Skew derivations with algebraic invariants of bounded degree. (Q897774)

Let \(A\) be a semiprime algebra over a field \(K\), \(\sigma\in\Aut(A/K)\), and \(\delta\in\Hom_K(A)\) so that for some \(0\neq q\in K\) and all \(x,y\in A\), \(\delta(xy)=\delta(x)y+\sigma(x)\delta(y)\) and \(\delta(\sigma(x))=q\sigma(\delta(x))\). Call \((\delta,\sigma)\) a \(q\)-skew derivation of \(A\), and assume in the results below that \(\delta\) and \(\sigma\) are algebraic in \(\mathrm{End}_K(A)\). Denote \(\ker(\delta)\) by \(A^\delta\). If \(a\in A^\delta\) is algebraic let the degree of its minimal \(K\)-polynomial be \(\deg(a)\), and if \(A^\delta\) is algebraic of bounded degree let \(\deg(A^\delta)\) be the maximal degree of any \(a\in A^\delta\). The main result in the paper is that for \((\delta,\sigma)\) a \(q\)-skew derivation, and with the assumptions above, if \(A^\delta\) is algebraic of bounded degree with \(\deg A^\delta+1<\mathrm{card\,}K\), then \(A\) is algebraic of bounded degree and is Artinian; when \(K\) is a perfect field then \(A\) is finite dimensional over \(K\). One consequence is that if \(A\) is a Banach algebra over \(\mathbb R\) or \(\mathbb C\) with \((\delta,\sigma)\) a \(q\)-skew derivation and \(\delta\) continuous, then \(A^\delta\) algebraic is equivalent to \(A\) finite dimensional.