Hereditary C*-subalgebra

Let $B\subseteq A$ be a $C^*$ -subalgebra. It is hereditary if, whenever $a\in A$ and $b\in B$ are positive elements satisfying

0\leq a\leq b,

then $a\in B$ . Thus $B$ is downward closed inside the positive cone of the ambient algebra $A$ . Heredity depends on the specified inclusion $B\subseteq A$ , not only on the abstract isomorphism type of $B$ . Every closed two-sided ideal is hereditary, but a hereditary subalgebra need not absorb multiplication by arbitrary elements of $A$ .

Equivalent characterizations

For a $C^*$ -subalgebra $B\subseteq A$ , the following are equivalent:

$B$ is hereditary;
$BAB\subseteq B$ ; and
$b^*ab\in B$ for every $a\in A$ and $b\in B$ .

These formulations connect order in the positive cone with algebraic compression. They also show that $\overline{bAb}$ is a hereditary $C^*$ -subalgebra whenever $b\in A_+$ .

Corners and generated hereditary subalgebras

If $p$ is a projection in $A$ , the corner

pAp=\{pap:a\in A\}

is hereditary. More generally, for $S\subseteq A_+$ , the hereditary subalgebra generated by $S$ is the smallest hereditary $C^*$ -subalgebra containing $S$ ; for a single positive element $h$ , it is $\overline{hAh}$ . Functional calculus gives the same subalgebra as the closure of $h^{1/2}Ah^{1/2}$ .

Hereditary subalgebras correspond, in the bidual von Neumann algebra $A^{**}$ , to open projections. This correspondence is a principal reason they are useful for support, Morita equivalence, and the internal geometry of ideals.

Examples and distinctions

Every closed two-sided ideal $I\triangleleft A$ is hereditary. The corner $pM_n(\mathbb C)p$ is hereditary in $M_n(\mathbb C)$ , but for a noncentral projection $p$ it is not an ideal. This demonstrates that heredity is strictly weaker than the two-sided absorption property.

The scalar subalgebra $\mathbb C1\subseteq M_n(\mathbb C)$ , for $n>1$ , is not hereditary: a nontrivial projection $q$ satisfies $0\leq q\leq1$ , yet $q\notin\mathbb C1$ . Hence being unital, closed, or stable under functional calculus does not by itself imply heredity in the ambient algebra.

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.5 on hereditary $C^*$ -subalgebras and positive order.
Bruce Blackadar, Operator Algebras: Theory of C-Algebras and von Neumann Algebras*, Springer, 2006. DOI record. Relevant: §II.3 on hereditary subalgebras, open projections, and corners.