# Banach-Jordan pair

A real or complex Jordan pair $V = ( V ^ { + } , V ^ { - } )$ (i.e. a pair $V = ( V ^ { + } , V ^ { - } )$ of real or complex vector spaces together with trilinear mappings $V ^ { \pm } \times V ^ { \mp } \times V ^ { \pm } \rightarrow V ^ { \pm }$, $( x , y , z ) \mapsto \{ x y z \}$, symmetric in the outer variables, which satisfy:

\begin{equation*} \{ u x \{ v y w \} \} - \{ v y \{ u x w \} \} = \{ \{ u x v \} y w \} - \{ v \{ x u y \} w \} \end{equation*}

for all $u , v , w \in V ^ { \pm }$, $x , y \in V ^ { \mp }$; cf. also Vector space), where the spaces $V ^ { \pm }$ are endowed with complete norms (cf. also Norm) making continuous the triple products $\{ x y z \}$, $x , z \in V ^ { \pm }$, $y \in V ^ { \mp }$. Banach–Jordan triple systems (cf. also Jordan triple system) can be viewed as Banach–Jordan pairs $V$ with $V ^ { + } = V ^ { - }$. For the general theory of Jordan pairs, see [a16]. In the notation of quadratic Jordan theory, $Q _ { x } y = \{ x y x \} / 2$. A classic example of a Banach–Jordan pair is that given by $( \operatorname{BL} ( X , Y ) , \operatorname{BL} ( Y , X ) )$, the space of continuous linear operators between two Banach spaces $X$, $Y$, with $\{ a b c \} = a b c + c b a$ (cf. also Banach space; Linear operator). Any Banach–Jordan algebra $J$ gives rise to a Banach–Jordan pair $( J , J )$, by taking the usual Jordan triple product. Moreover, Banach–Jordan pairs are interesting in themselves by their connection with bounded symmetric domains [a17]. Jordan–Banach triple systems with additional geometric requirements emerged in the work of W. Kaup. See [a13], [a26] or $\operatorname{JB} ^ { * }$-triple for a general report on the theory of $\operatorname{JB} ^ { * }$-triples (prime $\operatorname{JB} ^ { * }$-triples have been classified in [a23]); see [a24] for the structure theory of Hilbert triples, Hermitian Hilbert triples and atomic $\operatorname {JBW} ^ { * }$-triples, and see [a8], [a9] for so-called $H ^ { * }$-triple theory. A general account of both the geometric and the non-geometric theory of Jordan–Banach triple systems can be found in [a25].

Given a Banach–Jordan pair $V$ and an element $y \in V ^ { - \sigma }$, $\sigma = \pm$, one can define a Banach–Jordan algebra on $V ^ { \sigma }$, called the $y$-homotope of $V ^ { \sigma }$ and denoted by $V ^ { \sigma } ( y )$, by taking $x.z = \{ x y z \} / 2$ as product, and the original norm $|.|$ of $V ^ { \sigma }$. Of course, $V ^ { \sigma } ( y )$ can be given an equivalent norm $\| \, . \, \| ^ { \prime }$ so that $\| x. z \| ^ { \prime } \leq \| x \| ^ { \prime } \| z \| ^ { \prime }$. The set $\operatorname { Ker } ( y ) = \{ x \in V ^ { \sigma } : Q _ { y } x = 0 \}$ is a closed ideal of $V ^ { \sigma } ( y )$, and so the quotient $V ^ { \sigma ( y ) } / \operatorname { Ker } ( y )$ is a Banach–Jordan algebra, called the local algebra of $V$ at $y$ and usually denoted by $V _ { y } ^ { \sigma }$. Important notions in the theory of Jordan pairs, such as the Jacobson radical, are defined in terms of homotopes (the adverb "properly" usually refers to properties that hold in all homotopes); on the other hand, local algebras have become a prominent tool in the study of Jordan pairs [a2]. Local algebras play a fundamental role in proving the automatic continuity of derivations on Banach–Jordan pairs [a12], thus extending the extension, by A.R. Villena [a27], of the Johnson–Sinclair theorem to Banach–Jordan algebras.

## Finiteness conditions in Banach–Jordan pairs.

A basic finiteness condition in Jordan pairs is having finite capacity, i.e. the ascending and descending chain conditions on principal inner ideals (cf. also Chain condition). For a non-degenerate Jordan pair $V$ ($Q _ { x } V ^ { \mp } = 0$ implies $x = 0$, $x \in V ^ { \pm }$), the socle of $V$ can be characterized as the ideal $\operatorname{ Soc } ( V )$ of all $y \in V ^ { - \sigma }$ such that $V _ { y } ^ { \sigma }$ has finite capacity [a22] (cf. also Socle). Notice that for a Jordan pair of type $( \operatorname{BL} ( X , Y ) , \operatorname{BL} ( Y , X ) )$, its socle is the ideal $( \operatorname{FBL} ( X , Y ) , \operatorname{FBL} ( Y , X ) )$ of all continuous linear operators having finite rank (cf. also Linear operator). Moreover, it has finite capacity if and only if one of the two vector spaces is finite dimensional.

