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Triviality and continuity of pseudocharacters and pseudorepresentations. (English) Zbl 0945.43002
The paper states a number of theorems about pseudocharacters and pseudorepresentations. These concepts were defined and studied in detail by the author [Russ. J. Math. Phys. 2, No. 3, 353-382 (1994; Zbl 0907.22007)]. Briefly, a pseudorepresentation on a group \(G\) is a map \(f:G \to \mathbb{R}\) such that \(\{f(st)-f(s) -f(t):s, t\in G\}\) is bounded and \(f(x^n)=nf (x)\) for all \(x\in G\) and \(n\in{\mathbf N}\). A pseudorepresentation is a map from \(G\) to the bounded operators on a Banach space satisfying analogous conditions. If the second, power, condition is dropped the map is called a quasicharacter or quasirepresentation.
One of the theorems describes all continuous pseudocharacters on \(G\), thus correcting a mistake in an earlier publication of the author’s which was pointed out in M. Grosser’s review of that paper [Funkt. Anal. Appl. 27, No. 4, 87-90 (1993; MR 95g:22008)]. The pseudocharacters in the first Baire class are described in another theorem: they are of interest because of their relation to continuous quasicharacters. These theorems show that pseudocharacters which are not characters depend strongly on \(G/G_0\), where \(G_0\) is the connected component of the identity. Examples given indicate that the alternative types of pseudocharacters listed in the theorems in fact occur.
Quasirepresentations provide examples of almost multiplicative maps as studied by B. E. Johnson [J. Lond. Math. Soc. (2) 37, No. 2, 294-316 (1988; Zbl 0652.46031)]. Amenability of \(G\) is important for showing that quasirepresentations are close to representations and the paper describes a construction of the close by representation which uses an invariant mean rather than an approximate diagonal, as was used by Johnson. The paper concludes with the theorem that for almost connected groups amenability is a necessary condition for all quasirepresentations to be close to representations.

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
22D15 Group algebras of locally compact groups
43A07 Means on groups, semigroups, etc.; amenable groups
22D05 General properties and structure of locally compact groups