Minimal Primal Ideals in the Multiplier Algebra of a C ₀(X)-algebra

Abstract

Let A be a stable, σ-unital, continuous C ₀(X)-algebra with surjective base map φ :$\mathrm{Prim}(A)\;\rightarrow X$, where Prim(A) is the primitive ideal space of the C*-algebra A. Suppose that $\phi^{-1}(x)$ is contained in a limit set in Prim(A) for each $x \in X$ (so that A is quasi-standard). Let C_R(X) be the ring of continuous real-valued functions on X. It is shown that there is a homeomorphism between the space of minimal prime ideals of C_R(X) and the space MinPrimal(M(A)) of minimal closed primal ideals of the multiplier algebra M(A). If A is separable then MinPrimal(M(A)) is compact and extremally disconnected but if $X = \beta \bf{N} \setminus \bf{N}$ then MinPrimal(M(A)) is nowhere locally compact.

Mathematics Subject Classification (2010). Primary 46L05, 46L08, 46L45; Secondary 46E25, 46J10, 54C35.