Uniformly hyperfinite algebra: Difference between revisions

Revision as of 12:44, 5 May 2014

Revision as of 16:30, 12 May 2015

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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|volume=95|issue=2|pages=318–318318–340|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 ∞.