GENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS *

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1 J. ci. I. R. Ir Vol., No. 3, ummer 2000 GENERALIZED POITIVE DEFINITE FUNCTION AND COMPLETELY MONOTONE FUNCTION ON FOUNDATION EMIGROUP * M. Lshkrizdeh Bmi Deprtmet of Mthemtics, Istitute for tudies i Theoreticl physics d Mthemtics, Tehr, P. O. Box , Islmic Repulic of Ir Astrct A geerl otio of completely mootoe fuctiols o ordered Bch lger B ito proper H * -lger A ith itegrl represettio for such fuctiols is give. As pplictio of this result e hve otied chrcteriztio for the geerlized completely cotiuous mootoe fuctios o eighted foudtio semigroups. A geerlized versio of Bocher s theorem o foudtio semigroups is lso otied. Itroductio I the preset, pper e shll itroduce the cocept of completely mootoe fuctiol o ordered Bch lger B ito proper H * -lger A d e shll give itegrl represettio for such fuctiols ith respect to A-vlued mesures o (B, the spce of ll positive multiplictive lier fuctiols o B. As pplictio of the theory e shll oti itegrl represettio for the geerlized -ouded cotiuous completely mootoe A-vlued fuctios ith respect to positive A-vlued mesures o Γ, the spce of ll -ouded cotiuous oegtive semichrcters o foudtio semigroup ith Borel mesurle eight fuctio. We ill lso give geerliztio of our erlier versio of Bocher s theorem [4; Theorem 4.2]. Keyords: Loclly compct semigroups; Positive defiite fuctios; H * -lgers; pectrl mesures. Prelimiries Recll tht (see, [], [2], [3], [7] proper H * -lger is Bch lger A hose orm is Hilert spce orm d hich hs ivolutio: x x* o A such tht ( y, x *, z ( xy, z ( x, zy * for ll x, y, z A. Let τ ( A { xy : x, y A} e the trce clss of A. It is Bch lger ith respect to orm τ (. hich is relted to the orm. of A y τ ( * 2 for ll A tr( tr( (,. There is trce tr defied o τ (A such tht for ll, A, here (.,. deotes the sclr product o A. if * for some A the is clled positive d e rite 0. It is ovious tht 0 if d oly if ( x, 0 for ll x A. A right module H over A is clled Hilert module if there is τ (A -vlued fuctio (, o H H ith the folloig properties. ( ξ, ϕ ( ϕ (, ϕ for ll ξ,, ϕ H. 2. ( ξ, * (, ξ for ll ξ, H. 3. ( ξ, ( for ll H d ech A. 4. ( ξ 0 for ll ξ H d ( ξ, ξ 0 if d oly * 99 Mthemtics uject Clssifictio. Primry 43 A35, 43 A0; This reserch s i prt supported y grt from IPM. E-mil: lshkri@mth.ui.c.ir 245

2 Vol., No. 3, ummer 2000 Lshkrizdeh Bmi J. ci. I. R. Ir if ξ tr ( τ ( ξ τ (, for ll ξ, H. 6. H is complete i the orm ξ ( τ ( ξ, ξ 2. The fuctio (, is clled geerlized sclr product. There is lier structure o H such tht H is ordiry Hilert spce ith respect to the sclr product ξ, tr(, ξ. A A-lier opertor o H is dditive lier mppig T : H H such tht T ( ξ ( Tξ for ll ξ H, A; T is ouded i the i the sese tht Tξ M ξ for some M 0 d every ξ H. For ech ouded A-lier opertor T its djoit T * is A-lier d hs the property tht ( T ( T * for ll ξ, H. By rel ordered Bch lger e shll me rel Bch lger M r ith closed prtil order stisfyig the folloig: (i x y x z y z, for ll z Mr. (ii x 0, y 0 xy 0. (iii x 0 x 0, for ll oegtive rel umers. Note tht order o ordered Bch lger is clled closed if for every to sequeces (x d (y i B from x x d y y d x y ( it follos tht x y. A complex Bch lger B of