In mathematics, a hereditary C*-subalgebra of a C*-algebra is a particular type of C*-subalgebra whose structure is closely related to that of the larger C*-algebra.
[1] There is a bijective correspondence between closed left ideals and hereditary C*-subalgebras 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).
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.