Spectral synthesis in the multiplier algebra of a C_0(X)-algebra

Robert J Archbold, Douglas W. B. Somerset

Research output: Contribution to journalArticle

4 Citations (Scopus)
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Abstract

Let A be a C0(X)-algebra with continuous map φ from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M(A) is a C(β X)-algebra with continuous map Graphic: Prim(M(A)) → β X extending φ. For x ∈ Im(φ), let Jx = ⋂{P ∈ Prim(A): φ(P) = x} and Graphic. Then Graphic, the strict closure of Jx in M(A). Thus, Hx is strictly closed if and only if Graphic, and the ‘spectral synthesis’ question asks when this happens. In this paper, it is shown that, for σ-unital A, Hx is strictly closed for all x ∈ Im(φ) if and only if Jx is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.
Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalQuarterly Journal of Mathematics
Volume65
Issue number1
Early online date22 Jan 2013
DOIs
Publication statusPublished - 1 Mar 2014

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Spectral Synthesis
Multiplier Algebra
Algebra
Continuous Map
Strictly
Closed Map
If and only if
Primitive Ideal
Closed
Compact Hausdorff Space
Locally Compact
Unital
Closure
Graphics

Cite this

Spectral synthesis in the multiplier algebra of a C_0(X)-algebra. / Archbold, Robert J; Somerset, Douglas W. B.

In: Quarterly Journal of Mathematics, Vol. 65, No. 1, 01.03.2014, p. 1-24.

Research output: Contribution to journalArticle

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