Hereditary C*-subalgebra

There is a bijective correspondence between closed left ideals and hereditary C*-subalgebras of A. If L ⊂ A is a closed left ideal, let L* denote the image of L under the *-operation. The set L* is a right ideal and L* ∩ L is a C*-subalgebra. In fact, L* ∩ L is hereditary and the map L ↦ L* ∩ L is a bijection. It follows from this correspondence that every closed ideal is a hereditary C*-subalgebra. Another corollary is that a hereditary C*-subalgebra of a simple C*-algebra is also simple.

If p is a projection of A (or a projection of the multiplier algebra of A), then pAp is a hereditary C*-subalgebra known as a corner of A. More generally, given a positive a ∈ A, the closure of the set aAa is the smallest hereditary C*-subalgebra containing a, denoted by Her(a). If A is separable, then every hereditary C*-subalgebra has this form.

These hereditary C*-subalgebras can bring some insight into the notion of Cuntz subequivalence. In particular, if a and b are positive elements of a C*-algebra A, then ${\displaystyle a\precsim b}$ if b ∈ Her(a). Hence, a ~ b if Her(a) = Her(b).

If A is unital and the positive element a is invertible, then Her(a) = A. This suggests the following notion for the non-unital case: a ∈ A is said to be strictly positive if Her(a) = A. For example, in the C*-algebra K(H) of compact operators acting on Hilbert space H, a compact operator is strictly positive if and only if its range is dense in H. A commutative C*-algebra contains a strictly positive element if and only if the spectrum of the algebra is σ-compact. More generally, a C*-algebra contains a strictly positive element if and only if the algebra has a sequential approximate identity.