Compatible Peirce decompositions of Jordan triple systems - MSP

More documents

Recommendations

Info

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 79 (3.5) JMcJtf) (βT/rigid). In view of the Peirce decomposition J 2(e) = J (22> Θ t7 (21) 0 e7 (20) in (1.2), rigidity means J 2(e) = J (21), i.e., J (22) = J (20) = 0. From (3.4i) it suffices if either J m) or J m vanishes, since they are interchanged by P(e). In particular (3.6) e T / are rigid iff J m) = 0 , which shows rigidity is symmetric. The condition J (22) = 0 is that e and / do not "overlap" in J, in the sense that they have no common elements in their 2-spaces. An important situation where rigidity is automatic is the case of division tripotents, those e for which J 2(e) is a division system all of whose nonzero elements x are invertible in the sense that P(x) is an invertible operator. Such tripotents are found in abundance in systems satisfying the d.c.c. on inner ideals. Slightly more general are the domain tripotents, for which J 2(e) is a domain in the sense that all nonzero x are cancellable, i.e., P{x) is injective; this is equivalent to the condition that there be no zero divisors P{x)y = 0 for x, y Φ 0. At the other extreme from this case where J 2{β) is small is that where J 2(e) is large, namely the case of a full (maximal) tripotent e with J 0(e) — 0. PROPOSITION 3.7. If e is a full or domain tripotent, then any tripotent f collinear with e is automatically rigidly imbedded with e, If e is a domain tripotent then a nonzero tripotent f in J x(e) will be collinear with e as soon as P{f)e = 0. Proof. Suppose e and / are collinear. If e is full then J0(e)=0 implies Jm) = 0 and e, f are rigid. Suppose now that e is a domain tripotent, J2(e) is a domain. But P(/ (22))0 c P(/ 2(/)) J^/) = 0 by (1.2), so J (22) = 0 and hence e, f are rigid. Now suppose e is a domain tripotent and / a tripotent in Jx(e) with P(f)e = 0. Then collinearity reduces to {ffe} = e. Writing e = x + x + x for Peirce elements x 6 J^f), we have x — P(f) i 1 0 t 2 2 e = 0 by hypothesis, so x = {ffe} 6 {e7i(e)Ji(e)J (e)} c J {e), so also x — e — x 2 2 Q x eJ {e), x 2 yet P(x )Xi = 0 by (1.2). o Since J (e) is a domain this forces 2 one of x x to vanish. lf 0 Here x = 0 would imply e = cc is orthogonal x 0 to /, so instead it must be x that vanishes, and e = x — {ffe}. 0 λ • Rigidity is not an intrinsic property of the tripotents, it depends very much on the imbedding. For example, e — E n and / = E 12 are
KEVIN MCCRIMMON rigidly imbedded in M lt2 (D), but not in the larger system H Z {D, D o , j) (imbedded via e = H 12 , f = H 1Z ), since here e and / overlap on Grids An orthogonal-collinear family ί? = {e ) is rigidly imbedded or % rigid in J if each collinear pair ei9 ed is rigid (we observed that orthogonal pairs are always rigid). Thus for each i, j either J2(^)c /otei) or ^2te) c Jίiβj), according as et 1 eά or et T
Page 1 and 2: Pacific Journal of Mathematics COMP
Page 3 and 4: 58 KEVIN MCCRIMMON involution a-^a
Page 5 and 6: 60 KEVIN MCCRIMMON (k = j or i = j"
Page 7 and 8: 62 KEVIN MCCRIMMON (iii) [P{e)\ P(/
Page 9 and 10: 64 KEVIN MCCRIMMON patibility reduc
Page 11 and 12: 66 KEVIN MCCRIMMON elements z with
Page 13 and 14: 68 KEVIN MCCRIMMON of the underlyin
Page 15 and 16: 70 KEVIN MCCRIMMON For certain purp
Page 17 and 18: 72 KEVIN MCCRIMMON (U3) (U4) {J tjJ
Page 19 and 20: 74 KEVIN MCCRIMMON component in J t
Page 21 and 22: 76 KEVIN MCCRIMMON corresponding to
Page 23: 78 KEVIN MCCRIMMON ^(2001) = «Muoi
Page 27 and 28: 82 KEVIN MCCRIMMON algebra, hence t
Page 29 and 30: 84 KEVIN MCCRIMMON reduce to (*) =
Page 31 and 32: 86 KEVIN MCCRIMMON also vanish and
Page 33 and 34: 88 KEVIN MCCRIMMON lifted modulo ni
Page 35 and 36: 90 KEVIN MCCRIMMON a hierarchical f
Page 37 and 38: 92 KEVIN MCCRIMMON For example, tak
Page 39 and 40: 94 KEVIN MCCRIMMON pairs over arbit
Page 41 and 42: 96 KEVIN MCCRIMMON usual to simply
Page 43 and 44: 98 KEVIN MCGRIMMON S and S* preserv
Page 45 and 46: 100 KEVIN MCCRIMMON By (7.8) this i
Page 47 and 48: 102 KEVIN MCCRIMMON and triple syst
Page 49: Pacific Journal of Mathematics Vol.

Compatible Peirce decompositions of Jordan triple systems - MSP

Create successful ePaper yourself

Delete template?

Save as template?