Symmetries of Kirchberg algebras

David John Benson, A. Kumjian, N. C. Phillips

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Let Go and G, be countable abelian groups. Let gamma(i) be an automorphism of G(i) of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with {1A} = 0 in K-0(A), and an automorphism alpha is an element of Aut(A) of order two, such that K-0(A) Go, such that K-1(A) congruent to G(1), and such that a(*) : K-i(A) --> K-i(A) is gamma(i). As a consequence, we prove that every Z(2)-graded countable module over the representation ring R(Z(2)) of Z(2) is isomorphic to the equivariant K-theory K-Z2 (A) for some action of Z(2) on a unital Kirchberg algebra A. Along the way, we prove that every not necessarily finitely generated Z [Z(2)] -module which is free as a Z-module has a direct sum decomposition with only three kinds of summands, namely Z[Z(2)] itself and Z on which the nontrivial element of Z(2) acts either trivially or by multiplication by -1.

Original languageEnglish
Pages (from-to)509-528
Number of pages19
JournalCanadian Mathematical Bulletin
Volume46
Publication statusPublished - 2003

Keywords

  • C-STAR-ALGEBRAS
  • TOPOLOGICAL METHODS
  • CROSSED PRODUCTS
  • CSTAR-ALGEBRAS
  • FINITE-GROUPS
  • MODULES

Cite this

Benson, D. J., Kumjian, A., & Phillips, N. C. (2003). Symmetries of Kirchberg algebras. Canadian Mathematical Bulletin, 46, 509-528.

Symmetries of Kirchberg algebras. / Benson, David John; Kumjian, A.; Phillips, N. C.

In: Canadian Mathematical Bulletin, Vol. 46, 2003, p. 509-528.

Research output: Contribution to journalArticle

Benson, DJ, Kumjian, A & Phillips, NC 2003, 'Symmetries of Kirchberg algebras', Canadian Mathematical Bulletin, vol. 46, pp. 509-528.
Benson, David John ; Kumjian, A. ; Phillips, N. C. / Symmetries of Kirchberg algebras. In: Canadian Mathematical Bulletin. 2003 ; Vol. 46. pp. 509-528.
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AB - Let Go and G, be countable abelian groups. Let gamma(i) be an automorphism of G(i) of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with {1A} = 0 in K-0(A), and an automorphism alpha is an element of Aut(A) of order two, such that K-0(A) Go, such that K-1(A) congruent to G(1), and such that a(*) : K-i(A) --> K-i(A) is gamma(i). As a consequence, we prove that every Z(2)-graded countable module over the representation ring R(Z(2)) of Z(2) is isomorphic to the equivariant K-theory K-Z2 (A) for some action of Z(2) on a unital Kirchberg algebra A. Along the way, we prove that every not necessarily finitely generated Z [Z(2)] -module which is free as a Z-module has a direct sum decomposition with only three kinds of summands, namely Z[Z(2)] itself and Z on which the nontrivial element of Z(2) acts either trivially or by multiplication by -1.

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KW - MODULES

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