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Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko’s conjecture. (English. Russian original) Zbl 1187.22014
J. Math. Sci., New York 159, No. 5, 653-751 (2009); translation from Fundam. Prikl. Mat. 13, No. 7, 85-225 (2007).
A map $$T$$ from a group $$G$$ to the algebra $$L(E)$$ of bounded operators on a Banach space $$E$$ is called a quasirepresentation if there exists $$\varepsilon>0$$ such that $$\|(T(gh)-T(g)T(h))x\|\leq\varepsilon\|x\|$$ for any $$g,h\in G$$, $$x\in E$$. A description of the structure of all finite-dimensional, locally bounded quasirepresentations of arbitrary connected Lie groups is given. The paper is nicely written and contains a lot of examples and related results, including a generalization of the van der Waerden theorem on automatic continuity for group representations and the proof of Mishchenko’s conjecture on the oscillation of discontinuous representations at the identity element of a group.

MSC:
 22E46 Semisimple Lie groups and their representations 47D03 Groups and semigroups of linear operators
Keywords:
quasirepresentation; Lie group
Full Text:
References:
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