International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 7, Pages 421-442
Spectral integration and spectral theory for non-Archimedean Banach spaces
1Theoretical Department, Institute of General Physics, 38, Vavilov Street, Moscow 119991, Russia
2Laboratoire de Mathématiques Pures, Complexe Scientifique des Cézeaux, Aubière 63 177 , France
Received 22 January 2001; Revised 8 August 2001
Copyright © 2002 S. Ludkovsky and B. Diarra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Banach algebras over arbitrary complete non-Archimedean fields are considered such that operators may be nonanalytic. There are
different types of Banach spaces over non-Archimedean
fields. We have determined the spectrum of some closed commutative
subalgebras of the Banach algebra of the continuous linear operators on a free Banach space generated by projectors. We investigate the spectral integration of non-Archimedean Banach algebras. We define a spectral measure and prove several properties. We prove the non-Archimedean analog of
Stone theorem. It also contains the case of -algebras . We prove a particular case of a representation of a -algebra with the help of a -projection-valued measure. We consider spectral
theorems for operators and families of commuting linear continuous
operators on the non-Archimedean Banach space.