Automatic continuity of homomorphisms into normed quadratic algebras (Q2770596)

Let \(F\) be the field of the real or the complex numbers. Assume \(B\) is a complete normed quadratic algebra over \(F\), but not necessarily associative. A normed algebra \(B\) is said to have the property ACHR (Automatic Continuity of Homomorphisms into the right hand side of the arrow) if each homomorphism \(\varphi :A\to B\) is continuous for every complete normed algebra \(A\). The authors show the equivalence of the following three conditions: NEWLINENEWLINENEWLINE1. Homomorphisms from complete normed algebras over \(F\) into \(B\) are continuous.NEWLINENEWLINENEWLINE2. Homomorphisms from complete normed associative and commutative algebras over \(F\) into \(B\) are continuous. NEWLINENEWLINENEWLINE3. B has no isotropic element, i.e. a non-zero element with square zero. NEWLINENEWLINENEWLINEThus a real or complex complete normed quadratic algebra has property ACHR if and only if it has no non-zero element with zero square.NEWLINENEWLINENEWLINEAn example illustrates that the assumption of completeness is necessary.NEWLINENEWLINENEWLINEExamples of ACHR-algebras are spin JBW-factors, smooth normed algebras and complete normed noncommutative Jordan division algebras.