### Abstract

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 language | English |
---|---|

Journal | Houston Journal of Mathematics |

Publication status | Accepted/In press - 24 Jan 2019 |

### Fingerprint

### Cite this

*Houston Journal of Mathematics*.

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Minimal primal ideals in the inner corona algebra of a C_{0}(X)-algebra

AU - Archbold, Robert J.

AU - Somerset, Douglas W. B.

PY - 2019/1/24

Y1 - 2019/1/24

N2 - Let A = C(X) ⊗ K(H), where X is an infinite compact Hausdorff space andK(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.

AB - Let A = C(X) ⊗ K(H), where X is an infinite compact Hausdorff space andK(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.

M3 - Article

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

ER -