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

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Anatoly Kochubei

Non-Archimedean Operator Algebras


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.