Group representations in non-archimedean Banach spaces

Abstract

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Non-Archimedean Duality: Algebras, Groups, and Multipliers

“…The dual object 2 consists of functionals ω a (x) = (for an analog of Fubini's theorem for K-valued measures see [15]). With this multiplication, 2 is isomorphic to the group Banach algebra L(G 2 ) studied in [18] where, in particular, a bounded approximate identity for this algebra is constructed. Note that the right multiplier algebra of L(G 2 ) is isomorphic to the algebra of improper measures on G 2 ; see Exercise 8.B.v in [17].…”

“…Non-Archimedean harmonic analysis, the Fourier analysis of functions f : G → K where G is a group, K is a non-Archimedean valued field, was initiated in the thesis [19] by W. H. Schikhof; this subject should not be confused with the study of complex-valued functions on non-Archimedean structures started in another great thesis, by J. Tate. For Abelian groups admitting a K-valued Haar measure (this class was described by Monna and Springer [15]), Schikhof proved an analog of Pontryagin's duality theorem; see also [17,18]. Duality theorems for compact groups and related group algebras in the non-Archimedean setting were proved later by Schikhof [20] and Diarra [3]; for related subjects see [5].…”

On spectra of algebras over ultranormed fields

“…In their turn, non-archimedean analysis, functional analysis and representations theory of groups over non-archimedean fields develop fast in recent years [30,31,32,33,11,23,24]. This is motivated not only by needs of mathematics, but also their applications in other sciences such as physics, quantum mechanics, quantum field theory, informatics, etc.…”

Profinite and finite groups associated with loop and diffeomorphism groups of non‐Archimedean manifolds

“…The importance of such groups in the non-Archimedean functional analysis, representation theory, and mathematical physics is clear (see [1,8,10,11,14,18,19]). This paper is devoted to one aspect of such groups: their structure from the point of view of the p-adic compactification (see also about Banaschewski compactification in [18]).…”

“…On the other hand, there are irreducible continuous representations of compact groups in non-Archimedean Banach spaces [19]. Among them there are infinite-dimensional [3,4,16].…”