Regular representations of lattice ordered semigroups (Q2835247)

A lattice ordered group is a group \(G\) with a lattice order \(\leq\) compatible with \(G\), i.e. \(x\leq y\) implies \(gx\leq gy\) and \(xg\leq yg\) and a positive spanning cone \(P\), i.e. a subsemigroup \(P\) of \(G\) with \(P\cap P^{-1}=\{e\}\) and \(PP^{-1}=G\). Then any \(g\)\,\(\in\)\,\(G\) has a unique representation as \(g=g_+g_-^{-1},\,g_+,g_-\in P\). The author shows that a contraction \(T:\, P\to B(H)\) is regular in the sense that \(\widetilde T(g):=T(g_+)T(g_-)\) is completely positive definite on \(G\) if and only if for \(p_i\in P,\;i\leq n\) and \(g\in P\) with \(g\land p_i=e\) NEWLINE\[NEWLINE[T(g)^* \widetilde{T}(p_ip_j^{-1})T(g)]\leq[\widetilde{T}(p_ip_j^{-1})].NEWLINE\]NEWLINE As a consequence it is shown that Nica-covariant representations as introduced in \textit{A. Nica} [J. Oper. Theory 27, No. 1, 17--52 (1992; Zbl 0809.46058)] are regular. The paper generalizes a result of \textit{S. Brehmer} [Acta Sci. Math. 22, 106--111 (1961; Zbl 0097.31701)].