### Abstract

In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equalityφ{symbol} (ABA) = φ{symbol} (A) φ{symbol} (B) φ{symbol} (A)on their domains. We show that all those transformations originate from linear ^{*}-algebra isomorphisms and linear ^{*}-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented.

Original language | English |
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Pages (from-to) | 586-600 |

Number of pages | 15 |

Journal | Linear Algebra and Its Applications |

Volume | 419 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Dec 1 2006 |

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### Keywords

- Multiplicative Jordan triple isomorphism
- Positive operator
- Self-adjoint operator
- von Neumann algebra

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

**Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neumann algebras.** / Molnár, L.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 419, no. 2-3, pp. 586-600. https://doi.org/10.1016/j.laa.2006.06.007

}

TY - JOUR

T1 - Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neumann algebras

AU - Molnár, L.

PY - 2006/12/1

Y1 - 2006/12/1

N2 - In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equalityφ{symbol} (ABA) = φ{symbol} (A) φ{symbol} (B) φ{symbol} (A)on their domains. We show that all those transformations originate from linear *-algebra isomorphisms and linear *-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented.

AB - In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equalityφ{symbol} (ABA) = φ{symbol} (A) φ{symbol} (B) φ{symbol} (A)on their domains. We show that all those transformations originate from linear *-algebra isomorphisms and linear *-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented.

KW - Multiplicative Jordan triple isomorphism

KW - Positive operator

KW - Self-adjoint operator

KW - von Neumann algebra

UR - http://www.scopus.com/inward/record.url?scp=33748910514&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748910514&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2006.06.007

DO - 10.1016/j.laa.2006.06.007

M3 - Article

AN - SCOPUS:33748910514

VL - 419

SP - 586

EP - 600

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 2-3

ER -