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Hereditary C*-subalgebra
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In operator algebras, a hereditary C*-subalgebra of a C*-algebra A is a particular type of C*-subalgebra whose structure is closely related to that of A. 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. If a C*-algebra A contains a projection p, then the C*-subalgebra pAp, called a corner, is hereditary. Slightly more generally, given a positive a ∈ A, the closure of the set aAa is the smallest hereditary C*-subalgebra containing a, denoted by Her. If A is unital and the positive element a is invertible, we see that Her = A. This suggests the following notion of strict positivity for the non-unital case: a ∈ A is said to be strictly positive if Her = A. For instance, in the C*-algebra K of compact operators acting on Hilbert space H, c ∈ K is strictly positive if and only if the range of c is dense in H. 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. [ - ]