CONTINUOUS REINHARDT DOMAINS PROBLEMS ON PARTIAL JB*-TRIPLES

Transcription

1 CONTINUOUS REINHARDT DOMAINS PROBLEMS ON PARTIAL JB*-TRIPLES László STACHÓ Bolyai Institute Szeged, Hungary Stacho 13/11/2008, Granada László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 1 / 19

2 PARTIAL JB*-TRIPLES (PJT) (E, E 0, {...}) E Banach with. E 0 closed subspace, {...} : E E 0 E E cont. D (a, b) : x {abx}, a, b E 0 {xay} symmetric bilinear in x, y, conjugate-linear in a id (a, a) Der(E, E 0, {...}), a E 0 Jordan identity {xa{xbx}} = {{aax}bx} D ( (a, a) Her(E, ). ) with Sp(D (a, a)) 0 E0, E 0, {...} E 3 JB*-triple 0 László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 2 / 19

6 CANONICAL PJT OF A CIRCULAR DOMAIN D = e it D, t R circular domain in E = [ inf.carathéodory norm of D at 0 ], E D := aut(d) 0 { } {{ } } compl. hol. vect. fields in D (without loss of gen.) Braun-Kaup-Upmeier 1976:! (E, E D, {...} D ) (Sp ( ) {aa. } D E { D 0 is proved only in 1983!) (a ) } {xax}d x : a E D = aut(d) { } Kaup-Vigué 1990: D E D = symm. points of D Stachó : Sp ( {aa. } D ) 0, Any PJT (E, E 0, {...}) is subtriple of some (E, E D, {...} D ) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 3 / 19

10 CONSTRUCTION (E, E 0, {...}) PJT, B := Ball 1 (E), B E 0 = D E 0 D ε := a E 0 [ (a ) ] (εb ) {xax} x For ε sufficiently small: D ε bounded circular domain (E, E 0, {...}) subtriple of ( E, E Dε, {...} Dε ) Ψ bounded subgroup in Aut(E, E 0, {...}) = D Ψ-invariant bounded circular domain: (E, E 0, {...}) subtriple of ( E, E D, {...}) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 4 / 19

13 PROBLEM OF BIDUAL FOR A PJT Dineen 1985: D = Ball 1 (E) = (E, ED w, {...} ) with contractive projection P : E U E Barton-Timoney 1985: D symmetric = {...} is separately weakly continuous Genaral D: the proof works for the middle variable only? (E, E w D {...} ) for (E, E D, {...} D ) in general? Continuity properties of {...}? László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 5 / 19

16 EXTENSION PROBLEM OF INNER DERIVATIONS 0 := Span R { D(a, b) D(b, a) : a, b ED } inner derivations Panou 1990: dim(e) < δ 0 vanishes on E D = δ = 0 Tool: Aut(E, E D, {...} D ) is a compact group (dim(e) < ) Stachó 1996: norm dense grid in E D = similar conclusion Same with quasi-grids (finite subsets of G generate finite dim. closed subtriples) Does every δ 0 0 ED admit a unique extension δ 0 0? In which cases is δ 0 δ 0 continuous? László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 6 / 19

20 TEST OBJECTS: REINHARDT DOMAINS D C n classical Reinhardt domain: D bounded open connected ( ) and (z 1,..., z n ) D { (u1,..., u n ) : u k z k, 1 k n } D Sunada 1974: Description of Aud(D), E D = [ Hilbert balls ] D, D holomorphically equiv. classical Reinhardt domains = L L(C n ) LD + = D + László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 7 / 19

21 TEST OBJECTS: REINHARDT DOMAINS D C n classical Reinhardt domain: D bounded open connected ( ) and (z 1,..., z n ) D { (u1,..., u n ) : u k z k, 1 k n } D Sunada 1974: Description of Aud(D), E D = [ Hilbert balls ] D, D holomorphically equiv. classical Reinhardt domains = L L(C n ) LD + = D + László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 7 / 19

22 GENERALIZED R-DOMAMAINS ( E, Re, ) complex vector lattice, x = max { Re(e it x) : t R } well-defined Complete R-domain: x D {f E : f x } D Barton-Dineen-Timoney 1983: Sunada type theorems in E =[separable Banach space with basis] László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 8 / 19

25 CONTINUOUS REINHARDT DOMAINS (CRD) Vigué 1998: E = C(Ω), Ω compact, r : Ω [1, M] lower semicontinuous D = { f E : f (ω) < r(ω), ω Ω } D is symmetric r is continuous, Aut(D)0 = {f E : f (S) = 0} where S := {ω Ω : r(ω) lim η ω r(η)} Definition (Stachó-Zalar, 2003). CRD bounded complete R-domains in commutative C -algebras Recall: talk in Taiwan Kaohsiang Now: later developments (but no counterexamples) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 9 / 19

29 CRD in C 0 (Ω) Henceforth Ω locally compact space E := C 0 (Ω), f = f := max f D CRD in E, (E, E D, {...} D ) PJT Remark: Sunada Ω = {1,..., n}; non-convex Thullen domains D = ( f (1) 2 + f (2) p < 1 ) (0 < p < 1)! Ω 0 open Ω : E D = {f E : f (Ω \ Ω 0 ) = 0}. Remark: Ω 0 = E D = {0} trivial Ω 0 = Ω E D = E symmetric László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 10 / 19

