Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-26T22:00:20.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant convolution algebras

Published online by Cambridge University Press:  20 January 2009

J. W. Baker
J. W. Baker
Affiliation:
University of Sheffield
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be a commutative, semi-simple, convolution measure algebra in the sense of Taylor (6), and let S denote its structure semigroup. In (2) we initiated a study of some of the relationships between the topological structure of A^ (the spectrum of A), the algebraic properties of S, and the way that A lies in M(S). In particular, we asked when it is true that A is invariant in M(S) or an ideal of M(S) and also whether it is possible to characterise those measures on S which are elements of A. It appeared from (2) that if A is invariant in M(S) then S must be a union of groups and that A^ must be a space which is in some sense “ very disconnected ”. In (3) we showed that if A^ is discrete then A is “ approximately ” an ideal of M(S). (What is meant by “ approximately ” is explained in (3); it is the best one can expect since algebras which are approximately equal have identical structure semigroups and spectra.) In this paper we round off some of the results of (2) and (3). We show that if A is invariant in M(S) then A^ is totally disconnected, and that if A^ is totally disconnected then S is an inverse semigroup (union of groups). From these two crucial facts it is fairly straight-forward to obtain a complete characterisation of algebras A (and their structure semigroups) for which (i) A^ is totally disconnected, (ii) A is invariant in M(S), or (iii) A is an ideal of M(S).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 4 , December 1973 , pp. 299 - 306
Copyright
Copyright © Edinburgh Mathematical Society 1973

References

REFERENCES

(1) Austin, C. W., Duality theory for commutative semigroups, Trans. Amer. Math. Soc. 109 (1963), 245–256.CrossRefGoogle Scholar
(2) Baker, J. W., Convolution algebras with discrete spectra, J. London Math. Soc. (2)5(1972), 193–201.CrossRefGoogle Scholar
(3) Baker, J. W., A characterisation of a certain class of convolution algebras, Proc. Edinburgh Math. Soc. (2) 18 (1973), 199–205.CrossRefGoogle Scholar
(4) Glicksberg, I., Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269–276.CrossRefGoogle Scholar
(5) Srinivasan, T. P. and Tewari, U. B., A Stone-Weierstrass theorem for semi-groups, Bull. Amer. Math. Soc. 74 (1968), 1125–1127.CrossRefGoogle Scholar
(6) Taylor, J. L., The structure of convolution measure algebras, Trans. Amer. Math. Soc. 119 (1965), 150–166.CrossRefGoogle Scholar
(7) Taylor, J. L., L-subalgebras of M(G), Trans. Amer. Math. Soc. 135 (1969), 105–113.Google Scholar
(8) Taylor, J. L., The cohomology of the spectrum of a measure algebra, Acta Math. 126 (1971), 195–225.CrossRefGoogle Scholar