### 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 language | English |
---|---|

Pages (from-to) | 509-528 |

Number of pages | 19 |

Journal | Canadian Mathematical Bulletin |

Volume | 46 |

Publication status | Published - 2003 |

### Keywords

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

### Cite this

*Canadian Mathematical Bulletin*,

*46*, 509-528.

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

Research output: Contribution to journal › Article

*Canadian Mathematical Bulletin*, vol. 46, pp. 509-528.

}

TY - JOUR

T1 - Symmetries of Kirchberg algebras

AU - Benson, David John

AU - Kumjian, A.

AU - Phillips, N. C.

PY - 2003

Y1 - 2003

N2 - 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.

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.

KW - C-STAR-ALGEBRAS

KW - TOPOLOGICAL METHODS

KW - CROSSED PRODUCTS

KW - CSTAR-ALGEBRAS

KW - FINITE-GROUPS

KW - MODULES

M3 - Article

VL - 46

SP - 509

EP - 528

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

ER -