# Journal of Operator Theory

Volume 44, Issue 1, Summer 2000 pp. 139-150.

Semigroup crossed products and the structure of Toeplitz algebras**Authors**: Sriwulan Adji

**Author institution:**Department of Mathematics, Bandung Institute of Technology, Bandung 40132, Indonesia

**Summary:**Suppose $\Gamma$ is a totally ordered discrete abelian group, and $I$ is an ordered ideal in $\Gamma$. We show that the crossed product $A\times \Gamma^+$ by an action inflated from one $\Gamma/I$ is isomorphic to the induced algebra $\ind^{\widehat\Gamma}_{I^\bot}A \times (\Gamma/I)^+$. Using this we show how the $\wgamma$-invariant ideals in the Toeplitz algebra of $\Gamma$ are determined by the order ideals in $\Gamma$.

**Keywords:**$C*$-algebra, endomorphism, ordered group, covariant representation, crossed product, semigroup of isometries, Toeplitz algebra

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