Spectra of Abelian C*-Subalgebra Sums

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate before. Within the whole framework, e.g., the one-point compactification of $X$ and the spectrum of the algebra of asymptotically almost periodic functions can be described.