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

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

Research output: Contribution to journalArticle

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
JournalHouston Journal of Mathematics
Publication statusAccepted/In press - 24 Jan 2019

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Corona
Equivalence relation
Closed
Algebra
Extremally Disconnected
Quotient Map
Ultrapower
Multiplier Algebra
Closed Ideals
P-point
Compact Hausdorff Space
Lindelöf
Zero set
Compact Operator
Faithful
Irreducible Representation
Continuous Function
Hilbert space
Norm

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Minimal primal ideals in the inner corona algebra of a C_{0}(X)-algebra. / Archbold, Robert J. (Corresponding Author); Somerset, Douglas W. B.

In: Houston Journal of Mathematics, 24.01.2019.

Research output: Contribution to journalArticle

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