TY - JOUR

T1 - Semigroup crossed products and the induced algebras of lattice-ordered groups

AU - Ahmed, Mamoon Ali

AU - Pryde, Alan James

PY - 2010

Y1 - 2010

N2 - Let (G, G(+)) be a quasi-lattice-ordered group with positive cone G(+). Laca and Raeburn have shown that the universal C*-algebra C*(G, G(+)) introduced by Nica is a crossed product BG+ x(alpha) G(+) by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G(+) we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly alpha-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) x (alpha) over tilde, G+ and B(G/H)+ x(beta) G(+). This leads to our main result that B(G/H)+ x(beta) G(+) is realized as an induced C*-algebra Ind(H(sic))((G) over cap) (B(G/H)+ x(tau) (G/H)(+)).

AB - Let (G, G(+)) be a quasi-lattice-ordered group with positive cone G(+). Laca and Raeburn have shown that the universal C*-algebra C*(G, G(+)) introduced by Nica is a crossed product BG+ x(alpha) G(+) by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G(+) we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly alpha-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) x (alpha) over tilde, G+ and B(G/H)+ x(beta) G(+). This leads to our main result that B(G/H)+ x(beta) G(+) is realized as an induced C*-algebra Ind(H(sic))((G) over cap) (B(G/H)+ x(tau) (G/H)(+)).

UR - http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WK2-4XNN5PR-5-1&_cdi=6894&_user=542840&_pii=S0022247X09009512&_orig=search&_coverDate=04%2F1

M3 - Article

VL - 364

SP - 498

EP - 507

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -