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An application of the Sakai’s theorem to the characterization of \(H^*\)- algebras. (English) Zbl 0830.46044

An \(H^*\)-algebra is a nonassociative algebra \({\mathcal A}\) over \(\mathbb{K}\) (\(\mathbb{K}= \mathbb{R}\) or \(\mathbb{C}\)) whose underlying vector space is a Hilbert \(\mathbb{K}\)-space with the inner product \(\langle \cdot, \cdot \rangle\), endowed with an involution \(x\mapsto x^*\), linear if \(\mathbb{K}= \mathbb{R}\) and semilinear if \(\mathbb{K}= \mathbb{C}\), such that \[ \langle xy, z\rangle= \langle x, zy^* \rangle= \langle y, x^* z\rangle, \] for any \(x,y, z\in{\mathcal A}\). The first classification of these has been given by W. Ambrose in 1945 in the associative complex case. Afterwards other classes of real or complex \(H^*\)-algebras have been determined (alternative, Jordan, noncommutative Jordan, Lie, Mal’cev …) by diverse authors to the extend of to obtain a Wedderburn-Zorn theorem in nonassociative case (J. A. Cuenca, A. Rodríguez).
A light generalization of \(H^*\)-algebras is due to P. P. Saworotnow as follows: A complex algebra \({\mathcal A}\) which is a Hilbert space is called a two-sided \(H^*\)-algebra if for any \(x\in {\mathcal A}\) there exist elements \(x^r, x^l\in {\mathcal A}\) such that \[ \langle xy, z\rangle= \langle y, x^l z\rangle, \qquad \langle yx, z\rangle= \langle y, zx^r \rangle, \] for every \(y,z\in {\mathcal A}\). Any complex \(H^*\)-algebra is a two-sided \(H^*\)-algebra but the converse is not true. Nevertheless there exists a strong relation between both. In fact, Saworotnow has proved that every associative two-sided \(H^*\)-algebra with zero annihilator is a complex \(H^*\)-algebra with the same product, the involution \(x\mapsto x^r\) and convenient inner product. An \(H^*\)- algebra \({\mathcal A}\) has zero annihilator if \(x{\mathcal A}= {\mathcal A}x =0\) implies \(x=0\).
In this work, another proof of Saworotnow’s theorem is given using a well known theorem of Sakai relative to derivations in von Neumann algebras. Saworotnow’s original proof is entirely complex and associative and this restriction has not been eliminated by the author, who poses this problem at the end of his paper. However, a real version of this reasonings seems not to be very hard.
Remark. An erratum was detected by this reviewer. In page 320 line 24, where it is read “\(\langle T(x) y,z \rangle\)”, must be read \(\langle T^* (x) y,z\rangle\).

MSC:

46K15 Hilbert algebras
46H20 Structure, classification of topological algebras
46L10 General theory of von Neumann algebras
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
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