the form Mr im r, here M r is rel ordered Bch lger, is clled ordered Bch lger. O ordered Bch lger B, e put P ( B { B : 0} d P ( B { P( B : }. A lier fuctiol f o B is clled positive if f ( 0 for ll P(B. I the cse here B is commuttive, e shll deote y (B the spce of ll ouded multiplictive lier fuctiols o B d y (B the spce of ll positive fuctiols i (B. Defiitio.. Let B e commuttive ordered Bch lger. For every (the set of oegtive itegers e defie the opertor o B * (the dul of B y 0 f ( f ( f ( ; 0 f ( 0 f ( f ( f ( d for every 2 f ( ;, K, ( f ( ;, K, f ( ;, K, f B *,,, K, B;,2, K. A lier fuctiol f B * is clled completely mootoe if f ( ;, K, 0 for ll d,, K, P ( B. A opertor-vlued trsformtio U : B L (H (the spce of ll ouded lier opertors o Hilert spce H is clled completely mootoe if for every ξ H the mppig ϕ ξ : U ξ, ξ ( B defies completely mootoe fuctiol o B. We o recll some defiitios cocerig topologicl semigroups. Throughout this pper ill deote loclly compct, Husdorff topologicl semigroup. Defiitio.2. O commuttive topologicl semigroup ith C ( (the spce of ouded cotiuous complex-vlued fuctios o iductive idetity, for ech e defie the opertor o y f ( f (, 0 x f ( x; h 0 f ( 0 f ( xh f ( f ( xh d for every 2 f ( x; h, K, h f ( x; h, K, h f ( xh ; K, h, ( f C (, x, h, K, h,,2, K. A fuctio f C ( is clled completely mootoe if f 0 ( (cf. [5; p. 43]. Defiitio.3. A opertor-vlued trsformtio T : L (H is clled completely mootoe if for every ξ H the mppig x Txξ, ξ ( x is completely mootoe o. Defiitio.4. Let B e ordered commuttive Bch lger d H e Hilert module over proper H * -lger A. A lier mppig f : B A is clled completely mootoe A-fuctiol if for every f ( ;,K, 0 for every ( -positive elemets,,, of B here, f ( f ( 0 f ( ; 0 f ( 0 f ( f ( f ( d for every 2 246

3 J. ci. I. R. Ir Lshkrizdeh Bmi Vol., No. 3, ummer 2000 f ( ;, K, f ( ;, K, f ( ;, K, Defiitio.5. Let e commuttive topologicl semigroup ith idetity. A mppig f : A is clled completely mootoe if ll oegtive itegers d ll 0 f ( f (. f ( x; h, K, h 0 x, h, K, h f ( x; h 0 f ( 0 f ( xh f ( f ( xh d for every 2 f ( x; h, K, h f ( x; h, K, h f ( xh ; h, K, h. for here Defiitio.6. Let B Bch * lger d A e proper H * -lger. A lier mppig clled positive A-fuctiol if i i j * f ( * 0 i j for ll,, i B d,, i A. j f : B A Defiitio.7. Let e * -semigroup. The mppig ϕ : A is clled positive defiite if i j i * ϕ ( x* x 0 i j j for ll x,,x i d,, i A. Recll tht Borel mesurle mppig : (the set of oegtive rel umers ith ( xy ( ( y ( x, y d such tht d re loclly ouded (i.e., ouded o compct susets of is clled eight fuctio o. A fuctio f : is clled -ouded if there is k > 0 such tht f ( k(, for ll x. Recll lso tht M(, deotes the set of ll complex, regulr, siged mesures (ot ecessrily ouded of the form 2 i ( 3 4 here i is positive regulr mesure o ith L (, i i,2,3,4 (see, for exmple [2], [7], [9]. Note tht for elemet M (, d Borel set B, (B is ell-defied heever B is reltively compct. For every M (,, the equtio is fd(. fd ( f C (, defies mesure. M (, the spce of ll ouded regulr complex mesures o. With the orm. ( M (,, here. deotes the totl vritio of., the spce M(, defies Bch lttice, d ith the covolutio product ( ν ( f f ( xy d( dν ( y (, ν, M (,, f C 00(, here C 00 ( deotes the set of ll