$p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Anatoly Kochubei
## Non-Archimedean Operator AlgebrasAbstract
We consider some classes of algebras of bounded linear operators on Banach spaces
over non-Archimedean fields.
In particular, we propose a possible way based on the notion of a Baer ring [1] to
introduce a non-Archimedean version of the notion of a von Neumann algebra. Two
methods of constructing examples of such algebras are given. We develop an analog of
the crossed product construction, one of the main methods of constructing von
Neumann algebras in the classical case. This results [2] in a class of non-trivial
non-Archimedean factors (algebras with a trivial center) or algebras close to
factors corresponding, through the reduction procedure, to type I Baer rings. In the
second construction, the algebras are generated by regular representations of
discrete groups. In this situation [3], algebras of types I and III are obtained.
[1] I. Kaplansky, Rings of Operators, W. A. Benjamin, New York, 1968. [2] A. N. Kochubei, On some classes of non-Archimedean operator algebras, Contemporary Math. 596 (2013), 133-148. [3] A. N. Kochubei, Non-Archimedean group algebras with Baer reductions, Algebras and Represent. Theory 17 (2014), 1861-1867. |