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On the structure of connected locally compact groups

✍ Scribed by H. Boseck; G. Czichowski

Publisher
John Wiley and Sons
Year
1976
Tongue
English
Weight
571 KB
Volume
75
Category
Article
ISSN
0025-584X

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✦ Synopsis

On the structure of connected locally compact groups Dedicated to the 100. anniversary of the birthday of Erhard Xchmidt By H. BOSECK and G. CZICHOWSKI in Greifswald (Eingegangen am 29.12.1975)

Let G denote a connected locally compact topological group. By the theorem of YAMABE the group G is a projective limit of LIE-groups. This yields the possibility assosiating to G a topological LIE-algebra L = L ( G ) defined by FIT. M. GLUSHKOW [5] or K. LASHOF [8]. The LIE-algebra of a connected locally compact group has in general infinite dimension as a projective limit of finite -dimensional LIE-algebras. W. M. GLUSHKOW [5] has proved, that the algebraic dimension of the LIE-algebra L equals the topological dimension of the group G . I n [2] we defined an L-group as an L P-group satisfying the property ( L ) any finite-dimensional quotient group is a LIE-group.

By definition an L-group need not to be locally compact, but a projective limit of LIE-groups (LP-group). Hence t o any L-group we may assosiate a topological LIE-algebra, which is the projective limit of finite dimensional LIEalgebras. It is evident, that finite-dimensional L-groups coincide with LIE-groups and so are locally compact.

I n [ 2 ] and [ 4 ] we proved

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