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Quadratric functionals and Jordan *-derivations. (English) Zbl 0761.46047

Let \(A\) be a real Banach \(*\)-algebra with identity. A Jordan \(*\)- derivation on \(A\) is a function \(D: A\to A\), not necessarily linear, with the properties \[ D(a+b)=D(a)+D(b), \qquad D(a^ 2)=aD(a)+D(a)a^* \] for all \(a,b\in a\). Now let \(X\) be a real vector space which is also an \(A\)- module. An \(A\)-quadratic functional on \(X\) is a function \(Q: X\to A\) with the properties \[ Q(x+y)+Q(x-y)=2Q(x)+2Q(y), \qquad Q(ax)=aQ(x)a^*, \] for all \(x,y\in X\) and all \(a\in A\).
The author’s main result is to represent an \(A\)-quadratic functional \(Q\) in terms of Jordan \(*\)-derivations as follows: \[ Q(\sum_{\alpha\in J}a_ \alpha x_ \alpha)=\sum_{\alpha,\beta\in J}a_ \alpha c_{\alpha\beta} a_ \beta^*+\sum_{\alpha<\beta}(D_{\alpha\beta}(a_ \beta a_ \alpha) -a_ \beta D_{\alpha\beta}(a_ \alpha)-D_{\alpha\beta}(a_ \alpha)a_ \beta^*). \] In this formula \(J\) is a well-ordered set, \(\{x_ \alpha\}_{\alpha\in J}\) is a family in \(X\) such that every \(x\in X\) may be (not necessarily uniquely) represented in the form \(x=\sum_{\alpha\in J} a_ \alpha x_ \alpha\) for some family \(\{a_ \alpha\}_{\alpha\in J}\) in \(A\) with all but finitely many \(a_ \alpha=0\), the \(c_{\alpha\beta}\in A\) are constant such that \(c_{\alpha\beta}=c_{\beta\alpha}\) for all \(\alpha,\beta\in J\), and the \(D_{\alpha\beta}\) are Jordan \(*\)-derivations. The formula holds for all \(x\in X\) and for every representation of \(x\) in the form \(\sum_{\alpha\in J}a_ \alpha x_ \alpha\). The result was originally proved by S. Kurepa for the special case \(A=\mathbb{R}\).

MSC:

46K99 Topological (rings and) algebras with an involution
39B52 Functional equations for functions with more general domains and/or ranges
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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