\(^*\)-multilinear polynomials with invertible values (Q1192072)

If \(R\) is an algebra with involution \(*\) over the field \(F\), then a \(*\)- polynomial is any \(p(x_ 1,\dots,x_ n\), \(y_ 1,\dots,y_ n) \in F\{x_ i,y_ j\}\), the free algebra over \(F\), which is evaluated on \((r_ 1,\dots,r_ n) \in R^ n\) by substituting \(r_ i\) for \(x_ i\) and \(r_ i^*\) for \(y_ i\). A \(*\)-polynomial \(p(x_ 1,\dots,y_ n)\) is called multilinear if exactly one of \(x_ i\) or \(y_ i\) appears in each of its monomials. Let \(R^ o\) be the opposite ring of \(R\). The main result of the paper determines the structure of \(R\) when \(\text{char }F \neq 2\), \(\text{card }F > 5\), \(R\) is semi-prime, and \(p(x_ 1,\dots,y_ n)\) is a \(*\)-multilinear polynomial with every evaluation in \(R\) either zero or invertible, and with some evaluation invertible. In this case, there is a division ring \(D\) so that either \(R = M_ k(D)\) with \(\dim_{Z(D)}D\) finite when \(k \geq 2\) and the values of \(p\) are central when \(k \geq 3\), or else \(R = M_ k(D) \oplus M_ k(D)^ o\) with \((A,B^ o)^* = (B,A^ o)\) and if \(k \geq 2\) then \(\dim_{Z(D)}D\) is finite and the values of \(p\) are central.