Hereditary C*-subalgebra

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(a). If A is unital and the positive element a is invertible, we see that Her(a) = 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) = A. For instance, in the C*-algebra K(H) of compact operators acting on Hilbert space H, c ∈ K(H) 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.

It follows from the correspondence between closed left ideals and hereditary C*-subalgebras that a closed ideal, which is a C*-subalgebra, is hereditary . Another corollary is that a hereditary C*-subalgebra of a simple C*-algebra is also simple.

A hereditary C*-subalgebra of an approximately finite-dimensional C*-algebra is also AF. This is not true for subalgebras that are not hereditary. For instance, every abelian C*-algebra can be embedded into an AF C*-algebra.

Two C*-algebras are stably isomorphic if they contain stably isomorphic hereditary C*-subalgebras. Also hereditary C*-subalgebras are those C*-subalgebras in which the restriction of any irreducible representation is also irreducible.

References

"Operator algebras: theory of C*-algebras and von Neumann algebras", B. Blackadar, Def II.3.4.1 p. 75

"A Course in Operator Theory", John B. Conway, def. 5.2 p. 21