The classical Real Nullstellensatz of Dubois and Risler states that given a real closed field \(R\) and an ideal \(I\) of the polynomial ring \(R[X_1,\ldots,X_n]\) then \(I=\mathcal{I}(V(I))\) if and only if \(I\) is real. Here \(V(S)=\{x\in R^n\mid f(x)=0\mathrm{ for all }f\in S\}\) is the zero set of \(S\subset R[X_1,\ldots,X_n]\) and \(\mathcal{I}(T)=\{f\in R[X_1,\ldots,X_n]\mid f(x)=0\mathrm{ for all }x\in T\}\) is the vanishing ideal of \(T\subset R^n\). An ideal \(I\) is real if \(\sum a_i^2\in I\) implies \(a_i\in I\) for all \(i\).
The article under review extends this classical Real Nullstellensatz to the setting of \(*\)-algebras. We introduce the setting. The authors follow the approach of [J. W. Helton, S. McCullough and M. Putinar, Math. Z. 255, No. 3, 579–596 (2007; Zbl 1117.47010)].
Let \(F\) be either \(\mathbb{R}\) or \(\mathbb{C}\) with complex conjugation as involution. Let \(\mathcal{A}\) be a \(*\)-algebra over \(F\). A \(*\)-representation of \(\mathcal{A}\) on an \(F\)-vector space \(V\) with inner product is a mapping \(\mathcal{A}\to \mathrm{Hom}_F(V,V)\) with some natural properties. Let \(\mathcal{R}\) be the class of all \(*\)-representations of the \(*\)-algebra \(\mathcal{A}\). Let \(\mathcal{C}\) be a fixed subclass of \(\mathcal{R}\) (for example the subclass \(\Pi\) of all finite-dimensional \(*\)-representations). A \(\mathcal{C}\)-point is a pair \((\pi,v)\) where \(\pi\in\mathcal{C}\) and \(v\in V_\pi\). The set of all \(\mathcal{C}\)-points is denoted by \(\mathrm{pt}_\mathcal{C}(\mathcal{A})\). Given a subset \(S\) of \(\mathcal{A}\) one defines \(V_{\mathcal{C}}(S):=\{(\pi,v)\in\mathrm{pt}_{\mathcal{C}}(\mathcal{A})\mid \pi(s)v=0\mathrm{ for all }s\in S\}\). Note that \(V_\mathcal{C}(S)=V_\mathcal{R}(S)\cap\mathrm{pt}_\mathcal{C}(\mathcal{A})\). Given a subset \(T\) of \(\mathrm{pt}_\mathcal{R}(\mathcal{A})\) one defines \(\mathcal{I}(T):=\{a\in\mathcal{A}\mid \pi(a)v=0\mathrm{ for all }(\pi,v)\in T\}\). Note that \(\mathcal{I}(T)\) is a left ideal of \(\mathcal{A}\). Given a left ideal \(I\) of \(\mathcal{A}\), the radical \(^\mathcal{C}\sqrt{I}:=\mathcal{I}(V_{\mathcal{C}}(I))\) is called the \(\mathcal{C}\)-saturation of \(I\). The ideal \(I\) is said to have the left nullstellensatz property for \(\mathcal{C}\)-points if \(^{\mathcal{C}}\sqrt{I}=I\). A left ideal \(I\) of \(\mathcal{A}\) is said to be real if \(\sum a_i^*a_i\in I+I^*\) implies \(a_i\in I\) for all \(i\).
The main result of the paper is now the following. Let \(\mathfrak{F}= F\langle x,x^*\rangle\) denote the free \(*\)-algebra on \(x=(x_1,\ldots,x_n)\). Then a finitely generated left ideal \(I\) of \(\mathfrak{F}\) satisfies the left nullstellensatz for \(\Pi\)-points if and only if \(I\) is real.