fuctios i C ( ith compct support, defies Bch lger. From prt (iii of Theorem 4.6 of [7], e coclude tht ( lso holds for every -ouded Borel mesurle fuctio f o. We lso recll (see, for exmple, [], [6], [8] tht ~ M ( (or L ( deotes the set of ll mesures M ( for hich the mppigs x δ x d x δ x (here δ x deotes the Dirc mesure t from ito M( re ekly cotiuous. As i [7], e c defie M(, (or L ~ (, s the set of mesures M (, for hich. M (. The, M(, is closed, to-sided L-idel of M(,. Filly, e cll foudtio semigroup if { supp( : M ( } is dese i. A mppig χ : is clled semichrcter if χ ( xy χ( χ( y for ll x, y. We deote y Γ the set of ll -ouded cotiuous semichrcters o, d y Γ the set of oegtive semichrcters i Γ. If is commuttive d foudtio, the Γ is homomorphic to ( M (, heever Γ hs the compct ope topology d ( M (, hs the Gelfd topology. I prticulr; Γ is loclly compct Husdorff spce (see, Theorem 2.0 of [8]. A opertor-vlued trsformtio U : L (H is clled -ouded (cotiuous, respectively if for every H the mp: x U x is -ouded (cotiuous, respectively. Filly if U : L (H is such tht U xy U xu y ( x, y, the U is clled represettio. For further iformtio o the represettio theory of topologicl semigroups d *-lgers the reder is referred to [7]. ( 247

4 Vol., No. 3, ummer 2000 Lshkrizdeh Bmi J. ci. I. R. Ir 2. Geerlized Represettios d Positive-Defiite Fuctios o Weighted Foudtio emigroups We strt this sectio ith the folloig result hich is ideed geerliztio of our erlier result (Theorem 4.4 of [7]. Theorem 2.. Let e foudtio *-semigroup ith idetity d ith Borel mesurle eight fuctio such tht ( x ( ( x. Let T e *- represettio of M(, y ouded A-lier opertors o Hilert module H over proper H * - lger A such tht for every 0 ξ H there exists mesure M (, such tht T ξ 0. The there exists uique -ouded cotiuous *-represettio V of y A-lier opertors o H such tht (, T ξ (, V ξ d( ( H, M (,. x Proof. Recll tht y Theorem of [] H ith the ier product.,. here ξ, tr(, ξ defies Hilert spce d y Theorem 4 of [], the djoit opertor T * of T defies ouded A-lier opertor o H. o y Theorem 5.4 of [7] there exists -ouded cotiuous *-represettio V of y ouded opertors o the Hilert spce ( H,.,. such tht (2 Tξ, V d( ( M (,, H. x (3 No let R(A deote the spce of the right cetrlizers of A. From Lemm 2 of [4] d Theorem of [] for every U R(A e hve tru (, T ξ tr( U, T ξ T U V U d( o y Theorem 2 of [6] tru (, V ξ d(. x x x (, T ξ (, V ξ d( ( M (,, H. x This proves formul (2. We shll o use formul (3 d prove tht if A-lier for every M (,, the Vx is A-lier for every x. To see this from (3 for every M (,, H, d A e hve ( ξ d(, T ( ξ tr( T ( ξ, V, x T is tr (( T, tr( T ξ, V ξ d( x,( V xξ d( x ice oth the mppigs: x, V ( d x ξ x, ( Vx ξ re -ouded d cotiuous d is foudtio semigroup, from Lemm 4.8 of [7] e coclude tht Vx ( ξ ( Vxξ ( x, A. The folloig result is ideed geerliztio of our erlier versio of Bocher s theorem [4; Theorem 4.2]. Theorem 2.2. (Geerlized Bocher s theorem o foudtio semigroups. Let e commuttive foudtio topologicl *-semigroup ith idetity d ith Borel mesurle eight fuctio. Let A e proper H * -lger over Hilert module H. The mppig ϕ : τ ( A is -ouded cotiuous d positive defiite if d oly if there exists uique positive A-vlued mesure λ o such tht ϕ Γ ϕ( χ( dλ ( χ ( x Γ * ϕ. Proof. ice ϕ is -ouded d cotiuous, y Theorem of [6] there exists -ouded ekly cotiuous *-represettio V of y ouded A-lier opertors o Hilert A-module K ith some ξ 0 K such tht ϕ( x ( ξ 0, Txξ0 d V x ( for every x. Usig the itegrtio theory o pge 20 of [3] d Lemm 2 of the sme referece, e coclude tht the mppig Φ : (, τ ( A give y M Φ( ϕ( d( ( ξ0, V ξ0 d( ( M, x is ell-defied. It is lso esy to see the Φ defies positive A-fuctiol o the Bch *-lger M (,. Therefore, y Theorem 3 of [5] there exists positive τ& (A -vlued mesure λ o ( (, such tht M Φ( ˆ( dλ(. ( M (, Usig Theorem 2.0 of [7], e coclude tht Φ ( χ( d( dλ( ( M (,. ( * Γ By Fuii s theorem ( χ( dλ( χ d( ( M (,. ϕ ( d( * Γ ice oth fuctios ϕ d Γ x * χ( dλ( χ. re 248

5 J. ci. I. R. Ir Lshkrizdeh Bmi Vol., No. 3, ummer ouded d ekly cotiuous d is foudtio semigroup, e ifer tht ϕ ( χ( dλ( χ ( x. Γ * The uiqueess of λ follos i the sme lies s those of Theorem 4.2 of [4]. 3. Completely Mootoe Fuctiols o Ordered Bch Algers Our strtig poit of this sectio is the folloig: Theorem 3.. Let B e commuttive ordered Bch lger ith ouded pproximte idetity ( e i P(B. Let k e the set of ll completely mootoe fuctiols f i B * such tht f. The K is covex d ek*-compct suset of A *. If f is extreme poit of K, the f ( 0 for ll P(B d f ( f ( f ( for ll, B. Proof. It is cler tht K is covex d ek*-closed suset of the uit ll of B * d so y the Bch Aloglu theorem is ek*-compct. Let f e extreme poit of K. The it is cler tht f ( 0 for ll P(B. ice P(B sps B, to prove tht f ( f ( f ( for ll, B, it suffices to sho tht f ( f ( f ( for ll, P ( B. For every B e defie f * B y f ( f ( ( B. It is esy to see tht ( f f ( ;, K, f ( ;, K,,, for ll, d,,, K, P ( B. Thus, f f is lso completely mootoe. o d ( f f ( e 0( f f ( e 0, ( f f ( e ( f f ( e ( f f ( e ; 0, for ll, P ( B. follos tht 0 ( f f ( e ( f f From these to iequlities it f ( e ( e f ( e f ( e for ll d ll, P ( B. ice (e is ouded pproximte idetity for B, it follos tht 0 ( f f ( f ( (, P ( B. (4 Usig the fct tht f is completely mootoe, e coclude tht d Thus 0 0 f ( f ( (, P ( B, 0 f ( ; f ( f ( (, P ( B. 0 f ( f ( (, P ( B. (5 We shll o cosider three cses. Cse. f ( 0. o y (5, f ( 0. Hece f ( 0 f ( f ( (, P ( B. Cse 2. f (. The y (4, f ( f ( (, P ( B d so f ( f ( f ( f ( (, P ( B. Cse 3. 0 < f ( <. I this cse e rite f f f f ( f ( f (. f ( f ( From (4 it follos tht ( f f ( f ( K, d (5 implies tht f f ( lso elogs to K. ice f is extreme poit of K, it follos tht f f ( f. o f ( f ( f ( for ll, P ( B. This completes the proof. Theorem 3.2. Let B e commuttive ordered Bch lger ith ouded pproximte idetity (e i P (B. The lier trsformtio U : B L(H (H is Hilert spce is completely mootoe if d oly if there is positive opertor-vlued mesure E o (B such tht Uξ, ( d E (. ( H, B. ( B (6 Moreover, U is represettio if d oly if E is spectrl mesure. Proof. Let U : B L(H e completely mootoe. Without loss of geerlity, e my ssume tht U ( B. For every ξ H ith ξ e defie the lier fuctiol L ξ o B y L ( Uξ, ξ ( B. ξ It is cler tht L ξ defies completely mootoe fuctiol o B ith L ξ. By the itegrl form of the Krei-Milm theorem [0; p. 6] d Theorem 3., there exists uique regulr proility mesure ξ o 249

