Characterizations of certain classes of hereditary $C^ *$-subalgebras

Abstract:

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