Hostname: page-component-6bf8c574d5-vmclg Total loading time: 0 Render date: 2025-02-21T19:53:25.962Z Has data issue: false hasContentIssue false
  • English
  • Français

MORE ON RESTRICTED BOUNDARIES

Part of: General theory of linear operators Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  17 February 2025

SONJA MOUTON Open the ORCID record for SONJA MOUTON [Opens in a new window]  and
ROBIN HARTE
SONJA MOUTON*
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, South Africa
ROBIN HARTE
Affiliation:
School of Mathematics, Trinity College, Dublin, Ireland e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In our paper [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745], we defined and used the restricted topology to establish certain connections among the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum, and in [‘The restricted connected hull: filling the hole’, Bull. Aust. Math. Soc. 109 (2024), 388–392], we presented further properties of the restricted connected hull. We now continue our investigation of the restricted boundary. In particular, we prove a number of mapping and regularity-type properties of the restricted boundary. In addition, we use this concept to provide a new characterisation of the Jacobson radical of a Banach algebra and a variation of Harte’s theorem. Finally, we establish spectral continuity results, in particular, in ordered Banach algebras.

Keywords

exponential and boundary spectraboundaries and hullsrestricted topology

MSC classification

Primary: 46H10: Ideals and subalgebras
Secondary: 47A10: Spectrum, resolvent
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 11
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author acknowledges, with thanks, financial support provided by the National Research Foundation (NRF) of South Africa (Grant No. 132669).

References

Aupetit, B., A Primer on Spectral Theory (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
Harte, R. E., ‘The exponential spectrum in Banach algebras’, Proc. Amer. Math. Soc. 58 (1976), 114–118.CrossRefGoogle Scholar
Harte, R. E. and Wickstead, A. W., ‘Boundaries, hulls and spectral mapping theorems’, Math. Proc. R. Ir. Acad. 81A (1981), 201–208.Google Scholar
Mouton, S., ‘On the boundary spectrum in Banach algebras’, Bull. Aust. Math. Soc. 74 (2006), 239–246.CrossRefGoogle Scholar
Mouton, S., ‘Mapping and continuity properties of the boundary spectrum in Banach algebras’, Illinois J. Math. 53 (2009), 757–767.CrossRefGoogle Scholar
Mouton, S., ‘A condition for spectral continuity of positive elements’, Proc. Amer. Math. Soc. 137 (2009), 1777–1782.CrossRefGoogle Scholar
Mouton, S., ‘Generalized spectral perturbation and the boundary spectrum’, Czechoslovak Math. J. 71 (2021), 603–621.CrossRefGoogle Scholar
Mouton, S. and Harte, R. E., ‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745.CrossRefGoogle Scholar
Mouton, S. and Harte, R. E., ‘The restricted connected hull: filling the hole’, Bull. Aust. Math. Soc. 109 (2024), 388–392.CrossRefGoogle Scholar
Müller, V., Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras (Birkhäuser, Basel, 2007).Google Scholar
Raubenheimer, H. and Rode, S., ‘Cones in Banach algebras’, Indag. Math. (N.S.) 7 (1996), 489–502.CrossRefGoogle Scholar
Raubenheimer, H. and Swartz, A., ‘Regularity-type properties of the boundary spectrum in Banach algebras’, Rocky Mountain J. Math. 49 (2019), 2747–2754.CrossRefGoogle Scholar