This paper characterizes the class of full hereditary
C
∗
{C^ * }
-subalgebras and the class of hereditary
C
∗
{C^ * }
-subalgebras that generate essential ideals in a given
C
∗
{C^ * }
-algebra in terms of a certain projection of norm one from the enveloping von Neumann algebra of the
C
∗
{C^ * }
-algebra onto the enveloping von Neumann algebra of a hereditary
C
∗
{C^ * }
-subalgebra. For a
C
∗
{C^ * }
-dynamical system
(
A
,
G
,
α
)
(A,G,\alpha )
, it is also shown that an
α
\alpha
-invariant
C
∗
{C^ * }
-subalgebra
B
B
of
A
A
is a hereditary
C
∗
{C^ * }
-subalgebra belonging to either of the above classes if and only if the reduced
C
∗
{C^ * }
-crossed product
B
×
α
r
G
B{ \times _{\alpha r}}G
is a hereditary
C
∗
{C^ * }
-subalgebra, of the reduced
C
∗
{C^ * }
-crossed product
A
×
α
r
G
A{ \times _{\alpha r}}G
, belonging to the same class as
B
B
. Furthermore similar results for
C
∗
{C^ * }
-crossed products are also observed.