6 Vol., No. 3, ummer 2000 Lshkrizdeh Bmi J. ci. I. R. Ir (B such tht for every, B e hve L ( ξ o if ( d ( (. (, B B ξ ξ 0 ξ H is ritrry, the there exists uique positive regulr mesure ξ ith d U ξ ( d ( (. (, B B ξ ξ By the polriztio idetity for every B e hve U Thus here U 4 ( U 4 2 ξ, ξ ξ H d ( ξ, ξ U ( ξ, ξ i U ( ξ i, ξ i i U ( ξ i, ξ i. ( d ( (,,, (, B ξ H B ξ ( i ξ, ξ, ξ ξ, ξ i ξ i, ξ i ξ i, ξ i No let B ( ( B deote the -lger of ll Borel susets of (B. E o B ( ( B y Defie the opertor-vlued mesure ( M, ( M ( H, M B( ( B. E ξ It is esy to see tht E is positive, i the sese tht E ( M ξ 0 for ll ξ H d M B( ( B. Moreover, Uξ, ( d E (. ( B ( B, H. For simplicity, e revite this equlity s U ( B ( de ( B. No for every B e deote y ˆ the restrictio of the Gelfd trsform of to (B, tht is ˆ ( ( for ll (B. ice y the Gelfd represettio theorem poits of (B, P { ˆ : B} seprtes the from the toe-weierstrss theorem it follos tht it is dese i C 0( ( B, the spce of ll cotiuous complex-vlued fuctios o (B vishig t ifiity. No if U is multiplictive, the. ( B ˆ( ˆ( de ( B ( B ( de ˆ( de U ( B U U ˆ( de ice for every fixed B, ech of the fuctios ˆ ( de d ( B ˆ ˆ( de ( B ( ( B, ˆ( de ( ˆ P re ouded d ( B lier o P, d P is dese i C 0( ( B, Borel suset M of (B e hve ( B M ( ˆ( de ( B M ( de ( B the for every ˆ( de here M deotes the chrcteristic fuctio of the set M. A similr rgumet shos tht for every to Borel susets M d N of (B ( B M ( ( de N ( B M ( de ( B N ( de. Tht is E ( M N E( M E( N. o E is spectrl mesure o (B. The proof is o complete. The folloig theorem gives chrcteriztio of the completely mootoe fuctiols o commuttive ordered Bch lgers. Theorem 3.3. Let B e commuttive ordered Bch lger ith ouded pproximte idetity i P (B. The ouded lier mppig U of B ito proper H * -lger A is completely mootoe if d oly if there is positive τ (A -vlued mesure E o (B such tht ( U ( ( d( E (.( B, H. ( B Moreover, U is positive homomorphism if d oly if E is geerlized spectrl mesure. Proof. For every ξ H ith tr( ξ e defie L ( tr( ξ, ξu ( ξu (, ξ ( B. ξ From ξ, ξu ( ;, K ( ξ U ( ;, K, ( (;, Β,. 250

7 J. ci. I. R. Ir Lshkrizdeh Bmi Vol., No. 3, ummer 2000 d the fct tht U is ouded d completely mootoe e coclude tht defies completely lier L ξ fuctiol o B. o y Theorem 3.2 there exists opertor-vlued mesure E y ouded opertors o the Hilert spce ( H,.,. such tht Lξ ( ξu (, ( B ( d E (. ( B. For every T R(A y Lemm 2 of [4] e hve trt(, ξu ( tr( T, ξu( ξu(, T ( B ( B ( d E (. T ( d trt(, E (. ξ. No it is esily see tht the mppig: τ (A give y: (, E( M ξ ( M B( ( B defies τ (A -vlued mesure o (B. Therefore, y Lemm 2 of [4] d Theorem 2 of [6] e hve (B M (, ξu ( ( d(, E (. ξ ( B. ( B Thus, the proof is complete. We re o i positio to stte d prove the mi result of this pper. Note tht if H is right Hilert module over proper H * -lger A, the mppig T : A is clled -ouded d cotiuous if for every H the mppig x tr( Tx is -ouded cotiuous complex-vlued fuctio o. Theorem 3.4. Let e commuttive foudtio semigroup ith idetity d ith Borel mesurle eight fuctio cotiuous t the idetity. Let H e Hilert module over proper H * -lger A. The mppig T : A is -ouded cotiuous d completely mootoe if d oly if there exists uique positive A-vlued mesure E o such tht Γ ( (. (, T χ( d E (. x, H ξ x Γ χ T is homomorphism if d oly if E is geerlized