Orderings, real places, and valuations on noncommutative integral domains
This paper may be viewed as a continuation of previous work of the authors (jointly with \textit{K. H. Leung} [J. Algebra 198, No. 2, 412-427 (1997; Zbl 0909.16027)]) on the real spectrum of noncommutative rings, its principal aim being the development of an associated theory of real places and valuations. In particular, the relation between the set of real places on \(A\) and the set \(X\) of support \(\{0\}\) orderings on \(A\) is carefully examined, where for any ordering \(P\) of \(A\) the support of \(P\) is the prime ideal \(P\cap-P\). Given a preordering \(T\) in \(A\) (a subset \(T\subseteq A\) with \(\sum A^2(T)\subseteq T\), where \(A^2(T)\) consists of all permuted products of elements \(a_1,\ldots,a_n\), \(s_1,\ldots,s_m\), where \(a_i\in A\) and \(s_j\in T\)) with \(T\cap-T=\{0\}\), one denotes by \(X_T\) the set of support \(\{0\}\) orderings on \(A\) containing \(T\) and by \(G_T\) the set of all restrictions of elements of \(G\) to \(X_T\), where \(G\) consists of all mappings of the type \(\overline a\colon X\to\{-1,1\}\) with \(\overline a(P)=1\) resp. \(\overline a(P)=-1\) for \(a\in P\) and \(a\in-P\), for some element \(a\in A\). Each ordering \(P\in X_T\) yields a character \(\chi_P\) on \(G_T\) given by \(\overline a\mapsto\overline a(P)\). One calls \(X_T\) a fan if every character \(\chi\) with values in \(\{-1,1\}\) on \(G_T\) satisfying \(\chi(-1)=-1\) is of the form \(\chi_P\) for some ordering \(P\in X_T\). Actually, one may show that this implies any subset \(P\) of \(A\) satisfying \(T\subseteq P\), \(PP\subseteq P\), \(P\cup-P=A\) and \(P\cap-P=\{0\}\) to be an ordering (i.e., \(P\) is automatically also closed under addition). Denoting by \(M\) the set of so-called order compatible real places, one of the main results of the paper states that the canonical map \(\lambda\colon X\to M\) defines a \(P\)-structure on the space of orderings \((X,G)\), in the sense that \(\lambda^{-1}(\alpha)\) is a fan in \(X\) for every \(\alpha\in M\) and that for each fan \(Y\subseteq X\), there exist \(\alpha,\beta\in M\) with \(Y\subseteq\lambda^{-1}(\alpha)\cup\lambda^{-1}(\beta)\). Moreover, endowing \(M\) with the induced quotient topology makes it a Hausdorff space. The authors also define valuations on an integral domain and prove a general version of Bröcker's trivialization theorem for fans. An illustrative example is given by considering the orderings and real places on certain quantum planes over \(\mathbb{R}\) (skew polynomial ring \(\mathbb{R}[x,y]\), with \(yx=axy\), with \(1\neq a>0\)).
scientific article; zbMATH DE number 1271650