Some functional equations in Banach algebras and an application. (English) Zbl 0623.46021
Some results concerning certain functional equations in complex Banach algebras are presented. One of these results is used to prove the following result:
Let A be a Banach *-algebra with identity e and let X be a vector space which is also a unitary left A-module. Suppose there exists a mapping \(Q: X\to A\) with the properties
(i) \(Q(x+y)+Q(x-y)=2Q(x)+2Q(y)\) for all pairs x,y\(\in X,(ii)\) \(Q(ax)=aQ(x)y^*\) for all \(x\in X\) and all normal invertible \(a\in A.\)
Under these conditions for the mapping B(.,.): \(X\times X\to A\) defined by the relation \[ B(x,y) = 1/4(Q(x+y)-Q(x-y))\quad +\quad i/4(Q(x+iy)- Q(x-iy)) \] the following statements are fulfilled:
1) B(.,.) is additive in both arguments;
2) \(B(ax,y)=aB(x,y)\), \(B(x,ay)=B(x,y)a^*\) for all pairs x,y\(\in X\) and all \(a\in A;\)
3) \(Q(x)=B(x,x)\) for all \(x\in X.\)
The result above is an abstract generalization of the classical Jordan- Neumann characterization of pre-Hilbert space. If A is the complex number field, then the result above reduces to a result first proved by S. Kurepa.
Let A be a Banach *-algebra with identity e and let X be a vector space which is also a unitary left A-module. Suppose there exists a mapping \(Q: X\to A\) with the properties
(i) \(Q(x+y)+Q(x-y)=2Q(x)+2Q(y)\) for all pairs x,y\(\in X,(ii)\) \(Q(ax)=aQ(x)y^*\) for all \(x\in X\) and all normal invertible \(a\in A.\)
Under these conditions for the mapping B(.,.): \(X\times X\to A\) defined by the relation \[ B(x,y) = 1/4(Q(x+y)-Q(x-y))\quad +\quad i/4(Q(x+iy)- Q(x-iy)) \] the following statements are fulfilled:
1) B(.,.) is additive in both arguments;
2) \(B(ax,y)=aB(x,y)\), \(B(x,ay)=B(x,y)a^*\) for all pairs x,y\(\in X\) and all \(a\in A;\)
3) \(Q(x)=B(x,x)\) for all \(x\in X.\)
The result above is an abstract generalization of the classical Jordan- Neumann characterization of pre-Hilbert space. If A is the complex number field, then the result above reduces to a result first proved by S. Kurepa.
MSC:
46H05 | General theory of topological algebras |
46K05 | General theory of topological algebras with involution |
39B52 | Functional equations for functions with more general domains and/or ranges |
46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
Keywords:
functional equations in complex Banach algebras; Banach *-algebra; left A-module; Jordan-Neumann characterization of pre-Hilbert spaceReferences:
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