Equivalent conditions for the convolution theorem on semisimple groups

Author:
Ronald L. Lipsman

Journal:
Proc. Amer. Math. Soc. **34** (1972), 497-503

MSC:
Primary 43A80; Secondary 22E20, 47G05

DOI:
https://doi.org/10.1090/S0002-9939-1972-0310557-X

MathSciNet review:
0310557

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Abstract: For certain semisimple Lie groups *G*, it is known that convolution by an function, , is a bounded operator on . This result is a consequence of the so-called ``analytic continuation of the principal series'' which has been carried out on these groups. However, this continuation procedure does not generalize readily to *arbitrary* semisimple groups. In an attempt to bypass the continuation and obtain the convolution theorem in an alternate manner, we derive in this paper several equivalent conditions for this convolution theorem.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0310557-X

Keywords:
Semisimple Lie groups,
Cartan subgroups,
parabolic groups,
operator-valued Fourier transform,
Plancherel measure,
unitary representations,
convolution theorem,
direct integrals

Article copyright:
© Copyright 1972
American Mathematical Society