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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.

##### 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)
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##### References:
 [1] Frank F. Bonsall and John Duncan, Complete normed algebras, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. · Zbl 0271.46039 [2] Svetozar Kurepa, The Cauchy functional equation and scalar product in vector spaces, Glasnik Mat.-Fiz. Astronom. Ser. II Društvo Mat. Fiz. Hrvatske 19 (1964), 23 – 36 (English, with Serbo-Croatian summary). · Zbl 0134.32601 [3] Svetozar Kurepa, Quadratic and sesquilinear functionals, Glasnik Mat.-Fiz. Astronom. Ser. II Društvo Mat. Fiz. Hrvatske 20 (1965), 79 – 92 (English, with Serbo-Croatian summary). · Zbl 0147.35202 [4] C. E. Rickart, Banach algebras, Kreiger, Hunington, N.Y., 1974. · Zbl 0275.46045 [5] Pavla Vrbová, Quadratic functionals and bilinear forms, Časopis Pěst. Mat. 98 (1973), 159 – 161, 213 (English, with Czech summary). · Zbl 0263.39008 [6] J. Vukman, A result concerning additive functions in Hermitian Banach *-algebras and an application, Proc. Amer. Math. Soc. 91 (1984), no. 3, 367 – 372. · Zbl 0514.46034 [7] J. Vukman, Some results concerning the Cauchy functional equation in certain Banach algebras, Bull. Austral. Math. Soc. 31 (1985), no. 1, 137 – 144. · Zbl 0559.46024
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