A Dauns-Hofmann theorem for TAF-algebras

Let $A$ be a TAF-algebra, $Z(A)$ the centre of $A, Id(A)$ the ideal lattice of $A$, and $Mir(A)$ the space of meet-irreducible elements of $Id(A)$, equipped with the hull-kernel topology. It is shown that $Mir(A)$ is a compact, locally compact, second countable, $T_0$-space, that $Id(A)$ is an algebraic lattice isomorphic to the lattice of open subsets of $Mir(A)$, and that $Z(A)$ is isomorphic to the algebra of continuous, complex functions on $Mir(A)$. If $A$ is semisimple, then $Z(A)$ is isomorphic to the algebra of continuous, complex functions on $Prim(A)$, the primitive ideal space of $A$. If $A$ is strongly maximal, then the sum of two closed ideals of $A$ is closed.