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In mathematics, particularly in the theory of C *-algebras, a uniformly hyperfinite, or UHF, algebra is a C *-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras. == Definition and classification == A UHF C *-algebra is the direct limit of an inductive system where each ''An'' is a finite-dimensional full matrix algebra and each ''φn'' : ''An'' → ''A''''n''+1 is a unital embedding. Suppressing the connecting maps, one can write : If : then ''r kn'' = ''kn'' + 1 for some integer ''r'' and : where ''Ir'' is the identity in the ''r'' × ''r'' matrices. The sequence ...''kn''|''kn'' + 1|''kn'' + 2... determines a formal product : where each ''p'' is prime and ''tp'' = sup , possibly zero or infinite. The formal product ''δ''(''A'') is said to be the supernatural number corresponding to ''A''. Glimm showed that the supernatural number is a complete invariant of UHF C *-algebras. In particular, there are uncountably many isomorphism classes of UHF C *-algebras. If ''δ''(''A'') is finite, then ''A'' is the full matrix algebra ''M''''δ''(''A''). A UHF algebra is said to be of ''infinite type'' if each ''tp'' in ''δ''(''A'') is 0 or ∞. In the language of K-theory, each supernatural number : specifies an additive subgroup of R that is the rational numbers of the type ''n''/''m'' where ''m'' formally divides ''δ''(''A''). This group is the ''K''0 group of ''A''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Uniformly hyperfinite algebra」の詳細全文を読む スポンサード リンク
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