Journal article
The automorphisms of convolution algebras of C∞ functions
Journal of mathematical analysis and applications, Vol.64(1), pp.25-47
1978
DOI: 10.1016/0022-247X(78)90018-5
Abstract
We find the automorphisms and the spectra of several different topological convolution algebras of
C
∞-functions on the real line. Starting with the convolution algebra of compactly supported
C
∞-functions, equipped with the usual
LF-topology, we define a corresponding convolution algebra of
C
∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree
r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally
m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of
C
∞-functions of infinitly fast exponential decay at ∞,
H J
, on the one hand, and the algebra of
C
∞-functions of only a finite degree
e
−r¦x¦
decay at ∞,
J
r
0, on the other hand, have quite different automorphisms, although
H J
= ∩
r
J
r
0. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of
J
r
0, and that this representation does not extend to a representation on the Banach algebra
L
1(
R
).
Details
- Title: Subtitle
- The automorphisms of convolution algebras of C∞ functions
- Creators
- Palle E.T Jørgensen - University of Pennsylvania
- Resource Type
- Journal article
- Publication Details
- Journal of mathematical analysis and applications, Vol.64(1), pp.25-47
- DOI
- 10.1016/0022-247X(78)90018-5
- ISSN
- 0022-247X
- eISSN
- 1096-0813
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 1978
- Academic Unit
- Mathematics
- Record Identifier
- 9984240867302771
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