View source for Banach-Jordan pair

A real or complex Jordan pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300301.png" /> (i.e. a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300302.png" /> of real or complex vector spaces together with trilinear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300304.png" />, symmetric in the outer variables, which satisfy:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300305.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300307.png" />; cf. also [[Vector space|Vector space]]), where the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300308.png" /> are endowed with complete norms (cf. also [[Norm|Norm]]) making continuous the triple products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b1300309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003011.png" />. Banach–Jordan triple systems (cf. also [[Jordan triple system|Jordan triple system]]) can be viewed as Banach–Jordan pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003013.png" />. For the general theory of Jordan pairs, see [[#References|[a16]]]. In the notation of quadratic Jordan theory, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003014.png" />. A classic example of a Banach–Jordan pair is that given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003015.png" />, the space of continuous linear operators between two Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003017.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003018.png" /> (cf. also [[Banach space|Banach space]]; [[Linear operator|Linear operator]]). Any [[Banach–Jordan algebra|Banach–Jordan algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003019.png" /> gives rise to a Banach–Jordan pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003020.png" />, by taking the usual Jordan triple product. Moreover, Banach–Jordan pairs are interesting in themselves by their connection with bounded symmetric domains [[#References|[a17]]]. Jordan–Banach triple systems with additional geometric requirements emerged in the work of W. Kaup. See [[#References|[a13]]], [[#References|[a26]]] or [[JB*-triple|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003021.png" />-triple]] for a general report on the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003023.png" />-triples (prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003024.png" />-triples have been classified in [[#References|[a23]]]); see [[#References|[a24]]] for the structure theory of Hilbert triples, Hermitian Hilbert triples and atomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003026.png" />-triples, and see [[#References|[a8]]], [[#References|[a9]]] for so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003028.png" />-triple theory. A general account of both the geometric and the non-geometric theory of Jordan–Banach triple systems can be found in [[#References|[a25]]].

Given a Banach–Jordan pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003029.png" /> and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003031.png" />, one can define a [[Banach–Jordan algebra|Banach–Jordan algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003032.png" />, called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003034.png" />-homotope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003035.png" /> and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003036.png" />, by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003037.png" /> as product, and the original norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003039.png" />. Of course, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003040.png" /> can be given an equivalent norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003041.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003042.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003043.png" /> is a closed [[Ideal|ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003044.png" />, and so the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003045.png" /> is a Banach–Jordan algebra, called the local algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003046.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003047.png" /> and usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003048.png" />. Important notions in the theory of Jordan pairs, such as the [[Jacobson radical|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 [[#References|[a2]]]. Local algebras play a fundamental role in proving the automatic continuity of derivations on Banach–Jordan pairs [[#References|[a12]]], thus extending the extension, by A.R. Villena [[#References|[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|Chain condition]]). For a non-degenerate Jordan pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003049.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003050.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003052.png" />), the socle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003053.png" /> can be characterized as the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003054.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003056.png" /> has finite capacity [[#References|[a22]]] (cf. also [[Socle|Socle]]). Notice that for a Jordan pair of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003057.png" />, its socle is the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003058.png" /> of all continuous linear operators having finite rank (cf. also [[Linear operator|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 [[#References|[a4]]]. Using this result and the fact that semi-primitivity is inherited by local algebras, it can be shown (see [[#References|[a19]]] and [[#References|[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 [[#References|[a15]]], [[#References|[a20]]] and [[#References|[a21]]]. Since von Neumann regular Banach–Jordan pairs are idempotent-finite, it follows that any von Neumann regular Banach–Jordan pair has finite capacity ([[#References|[a18]]] and [[#References|[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 ([[#References|[a10]]] and [[#References|[a11]]]). Using Zel'manov's structure theorem for strongly prime Jordan systems [[#References|[a28]]] (see also [[#References|[a1]]]), prime complex Banach–Jordan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003059.png" />-triples with non-zero socle and without nilpotent elements were classified in [[#References|[a6]]], getting as a consequence, among other results, the Bunce–Chu structure theorem [[#References|[a7]]] for compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003060.png" />-triples.

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

i) the [[Jacobson radical|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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Soc.'' , '''130'''  (2001)  pp. 25–36</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Bouhya,   A. Fernández,   "Jordan-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003061.png" />-triples with minimal inner ideals and compact JB<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003062.png" />-triples"  ''Proc. London Math. Soc.'' , '''68'''  (1994)  pp. 380–398</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L.J. Bunce,   C.H. Chu,   "Compact operations, multipliers and the Radon–Nikodym property in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130030/b13003063.png" />-triples"  ''Pacific J. Math.'' , '''53'''  (1992)  pp. 249–265</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Castellón,   J.A. Cuenca,   C. 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