Minimal primal ideals in the inner corona algebra of a C_{0}(X)-algebra

Robert J. Archbold (Corresponding Author), Douglas W. B. Somerset

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Abstract

Let A = C(X) ⊗ K(H), where X is an infinite compact Hausdorff space and
K(H) is the algebra of compact operators on a separable, infinite-dimensional Hilbert space. Let As be the norm-closed ideal of the multiplier algebra M(A) consisting of all the strong∗ -continuous functions from X to K(H). Then As/A is the inner corona algebra of A. We identify the space MinPrimal (As) of minimal closed primal ideals in As. If A is separable then MinPrimal (As) is compact and extremally disconnected. Using ultrapowers, we exhibit a faithful family of irreducible representations of As/A and hence show that if every point of X lies in the boundary of a zero set (i.e. if X has no P-points) then the minimal closed primal ideals of As/A are precisely the images under the quotient map of the minimal closed primal ideals of As. The map between MinPrimal (As) and MinPrimal (As/A) need not be continuous, however, and MinPrimal (As/A) is not weakly Lindelof. As an application, it is shown that if X = βN\N then the relation of inseparability on Prim(As/A) is an equivalence relation but not an open equivalence relation.
Original languageEnglish
Pages (from-to)117-150
Number of pages24
JournalHouston Journal of Mathematics
Volume46
Issue number1
Publication statusPublished - 2020

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