Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T06:21:20.226Z Has data issue: false hasContentIssue false
  • English
  • Français

Norm Decreasing Homomorphisms Between Ideals of Lp(G)

Published online by Cambridge University Press:  20 November 2018

N. J. Kalton  and
G. V. Wood
N. J. Kalton
Affiliation:
University College of Swansea, Swansea, U.K.SA2 8PP
G. V. Wood
Affiliation:
University College of Swansea, Swansea, U.K.SA2 8PP
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 G1 and G2 be compact groups and T : Lp(G1) → LP(G2) (1 ≦ P ≦ ∞ ) be an algebra homomorphism. If || T || ≦ 1 and T is either a monomorphism of an epimorphism then T can in many cases be explicitly characterized (see [4 ; 8 ; 9 ; 11 ; 13 ; 14]). Excluding p = 2, the outstanding cases are 1 < p < ∞ for monomorphisms and 2 < p < ∞ for epimorphisms (cf. [14]). One aim of the present note is to complete this work. We also consider the problem of extending these results in some form to homomorphisms on ideals of group algebras; the only known result in this area is for abelian groups [3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 1 , 01 February 1978 , pp. 102 - 111
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Dunford, N. and Schwartz, J. T., Linear operators I, (Interscience, New York, 1958).Google Scholar
2. Forelli, F., The isometries of HP, Can. J. Math. 16 (1964), 721–728.Google Scholar
3. Forelli, F., Homomorphisms of ideals in group algebras, Illinois J. Math. 9 (1965), 410–417.Google Scholar
4. Greenleaf, F. P., Norm decreasing homomorphisms of group algebras, Pacific J. Math. 15 (1965), 1187–1219.Google Scholar
5. Kalton, N. J. and Wood, G. V., Norm decreasing homomorphisms between ideals of C﹛G), to appear.Google Scholar
6. Lacey, H. E., The isometric theory of classical Banach spaces (Springer-Verlag, 1974).Google Scholar
7. Loomis, L. H., An introduction to abstract harmonic analysis (Van Nostrand, 1953).Google Scholar
8. Parrott, S. K., Isometric multipliers, Pacific J. Math. 25 (1968), 159–166.Google Scholar
9. Rigelhof, R., Norm-decreasing homomorphisms of group algebras, Trans. Amer. Math. Soc. 136 (1969), 361–372.Google Scholar
10. Rudin, W., Lp-isometries and equimea surability, Indiana Math. J. 25 (1976), 215–228.Google Scholar
11. Strichartz, R. S., Isomorphisms of group algebras, Proc. Amer. Math. Soc. 17 (1966), 858–862.Google Scholar
12. Wood, G. V., A note on isomorphisms of group algebras, Proc. Amer. Math. Soc. 25 (1970), 771–775.Google Scholar
13. Wood, G. V., Isomorphisms of Z-group algebras, J. London Math. Soc. 4 (1972), 425–428.Google Scholar
14. Wood, G. V., Homomorphisms of group algebras, Duke Math. J. 1+1 (1974), 255–261.Google Scholar