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We associate a Bratteli-type diagram to AH-algebras arising from generalizeddiagonal connecting maps. We use this diagram to give an explicit descriptionof the connected components of the spectrum of an associated canonicalC$^*$-diagonal. We introduce a topological notion on these connectedcomponents, that of being spectrally incomplete, and use it as a tool to showhow various classes of AI-algebras, including certain Goodearl algebras andAH-algebra models for dynamical systems $([0,1],\sigma)$, do not admit uniqueinductive limit Cartan subalgebras. We focus on a class of spectrally completeC$^*$-algebras, namely the AF-algebras, and prove that they admit uniqueinductive limit Cartan subalgebras. This is done via a method that generalizesElliott's proof of classification of AF-algebras to include Cartan pairs.

C$^*$-diagonals in AH-algebras arising from generalized diagonal connecting maps: spectrum and uniqueness