A-vlued spectrl mesure. Proof. From the cotiuity of t the idetity of it follos tht M (, hs ouded pproximte idetity i P ( M (, (see [9]. It is lso esy to see tht the equtio ( ξ, U ( ( T d( ( M (,, H x defies completely mootoe A-vlued ouded fuctiol o the ordered Bch lger M (,. Therefore, y Theorem 3.3 there exists uique positive A-vlued mesure E o ( (, such tht M ( (, U ( ˆ ( χ d E (. ( M (, ξ χ ( ξ, H, M (,. No pplictio of this equlity d Theorem 2.0 of [8] ith the id of Fuii s theorem gives ( ( U ( χ( d( d E (. Γ Γ ( (. χ( d E (. d x No sice oth mppigs x χ( d Eχ (. χ χ Γ d x Tx re -ouded d cotiuous d is lso foudtio semigroup, e coclude tht ( (. ( Tx χ( d E (. H, x χ Γ Remrk. The folloig exmple shos tht the coclusio of the precedig theorem is ot vlid i geerl for o-foudtio semigroups. Exmple 3.5. Let [0,]. The ith the usul topology of the rel lie d the multiplictio xy mi( x, y( x, y defies o-foudtio semigroup. If e choose o, the Γ { }, here deotes the fuctio hich is ideticlly oe o. It is cler tht the mppig T : L( L2 (, m (m deotes the Leesgue mesure o [0,] give y f xˆ f ( x, f L2 (, m, T x here xˆ deotes the chrcteristics fuctio o [0,x], defies completely mootoe opertor-vlued trsformtio of y opertors o the Hilert module L 2 (, m (see [3]. If the formul (6 is vlid for T, the e rrive t the cotrdictio tht T x I for every x i, here I deotes the idetity opertor o L 2 (, m. Refereces. Bker, A. C. d Bker, J. W. lger of mesures o loclly compct semigroups III, J. Lodo Mth. oc., 4, , ( Bker, J. W. Mesure lgers o semigroups I: The Alyticl d Topologicl Theory of emigroups- Treds d Developmets, (Ed., Hoffm, K. H. et l., de Gruyter, pp , (

8 Vol., No. 3, ummer 2000 Lshkrizdeh Bmi J. ci. I. R. Ir 3. Bker, J. W. d Lshkrizdeh Bmi, M. O the represettios of certi idempotet topologicl semigroups, emigroup Forum, 44, , ( Bker, J. W. d Lshkrizdeh Bmi, M. Represettios d positive-defiite fuctios o topologicl semigroups, Glsgo Mth. J., 38, 99-0, ( Dukl, C. F. d Rmirez, D. E. Represettios of commuttive semitopologicl semigroups, Lecture Notes i Mthemtics, 435 Berli, priger-verlg, Ne York, ( Dziotyieyi, H. A. M. The logue of the group lger for topologicl semigroups, Pitm, Bosto, Mss., Lodo, ( Lshkrizdeh Bmi, M. Represettios of foudtio semigroups d their lgers, Cdi J. Mth., 37, 24-47, ( Lshkrizdeh Bmi, M. Bocher s theorem d the Husdorff momet theorem o foudtio semigroups, Iid., 37, , ( Lshkrizdeh Bmi, M. Fuctio lgers o eighted topologicl semigroups, Mth. Jpoic, 47, (2, , ( Phelps, R. P. Lectures o Choquet s theorem, Priceto, (960.. oroto, P. P. A geerlized Hilert spce, Duke Mth. J., 35, 9-97, ( oroto, P. P. Represettio of topologicl group o Hilert module, Iid., 37, 45-50, ( oroto, P. P. Bocher-Rikov theorem for geerlized positive defiite fuctio, Iid., 38, 7-2, ( oroto, P. P. Geerlized positive lier fuctiols o Bch lger ith ivolutio, Proc. Amer. Mth. oc., 3, , ( oroto, P. P. Itegrl s certi type of positive defiite fuctio, Iid., 35, 93-95, ( oroto, P. P. emigroups ith positive defiite structure, Iid., 40, , ( oroto, P. P. d Friedell, J. C. Trce-clss for ritrry H * -lger, Iid., 26, 95-00, ( leijpe, G. L. G. Covolutio of mesure lgers o semigroups, Ph. D. Thesis, Ctholic Uiversity, Toerooiveld, Nijmege, The Netherlds, (