CONTINUOUS REINHARDT DOMAINS PROBLEMS ON PARTIAL JB*-TRIPLES László STACHÓ Bolyai Institute Szeged, Hungary stacho@math.u-szeged.hu www.math.u-szeged.hu/ Stacho 13/11/2008, Granada László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 1 / 19
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
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
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
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
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 1990-91: 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
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 1990-91: 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
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 1990-91: 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
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 1990-91: 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 2006. Now: later developments (but no counterexamples) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 9 / 19
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 2006. Now: later developments (but no counterexamples) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 9 / 19
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 2006. Now: later developments (but no counterexamples) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 9 / 19
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 2006. Now: later developments (but no counterexamples) László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 9 / 19
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
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
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
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
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 + 1 2 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
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 + 1 2 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
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 + 1 2 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
OWN REFERENCES (CRD) L.L. Stachó, Continuous Reinhardt-domains from a Jordan view point, Studia Math. 185(2), 177-199 (2008). L.L. Stachó, Banach Stone type theorem for lattice norms in C 0 -spaces, Acta Sci. Math. (Szeged) 73, 193-208 (2007). J.M. Isidro - L.L. Stachó, Holomorphic invariants of continuous bounded symmetric Reinhardt domains, Acta Sci. Math. (Szeged) 71, 105-119 (2004). L.L. Stachó and B. Zalar, Symmetric continuous Reinhardt domains, Archiv der Math.(Basel) 81, 50-61 (2003). László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 18 / 19
OWN REFERENCES (PJT) L.L. Stachó, On the classification of bounded circular domains, Proc. R. Ir. Acad. 91A(2), 219-238 (1991). L.L. Stachó, On the spectrum of inner derivations in partial Jordan triples, Math. Scandinavica 66, 242-248 (1990). L.L. Stachó, On the structure of inner derivations in partial Jordan-triple algebras, Acta Sci. Math.(60), 619-636 (1995). László STACHÓ () CONTINUOUS REINHARDT DOMAINS 13/11/2008, Granada 19 / 19