On one-sided division infinite-dimensional normed real algebras

Publicacions Ma emá iques, Vol 36 (1992), 485-488 . Abs ac ON ONE-SIDED DIVISION INFINITE-DIMENSIONAL NORMED REALALGEBRAS JOSÉ ANTONIO CUENCA MIRA Dedica ed o he memo y o Pe e Menal In his no e we in oduce he concep o Cayley homomo phism which is closely ela ed wi h hose o composi ion algeb a and no malized o hogonal mul iplica ion . The key esul shows he exis en e o ce ain ypes o Cayley homomo phisms o in ini e dimension . As an applica ion we p o e he exis en e o le di ision in ini e-dimensional comple e no med eal algeb as wi h le uni y . Le K be a ield o cha ac e is ic di e en om 2, V and W ec o K-spaces (no necessa ily wi h ini e dimension) e e yone endowed wi h a nondegene a e symme ic bilinea o m . These o ms will be deno ed by ( 1 ) . We shall say ha (V, -, e) is a Cayley iad (o e V) i - is an in olu i e isome y o V and e an elemen in V such ha é = e . We deno e by G(W) he ec o subspace o he linea maps T in EndK (W ) which ha e an adjoin map T* wi h espec o ( 1 ) . The linea mapS V -j G(W) ca ying e e y x o S x is said o be a Cayley homomo phism (b ie ly, homomo phism) om he Cayley iad o W i he ollowing condi ions a e sa is ied : 1) S y o5 x = (xix) Id o any x in V (whe e Id deno es he iden i y ope a o on W) . 2) Sx = S, 3) S, = Id . I is easy o show ha 2) is equi alen o he ollowing condi ion : 2') Fo any x, y, z in V we ha e (Sx(y)1 Sx(z)) _ (x1 x) (y1 z) . Linea izing equali y 1) we ob ain S x oS y + S g oS x = 2(xl y) Id 486  J . A . CUENCA MIRA o any x, y in V . On he o he hand 1), 2) and 3) yield o (eje) = 1 . Assume ha V= W is a composi ion K-algeb a whose symme ic bilinea o m is (1) and wi h Cayley an iau omo phis n - . I e is he uni y o V hen he map sending e e y elemen x in V o he ig h mul iplica ion ope a o R ., is a Cayley homomo phism om he Cayley iad (V,-, e) o V . Con e sely, le V be a ini o-dimensional ec o space endowed wi h a nondegene a e symme ic bilinea o m, assume (V, -, e) a Cayley iad o e V and S : V -> G(V) = EndK(V) a Cayley homomo phism om he iad o V . As in [5, p oo o P oposi ion 1 in p . 25], we can de ine a p oduc on V o which V becomes a composi ion algeb a . Cayley homomo phisms a e also closely ela ed wi h no malized o hogonal mul iplica ions [1, p . 140-158] . Theo em 1 . Le K be an o de ed ield whe e e e y posi i e elemen has a squa e oo , H a ec o K-space wi h coun able in ini e dimension which is endowed wi h a posi i e de ini e symme ic bilinea o m . Then he e a e a Cayley iad (H, -, e) o e H and an homomo phism om his iad o H . Mo eo e - only ixes he ec o line spanned by e . P oo . I is well known ha H has an o hono mal basis eo, el) . . . , en ,. . . (see o ins an e [2, Theo em 29]) . Fo any in ege n> 0 le V n ( esp . W,) be he subspace o H spanned by he n + 1 ( esp . 2n) i s elemen s o his basis . Le e = eo . Deno e by - he linea ope a o on H ixing eand sending he emainde s elemen s o his basis in o hei opposi es . Fo each n he map - can be es ic ed o an in olu i e au omo phism o V nwhich we shall also deno e in he same way . E e y (V n , -, e) is a Cayley iad . We shall p o e by induc- ion ha o e e y n he e exis s a Cayley homomo phism S(n) om his iad o W  , which sa is ies he ollowing p ope y : o all m < n and any x, E V, yE W, n we ha e S( n1 (y) = SX )(y) . Ob iously he s a e- men is ue o n = 0 . Assume now ha he e exis s an homomo phism S(n) om (V n , -, e) o W n sa is ying he equi ed p ope y .  We shall de ine S(n4- ) . Fo his, i s ly we obse e ha he e exis s an isome y b n om W  , on o he o hogonal subspace o Wn ela i e o W, + , . So e e y elemen in Wn+ can be w i en in a unique way as x + bn (y) wi h x, y E W n . An a bi a y elemen in Vn+1 is a sum u+Ae n+ wi h u E V n and A E K . We deno e by S(+áe) +1 he linea map om Vn+ o Wn+ de ined in he ollowing way Su+áe .