Strictly positive element of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A positive element $h\in A_+$ is strictly positive if the hereditary $C^*$ -subalgebra it generates is all of $A$ :

\overline{hAh}=A.

Equivalently, every nonzero positive linear functional $\omega$ on $A$ satisfies $\omega(h)>0$ . Strict positivity therefore means that $h$ has full support throughout $A$ ; it is stronger than $h\geq0$ and does not, in a nonunital algebra, mean that $h$ is invertible inside $A$ .

Equivalent characterizations

After rescaling so that $0\leq h\leq1$ , the following are equivalent: $h$ is strictly positive; $\overline{hA}=\overline{Ah}=A$ ; and the sequence

e_n=f_n(h),\qquad f_n(t)=\min\{nt,1\},

is an approximate identity for $A$ . Consequently, $A$ contains a strictly positive element exactly when it is $\sigma$ -unital Pedersen, §1.4.

Examples and non-examples

In a unital $C^*$ -algebra, a positive element is strictly positive exactly when it is positive and invertible. In $C_0(\mathbb R)$ , the function

h(x)=e^{-x^2}

is strictly positive even though it is not invertible as an element of $C_0(\mathbb R)$ . More generally, a positive function in $C_0(X)$ is strictly positive exactly when it is nonzero at every point of $X$ . The function $x^2e^{-x^2}$ is positive but not strictly positive because evaluation at $0$ annihilates it.

For $K(H)$ , a positive compact operator is strictly positive precisely when it has trivial kernel and dense range. Such an operator exists only when $H$ is separable.

Conventions and scope

“Strictly positive” has a specialized meaning here. It does not mean that there is an $\varepsilon>0$ with $h\geq\varepsilon1$ in the unitization; that stronger inequality would make $h$ invertible and cannot hold for typical nonunital examples. Likewise, “full positive element” can refer to generation of the whole closed two-sided ideal, a weaker condition than generating the whole hereditary subalgebra.

References

Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.4 on strictly positive elements, hereditary subalgebras, and countable approximate units.