International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 7, Pages 421-442
doi:10.1155/S016117120201150X

Spectral integration and spectral theory for non-Archimedean Banach spaces

S. Ludkovsky1 and B. Diarra2

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.

Abstract

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 ℒ(E) of the continuous linear operators on a free Banach space E 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 C-algebras C∞(X,𝕂). We prove a particular case of a representation of a C-algebra with the help of a L(Aˆ,μ,𝕂)-projection-valued measure. We consider spectral theorems for operators and families of commuting linear continuous operators on the non-Archimedean Banach space.