed set in a normed vector space, $A^+$ is closed.

$\blacksquare$

Sources

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