Uniformly hyperfinite algebra: Difference between revisions

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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 [[Operator norm|norm topology]], of an increasing union of finite -dimensional full [[matrix ring|matrix algebras]].

== Definition and classification ==

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>

== Classification ==

If

If

:<math>A_n \simeq M_{k_n} (\mathbb C),</math>

then ''r krk''_''n'''' = ''k''_{''n'' + 1} for some integer ''r'' and

:<math>\phi_n (a) = a \otimes I_r,</math>

where ''I_r'' is the identity in the ''r'' &times; ''r'' matrices. The sequence ...''k_n''|''k''_{''n'' + 1}|''k''_{''n'' + 2}... determines a formal product

:<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=318–318318–340|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|doi-access=free}}</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 ∞.

In the language of [[K-theory]], each [[supernatural number ]]

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

specifies an additive subgroup of '''RQ''' 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''. <citeref name=Rordam00 />

=== AnCAR algebra example===

One example of a UHF C*-algebra is the [[CAR algebra]]. It is defined as follows: let ''H'' be a separable complex Hilbert space ''H'' with orthonormal basis ''f_n'' and ''L''(''H'') the bounded operators on ''H'', consider a linear map

:<math>\alpha : H \rightarrow L(H)</math>

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\{ \alpha(f_n), \alpha(f_m) \} = 0 \quad \mbox{and} \quad \alpha(f_n)^*\alpha(f_m) + \alpha(f_m)\alpha(f_n)^* =

\langle f_m, f_n \rangle I.

</math>

The CAR algebra is the C*-algebra generated by

:<math>\{ \alpha(f_n) \}\;.</math>

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can be identified with the multiplicity 2 embedding

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

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

== References ==

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[[Category:C*-algebras]]

[[de:UHF-Algebra]]