Spectra of abelian C-subalgebra sums

Let Cb(X) be the C∗-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space X. Moreover, let A0 be some ideal and A1 be some unital C-subalgebra of Cb(X). For A0 and A1 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.