Uniformly hyperfinite algebra: Difference between revisions

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In [[operatormathematics]], particularly in the theory of [[C*-algebras]], a '''uniformly hyperfinite''', or '''UHF''', algebra is onea C*-algebra that iscan be written as the closure, in the appropriate[[Operator norm|norm topology]], of an increasing union of finite dimensional full [[matrix ring|matrix algebras]].

{{Noref|date=June 2011}}

In [[operator algebras]], a '''uniformly hyperfinite''', or '''UHF''', algebra is one that is the closure, in the appropriate topology, of an increasing union of finite dimensional full matrix algebras.

== Definition and classification ==

== C*-algebras ==

A UHF [[C*-algebra]] is the [[direct limit]] of an inductive system {''A_n'', ''φ_n''} where each ''A_n'' is a finite dimensional full matrix algebra and each ''φ_n'' : ''A_n'' → ''A''_''n''+1 is a unital embedding. Suppressing the connecting maps, one can write

:<math>A = \overline {\cup_n A_n}.</math>

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:<math>\delta(A) = \prod_p p^{t_p}</math>

where each ''p'' is prime and ''t_p'' = sup {''m'' &nbsp; | &nbsp; ''p^m'' divides ''k_n '' for some ''n''}, possibly zero or infinite. The formal product ''&delta;''(''A'') is said to be the [[supernatural number]] corresponding to ''A''.<ref name=Rordam00>{{cite book|last=Rørdam|first=M.|last2=Larsen|first2=F.|last3=Laustsen|first3=N.J.|title=An Introduction to K-Theory for C*-Algebras|year=2000|publisher=Cambridge University Press|___location=Cambridge|isbn=0521789443}}</ref> [[James Glimm|Glimm]] showed that the supernatural number is a complete invariant of UHF C*-algebras.<ref name=glimm60>{{cite journal|last=Glimm|first=James G.|title=On a certain class of operator algebras|journal=Transactions of the American Mathematical Society|date=1 February 1960|year=1960|volume=95|issue=2|pages=318–318|doi=10.1090/S0002-9947-1960-0112057-5|url=http://www.ams.org/journals/tran/1960-095-02/S0002-9947-1960-0112057-5/S0002-9947-1960-0112057-5.pdf|accessdate=2 March 2013}}</ref> In particular, there are uncountably many isomorphism classes of UHF C*-algebras.

If ''&delta;''(''A'') is finite, then ''A'' is the full matrix algebra ''M''_{''&delta;''(''A'')}. A UHF algebra is said to be of ''infinite type'' if each ''t_p'' in ''&delta;''(''A'') is 0 or ∞.

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:<math>\delta(A) = \prod_p p^{t_p}</math>

specifies an additive subgroup of '''R''' that is the rational numbers of the type ''n''/''m'' where ''m'' formally divides ''&delta;''(''A''). This group is the [[Operator K-theory|''K''₀ group]] of ''A''. <cite name=Rordam00 />

=== An example===

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:<math>M_{2^n} \hookrightarrow M_{2^{n+1}}.</math>

Therefore the CAR algebra has supernatural number 2^∞. This identification also yields that its ''K''₀ group is the [[dyadic rational]]s.<ref name="Davidson97">{{cite book|last=Davidson|first=Kenneth|title=C*-Algebras by Example|year=1997|publisher=Fields Institute|isbn=0-8218-0599-1|pages=166,218-219,234}}</ref>

== References ==

<references />

[[Category:C*-algebras]]