Dilations and full corners on fractional skew monoid rings. (Q2438271)

For both \(C^*\)-algebras and rings, crossed products by semigroups of endomorphisms have been constructed in various ways. In certain cases, such crossed products have been proved to be isomorphic to full corners in crossed products by groups; results of this type are known as dilation theorems. For example, Laca proved that if \(S\) is a cancellative left Ore monoid with enveloping group \(G\) and \(\alpha\) is an action of \(S\) on a unital \(C^*\)-algebra \(A\) such that the induced maps \(\alpha(s)\) are injective with hereditary images, then the semigroup \(C^*\)-crossed product \(A\times_\alpha S\) is isomorphic to a full corner of the group \(C^*\)-crossed product \(A_S\times_{\widehat\alpha}G\), where \(A_S\) is a certain inductive limit \(C^*\)-algebra associated to \((A,\alpha)\) and \(\widehat\alpha\) is an induced action of \(G\) on \(A_S\) [\textit{M. Laca}, J. Lond. Math. Soc., II. Ser. 61, No. 3, 893-904 (2000; Zbl 0973.46066)]. An algebraic construction analogous to semigroup \(C^*\)-crossed products, denoted \(S^{\text{op}}*_\alpha A*_\alpha T\), where \(\alpha\) is an action of a monoid \(T\) on a unital ring \(A\) by ring endomorphisms and \(S\) is a submonoid of \(T\) satisfying the left denominator conditions, was constructed by \textit{P. Ara, M. A. González-Barroso}, the author, and the reviewer [J. Algebra 278, No. 1, 104-126 (2004; Zbl 1063.16033)]. They established a dilation theorem under the assumptions that \(S=T\) is a cancellative left Ore monoid with enveloping group \(G\) and each \(\alpha(s)\) maps \(A\) isomorphically onto a full corner of \(A\). In this case, there exist a formal ring of fractions \(S^{-1}A\), a nonzero idempotent \(e\in S^{-1}A\), and an action \(\widehat\alpha\) of \(G\) on \(S^{-1}A\) such that \(S^{\text{op}}*_\alpha A*_\alpha S\cong e\bigl((S^{-1}A)*_{\widehat\alpha}G\bigr) e\) [op. cit.]. The main theorem of the present paper removes the restriction in the last-mentioned theorem that \(S\) acts on \(A\) by corner isomorphisms, yielding a dilation isomorphism \(S^{\text{op}}*_\alpha A*_\alpha S\cong e(B*_{\widehat\alpha}G)e\) for a suitable ring \(B\). In case \(S\) acts on \(A\) by injective unital ring endomorphisms, the latter isomorphism simplifies to \(S^{\text{op}}*_\alpha A*_\alpha S\cong B*_{\widehat\alpha}G\). Similar results are obtained for semigroup \(C^*\)-crossed products, thus removing from Laca's theorem the restriction that the maps \(\alpha(s)\) have hereditary images. In the final section of the paper, both the ring and the \(C^*\)-algebra dilation theorems are generalized to actions by monoids \(S\) which satisfy the left denominator conditions but are not necessarily cancellative.