Characterizations of certain classes of hereditary 𝐶*-subalgebras

Abstract

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.