Hereditary C*-subalgebra
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. A C*-subalgebra "B" of "A" is a hereditary C*-subalgebra if for all "a" ∈ "A" and "b" ∈ "B" such that 0 ≤ "a" ≤ "b", we have "a" ∈ "B". Correspondence with closed left ideals. 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. Connections with positive elements. 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 formula_1 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.