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 0 ≤ "a" ≤ "b", where "b" ∈ "B" and "a" ∈ "A", 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" $mapsto$ "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 (AF) C*-algebra is also AF. This is not true in general. 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.