It is well known that under certain assumptions (von Neumann regularity, finite spectrum, algebraicity or coincidence with the socle) a semi-primitive associative Banach algebra is finite dimensional. This is no longer true for Jordan–Banach algebras, but, under any of the above requirements (in the appropriate Jordan versions), semi-primitive Jordan–Banach algebras have finite capacity [a4]. Using this result and the fact that semi-primitivity is inherited by local algebras, it can be shown (see [a19] and [a14]) that for a semi-primitive complex Banach–Jordan pair, the socle, the set of properly algebraic elements, the largest properly algebraic ideal, the largest properly spectrum-finite ideal, and the largest von Neumann regular ideal all coincide. Related results can be found in [a15], [a20] and [a21]. Since von Neumann regular Banach–Jordan pairs are idempotent-finite, it follows that any von Neumann regular Banach–Jordan pair has finite capacity ([a18] and [a10]). Loos' classification of simple Jordan pairs of finite capacity (any Jordan pair of finite capacity is a direct sum of finitely many simple ones) can be refined in the complex Banach setting: A simple complex Banach–Jordan pair having finite capacity is either finite dimensional, the Jordan pair defined by a continuous symmetric bilinear form on a complex Banach vector space, or a Jordan pair of continuous linear operators between two Banach pairings, one of them being finite dimensional ([a10] and [a11]). Using Zel'manov's structure theorem for strongly prime Jordan systems [a28] (see also [a1]), prime complex Banach–Jordan $*$-triples with non-zero socle and without nilpotent elements were classified in [a6], getting as a consequence, among other results, the Bunce–Chu structure theorem [a7] for compact $\operatorname{JB} ^ { * }$-triples.

An associative algebra is Noetherian if it satisfies the ascending chain condition on left ideals (cf. also Associative rings and algebras; Noetherian ring). A.M. Sinclair and A.W. Tullo showed [a29] that a complex Noetherian Banach associative algebra is finite dimensional. For a Jordan algebra or pair, the suitable Noetherian condition is the ascending chain condition on inner ideals. M. Benslimane and N. Boudi [a3] proved that a complex Noetherian Banach–Jordan algebra is finite dimensional. This result has been extended, [a5], to Banach–Jordan pairs. Actually, the following has been proved:

i) the Jacobson radical of a Noetherian Banach–Jordan pair is finite dimensional;

ii) non-degenerate Noetherian Banach–Jordan pairs have finite capacity; and

iii) complex Noetherian Banach–Jordan pairs are finite dimensional.

#### References

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[a4] | M. Benslimane, A. Fernández, A. Kaidi, "Caractérisation des algèbres de Jordan Banach de capacitè finie" Bull. Sci. Math. , 112 : 2 (1988) pp. 473–480 |

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[a6] | K. Bouhya, A. Fernández, "Jordan-$\square ^ { * }$-triples with minimal inner ideals and compact JB$\square ^ { * }$-triples" Proc. London Math. Soc. , 68 (1994) pp. 380–398 |

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[a8] | A. Castellón, J.A. Cuenca, C. Martin, "Jordan $H ^ { * }$-triple systems" S. González (ed.) , Non-Associative Algebra and Its Applications , Kluwer Acad. Publ. (1994) pp. 66–72 |

[a9] | A. Castellón, J.A. Cuenca, C. Martin, "Special Jordan $H ^ { * }$-triple systems" Commun. Algebra , 28 (2000) pp. 4699–4706 |

[a10] | A. Fernández, E. Garcia, E. Sánchez, "Von Neumann regular Jordan Banach triple systems" J. London Math. Soc. , 42 (1990) pp. 32–48 |

[a11] | A. Fernández, E. Garcia, E. Sánchez, M. Siles, "Strong regularity and hermitian Hilbert triples" Quart. J. Math. Oxford , 45 (1994) pp. 43–55 |

[a12] | A. Fernández, H. Marhnine, C. Zarhouti, "Derivations on Banach Jordan pairs" Quart. J. Math. Oxford (to appear) |

[a13] | Y. Friedman, "Bounded symmetric domanin and the $J B ^ { * }$-triple structure in physics" Proc. 1992 Oberwolfach Conf. Jordan algebras , de Gruyter (1994) pp. 61–82 |

[a14] | G. Hessenberger, "A spectral characterization of the socle of Banach Jordan systems" Math. Z. , 223 (1996) pp. 561–568 |

[a15] | G. Hessenberger, "Inessential and Riesz elements in Banach Jordan systems" Quart. J. Math. Oxford , 47 (1996) pp. 337–347 |

[a16] | O. Loos, "Jordan pairs" , Lecture Notes in Mathematics , 460 , Springer (1975) |

[a17] | O. Loos, "Bounded symmetric domains and Jordan pairs" , Lecture Notes , Univ. California (1977) |

[a18] | O. Loos, "Recent results on finiteness conditions in Jordan pairs" , Proc. 1992 Oberwolfach Conf. Jordan algebras , de Gruyter (1994) pp. 83–95 |

[a19] | O. Loos, "Properly algebraic and spectrum-finite ideals in Jordan systems" Math. Proc. Cambridge Philos. Soc. , 114 (1993) pp. 149–161 |

[a20] | O. Loos, "Nuclear elements in Banach Jordan pairs" , Proc, 1997 Internat. Conf. Jordan structures , Dept. Álgebra, Geom. y Topol. Univ. Málaga (1999) pp. 116–117 |

[a21] | A. Maouche, "Caractérisation spectrales du radical et socle d'une paire de Jordan–Banach" Canad. Math. Bull. , 40 (1997) pp. 488–497 |

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[a25] | A. Rodriguez Palacios, "Jordan structures in analysis" , Proc. 1992 Oberwolfach Conf. Jordan Algebras , de Gruyter (1994) pp. 97–186 |

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[a28] | E. Zel'manov, "On prime Jordan triple systems III" Sib. Math. J. , 26 (1985) pp. 71–82 |

[a29] | A.M. Sinclair, A.W. Tullo, "Noetherian Banach algebras are finite dimensional" Math. Ann. , 211 (1974) pp. 151–153 |

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Banach–Jordan pair.

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