Homomorphisms between Lie \(\mathrm{JC}^*\)-algebras and Cauchy-Rassias stability of Lie \(\mathrm{JC}^*\)-algebra derivations.

*(English)*Zbl 1091.39006This is a joint review of the above-mentioned paper, to be called (I) below, and of the author’s paper [Bull. Braz. Math. Soc. (N.S.) 36, No. 1, 79–97 (2005; Zbl 1091.39008)], called (II).

The author considers a \(C^*\)-algebra \(A\) equipped with its canonical Jordan product \(x\circ y= {1\over2} (xy+yx)\) and with another continuous bilinear mapping \((x,y)\mapsto [x,y]\) from \(A\times A\to A\). Specialising to the canonical Lie product \([x,y]={1\over2} (xy-yx)\) in (I), he refers to \(A\) as a Lie \(\mathrm{JC}^*\)-algebra, and specialising to just some (continuous) Lie algebra product satisfying the Poisson identity in (II), he refers to \(A\) as a Poisson \(\mathrm{JC}^*\)-algebra. (Actually, he doesn’t care to give this definition explicitly and leaves it to the reader to guess the meaning of his notions.)

These papers deal with linear mappings that preserve the additional products, called Lie or Poisson \(\mathrm{JC}^*\)-homomorphisms. The results are of the following type: (1) Under some technical assumptions, an “almost” JC\(^*\)-homomorphism in fact is a JC\(^*\)-homomorphism. (2) Under some other technical assumptions, given an “almost” \(\mathrm{JC}^*\)-homomorphism, there is an actual JC\(^*\)-homomorphism “close” to it. The precise formulations are too technical to be reproduced here; the assumptions are just custom-made to be able to quote appropriate theorems from the literature. The paper (I) also considers derivations.

It has to be pointed out that apart from the section on derivations the paper (II) is a verbatim copy of (I), replacing the word “Lie” by “Poisson” where needed. However, for some reason that is not intelligible to me, a \(\mathrm{JC}^*\)-homomorphism is stipulated to be involutive in (I), but not in (II). In spite of all the similarities between (I) and (II), there is no reference in either of the papers to the other.

As already noted, the writing is mysterious at times since key notions are not or not satisfactorily defined and key properties, such as the continuity of the bracket product in (II), are not spelt out.

Another puzzling fact is that the author never makes use of the additional Lie or Poisson type properties of the bracket product; only bilinearity and continuity are essential. Therefore, the two papers are not only word for word practically identical; mathematically they are absolutely identical.

The author considers a \(C^*\)-algebra \(A\) equipped with its canonical Jordan product \(x\circ y= {1\over2} (xy+yx)\) and with another continuous bilinear mapping \((x,y)\mapsto [x,y]\) from \(A\times A\to A\). Specialising to the canonical Lie product \([x,y]={1\over2} (xy-yx)\) in (I), he refers to \(A\) as a Lie \(\mathrm{JC}^*\)-algebra, and specialising to just some (continuous) Lie algebra product satisfying the Poisson identity in (II), he refers to \(A\) as a Poisson \(\mathrm{JC}^*\)-algebra. (Actually, he doesn’t care to give this definition explicitly and leaves it to the reader to guess the meaning of his notions.)

These papers deal with linear mappings that preserve the additional products, called Lie or Poisson \(\mathrm{JC}^*\)-homomorphisms. The results are of the following type: (1) Under some technical assumptions, an “almost” JC\(^*\)-homomorphism in fact is a JC\(^*\)-homomorphism. (2) Under some other technical assumptions, given an “almost” \(\mathrm{JC}^*\)-homomorphism, there is an actual JC\(^*\)-homomorphism “close” to it. The precise formulations are too technical to be reproduced here; the assumptions are just custom-made to be able to quote appropriate theorems from the literature. The paper (I) also considers derivations.

It has to be pointed out that apart from the section on derivations the paper (II) is a verbatim copy of (I), replacing the word “Lie” by “Poisson” where needed. However, for some reason that is not intelligible to me, a \(\mathrm{JC}^*\)-homomorphism is stipulated to be involutive in (I), but not in (II). In spite of all the similarities between (I) and (II), there is no reference in either of the papers to the other.

As already noted, the writing is mysterious at times since key notions are not or not satisfactorily defined and key properties, such as the continuity of the bracket product in (II), are not spelt out.

Another puzzling fact is that the author never makes use of the additional Lie or Poisson type properties of the bracket product; only bilinearity and continuity are essential. Therefore, the two papers are not only word for word practically identical; mathematically they are absolutely identical.

Reviewer: Dirk Werner (Berlin)

##### MSC:

39B52 | Functional equations for functions with more general domains and/or ranges |

17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |

17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |

46L05 | General theory of \(C^*\)-algebras |