22/12/31 23:57:38.70 rNlYJ3SK.net
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他にも非可換群に対する双対理論の類似物は存在していて、いくつかは作用素環論の言葉で定式化されている。基本的な出発点は群 G の群環と双対群 G^ の関数環とが同型になっているということである。
URLﾘﾝｸ(en.wikipedia.org)
Pontryagin duality
Dualities for non-commutative topological groups
For non-commutative locally compact groups {\displaystyle G}G the classical Pontryagin construction stops working for various reasons, in particular, because the characters don't always separate the points of {\displaystyle G}G, and the irreducible representations of {\displaystyle G}G are not always one-dimensional. At the same time it is not clear how to introduce multiplication on the set of irreducible unitary representations of {\displaystyle G}G, and it is even not clear whether this set is a good choice for the role of the dual object for {\displaystyle G}G. So the problem of constructing duality in this situation requires complete rethinking.
Theories built to date are divided into two main groups: the theories where the dual object has the same nature as the source one (like in the Pontryagin duality itself), and the theories where the source object and its dual differ from each other so radically that it is impossible to count them as objects of one class.
The second type theories were historically the first: soon after Pontryagin's work Tadao Tannaka (1938) and Mark Krein (1949) constructed a duality theory for arbitrary compact groups known now as the Tannaka?Krein duality.[17][18] In this theory the dual object for a group {\displaystyle G}G is not a group but a category of its representations {\displaystyle \Pi (G)}{\displaystyle \Pi (G)}.
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