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Abstract index group : ウィキペディア英語版
Abstract index group
In operator theory, every Banach algebra can be associated with a group called its abstract index group.
== Definition ==
Let ''A'' be a Banach algebra and ''G'' the group of invertible elements in ''A''. The set ''G'' is open and a topological group. Consider the identity component
:''G''0,
or in other words the connected component containing the identity 1 of ''A''; ''G''0 is a normal subgroup of ''G''. The quotient group
''A'' = ''G''/''G''0
is the abstract index group of ''A''. Because ''G''0, being the component of an open set, is both open and closed in ''G'', the index group is a discrete group.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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