+1 (x + bn(y)) = S(n) (x) - Ay + bn(S(n) (y) + ñx) . The map S(n+ ) : Vn+ -> G(Wn+1) = End K(W . +1 ) gi en by u + Aen+ --> Su+Aen +1 is an homomo phism om he Cayley iad ONE-SIDED DIVISION NORMED REALALGEBRAS  487 (V,+1, -, e) o Wn+i which sa is ies he equi ed p ope y . Fo any ele- men s x, y E H he e exis a V n such ha x, yE V n . The elemen S~ n l (y) does no depend o he chosen ec o space V n and will be deno ed by Sx(y) . E e y S ., is a linea ope a o on H which has adjoin ope a o equal o S, and he map x -, S x om H o £(H) is an homomo phism om he Cayley iad (H, -, e) o H . a Theo em 2 . Le H be an in ini e-dimensional sepa able eal Hilbe space . Then he e exis a Cayley iad (H, -, e) and an homomo phism om his iad o H . P oo . The e exis s in H a comple e o hono mal sys em o coun able in ini e ca dinal . Le H' be he ec o subspace spanned by his se . By heo em 1, he e a e a Cayley iad (H', -, e) and an homomo phism S' om his iad o H' . We also deno e by - he con inuous ex ension o - o H . I x, y E H and {xn}, {yn} a e sequences o elemen s in H' such ha x =  lim xn , y =  lim y, hen we ha e ha {S,', :, (Y .)} n-ce n-oo is a Cauchy sequence in H . The limi does no depend o he chosen Cauchy sequences and is deno ed by Sx(y) . Fo any x, yE H we ha e 115.(y)jj = IIxil IIy1I . Mo eo e all S x is con inuous wi h adjoin map S~ ú . So he linea mapS : H -~ £(H) gi en by S : x --> S~, is an homomo phism om he Cayley iad (H, -, e) o H . a We ecall ha a (nonassocia i e) algeb a V is a le di ision algeb a i o any nonze o elemen x in V he le mul iplica ion ope a o L ', is in e sible . In a simila way igh di ision algeb as can be de ined . The eal algeb a V is said o be an absolu e alued algeb a i i is no med and i sa is ies 11xyjj = lixil jjyjj o any x, yE V . I H is a eal Hilbe space, (H, -, e) a Cayley iad o e H and S a Cayley homomo phism om he iad o H, hen H wi h he p oduc de ined by xy = S,,(y) becomes a le di ision algeb a wi h le uni y e which is absolu e alued . So we ha e Theo em 2' . In e e y sepa able eal Hilbe space wi h in ini e di- mension can be de ined a (nonassocia i e) p oduc wi h which H becomes a le di ision absolu e alued algeb a wi h le uni y . Rema k 1 . Theo em T shows he exis en e o eal le di ision com- ple e no med algeb as wi h in ini e dimension . I was conjec u ed by F . B . W ig h [4] ha e e y no med di ision algeb a o e he eals is ini e-dimensional . Rema k 2 . Le M be a se , L an ul a il e on M and {Hy}7Em a amily o eal Hilbe spaces which a e le di ision absolu e alued 48 8  J . A . CUENCA MIRA algeb as wi h le uni y . Le H be he l°°-sum o his amily and N he ideal o he elemen s (x j ., F m such ha lim 11x711 = 0 . Then N is closed and HIN is a no med algeb a which will be deno ed by (H i)u and called he no med ul ap oduc o {H . y } wi h espec o 7d (see [6]) . Mo eo e o any elemen [(x7)] in he ul ap oduc we ha e 11[(x7)]11 = lim ~Ix711- u So his ul ap oduc is a Hilbe space . I is easy o show ha (H7)u is a le di ision absolu e alued algeb a wi h le uni y . We can ob ain om his ha he e exis eal Hilbe spaces wi h in ini e Hilbe dimen- sion big enough o e which we can de ine a p oduc endowing i wi h a s uc u e o le di ision absolu e alued algeb a wi h le uni y . Finally we obse e ha A . Rod íguez [3] has gi en he s uc u e heo y o le di ision absolu e alued algeb as wi h le uni y . Re e en es 1 .  D . HUSEMOLLER, Vib e bundles," Sp inge -Ve lag, New Yo k, sec- ond edi ion, 1975 . 2 .  I . KAPLANSKY, "Linea algeb a and geome y," Chelsea, New Yo k, 1974 . 3 .  A . RODRÍGUEZ, One-sided di ision absolu e alued algeb as, o ap- pea . 4 .  F . B . WRIGHT, Absolu e alued algeb as, P oc . N .A .S . 39 (1953), 330-332 . 5 .  K . A . ZHEVLAKOV, A . M . SLIN'KO, I . P . SHESTAKOV, A . I . SHIR- SHOV, "Rings ha a e nea ly associa i e," Academic P ess, New Yo k, 1982 . 6 . H . EINRICH, Ul ap oduc s in Banach space heo y, T . Reine Angew . Ma h . 313 (1980), 72-104 . Depa amen o de Álgeb a, Geome ía y Topología Uni e sidad de Málaga Apa ado 59 29080 Málaga SPAIN Rebu el 13 de Gene de 1992