Everything about Hereditary C -subalgebra totally explained
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've
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 (AF) C*-algebra is also AF. This isn't 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.
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