A generalization of the original Jordan-von Neumann theorem. (English) Zbl 1243.47010
Motivated by the original form of the well-known Jordan-von Neumann theorem [P. Jordan and J. von Neumann, Ann. of Math. (2) 36, 719–723 (1935; Zbl 0012.30702, JFM 61.0435.05)], the author explores extensions in the setting of sesquilinear forms on modules over *-algebras, in particular, the problem of representability of quadratic functionals. One of the main results is Corollary 4.3, stating that a mapping \(Q\), defined on a unitary right \(A\)-module \(M\) over a unital semiprime complex *-algebra \(A\) without nonzero central ideals with values in a unitary \(A\)-bimodule \(B\) satisfying some mild algebraic conditions that fulfils the parallelogram law and the homogeneity condition \(Q(fa)=a^*Q(f)a\) for all \(f\in M\) and all \(a\in A\) which are hermitian or skew-hermitian, always is an \(A\)-quadratic functional, that is, \(Q(f)=S(f,f)\), where the \(A\)-sesquilinear form \(S: M\times M\to B\) is given by the canonical polarisation identity.
Reviewer: Martin Mathieu (Belfast)
MSC:
| 47A07 | Forms (bilinear, sesquilinear, multilinear) |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 46C15 | Characterizations of Hilbert spaces |
Keywords:
quadratic functional; homogeneity equation; sesquilinear form; \(*\)-ring; complex \(*\)-algebra; ideal; (bi)module, complex vector space; Jordan derivationReferences:
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