33 FORMULA FOR {...} D THEOREM.! [ Ω ω µ ω ] : µ ω 0 Radon measure on Ω 0 total mass of µ ω M := sup 0 x,a,y 1 max{xay} {xay}(ω) = 1 2 x(ω) for x, y E, a E D. ay dµ ω y(ω) ax dµ ω Remark: bidual-free. Nevertheless for fine structure: Stachó-Zalar 2003 (symmetric case with bidualization) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 11 / 19

36 INGREDIENTS OF PROOF ( C0 (Ω), E } {{ } 0, {...} ) with Reinhardt property: E Ψ Aut(E, E 0, {...}) Ψ := { ψ : ψ = 1 } [ f e iφ f ] : D D, φ bounded C(Ω, R) E D C 0 (Ω 0 ) for some Ω 0 open Ω ψ{xax} D = 2{(ψx)ax} D + {x(ψ)a)x} D {xay} D (ω) = 0 if x(ω) = y(ω) = 0. {xay} 0 if x, a, y 0 Villanueva 2002, Riesz type representation for 3-linear φ : E E 0 E E φ = φ 1 φ 2 φ 1, φ 2 0 László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 12 / 19

39 SYMMETRIC PART Stachó-Zalar 2003: D E D = { f C 0 (Ω 0 ) : } ω Ω i m(ω) f (ω) 2 < 1, i I Π D := { Ω i : i I } partition of Ω, m : Ω R + sup #Ω i <, 0 < inf m sup m(η) < i I i I η Ω i µ ω = η Ω i(η) m(η)δ η for ω Ω 0 Remark: Sunada for Ω 0 finite, D E D = [Hilbert balls] László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 13 / 19

42 EXTENSION OF INNER DERIVATIONS THEOREM. (E, E 0, {...}) PJT with Reinhardt property = Proof: M δ M δ E0 (δ 0 ) δx := N {a k b k x} k=1 [ N ] N [ ] 2δx Ωi = m(η) a k (η)b k (η) x Ωi + m(η)x(η)b k (η) a k Ωi η Ω i k=1 k=1η Ω i N δx(ω) = a k (ζ)b k (ζ) dµ ω (ζ) x(ω) for ω Ω \ Ω 0 ζ Ω 0 k=1 Continuity of {...} = sup ω Ω µ ω (Ω 0 ) < N δ E0 It suffices: sup a k (ζ)b k (ζ) Const. inf m ζ Ω 0 k=1 László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 14 / 19

46 PROOF For ω Ω i (i fixed) choose 0 e ω E 0 = C 0 (Ω 0 ): ( 1 = e ω (ω) = max e ω, e ω Ωi \ {ω} ) = 0 2 N δe ω (ω) = [#Ω } {{ } i + 1] m(ω) a ω Ω i } {{ } k (ω)b k (ω) δ E0 ω Ω i k=1 inf m N m(ζ) b k (ζ)a k (ζ) = 2[ e ζ ](ζ) N m(ω) a k (ω)b k (ω) ω Ω i k=1 k=1 k=1 N 2 m(ω) a k (ω)b k (ω) 2 E 0 + E 0 #Ω i + 1 ω Ω i László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 15 / 19

50 FINE STRUCTURE E = C 0 (Ω), E 0 = C 0 (Ω 0 ), m : Ω 0 R +, 0 < inf m sup m < Π = {Ω i : i I } partition of Ω 0, κ ω sup i I #Ω i < ( 0 Radon measure on I K I open: i K Ω ) i open Ω 0 {xax}(ω) := x(ω) m(ζ)a(ζ)x(ζ) dκ ω (i) i I ζ Ω i { }} { E(Ω, Ω 0, m, Π, κ) := (E, E 0, {...}) with κ ω = δ i(ω) for ω Ω 0 THEOREM. E(Ω, Ω 0, m, Π, κ) subtriple of E, E D, {...} D ) for some CRD D iff 1) Ω 0 ω η Ω i(ω) m(η)f (η) continuous f E 0, 2) ω κ ω weakly continuous László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 16 / 19

57 BIDUAL EMBEDDING Recall: E = C 0 (Ω) C( Ω), Ω Ω compact hyperstonian THEOREM. Let D be a CRD in E = C 0 (Ω). Then there exists a CRD D in E := E C( Ω) such that ( ) 1) (E, E D, {...} D ) is a subtriple of E, ED, {...} D 2) E D is the weak*-closure of E in E 3) {...} D is the separately weak*-continuous extension of {...} D. László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 17 / 19

62 OWN REFERENCES (CRD) L.L. Stachó, Continuous Reinhardt-domains from a Jordan view point, Studia Math. 185(2), (2008). L.L. Stachó, Banach Stone type theorem for lattice norms in C 0 -spaces, Acta Sci. Math. (Szeged) 73, (2007). J.M. Isidro - L.L. Stachó, Holomorphic invariants of continuous bounded symmetric Reinhardt domains, Acta Sci. Math. (Szeged) 71, (2004). L.L. Stachó and B. Zalar, Symmetric continuous Reinhardt domains, Archiv der Math.(Basel) 81, (2003). László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 18 / 19

63 OWN REFERENCES (PJT) L.L. Stachó, On the classification of bounded circular domains, Proc. R. Ir. Acad. 91A(2), (1991). L.L. Stachó, On the spectrum of inner derivations in partial Jordan triples, Math. Scandinavica 66, (1990). L.L. Stachó, On the structure of inner derivations in partial Jordan-triple algebras, Acta Sci. Math.(60), (1995). László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 19 / 19