Bull. Kyushu Inst. Tech. (Math. Natur. Sci.) No. 37, 1990, pp. 1-20

REPRODUCING KERNELS IN MODULES OVER C*-ALGEBRAS AND THEIR APPLICATIONS

Shigeru IToH

(Received November 7, 1989)

1. Introduction Sz.-Nazy et al. gave dilation theorems for positive definite operator valued mappings in Hilbert spaces (cf. Sz.-Nazy [34], Sz.-Nazy and Foias [35], Masani [19, 20], Mlak [22], Evans and Lewis [7]). Szafraniec [30, 31, 32, 33] obtained simple characterizations of boundedness condition. Loynes [18] extended dilation theorems to the case of VH-spaces. Stinespring [27] investigated similar results for completely positive maps. Paschke [25] also considered Stinespring type theorems in Hilbert C*- modules. Itoh [14] combined these results in the frame of Hilbert C"-modules and gave generalizations of dilation theorems and characterizations of boundedness condition. G6rniak, Masani, Weron et al. investigated positive definite B(X,X*)-valued mappings, where X is a Banach space, and obtained dilation theorems for them ([5], [8], [9], [12], [19], [20], [22], [23], [24], [37], [38]). We also refer the reader to the papers [10], [13] for corresponding results in locally convex spaces. Aronszajn [1] described the theory of reproducing kernel Hilbert spaces for numerical-valued positive definite kernels. Later, this theory was extended to the case of (Hilbert space) operator-valued kernels (cf. Evans and Lewis [7]). Kakihara [15] further extended the theory to the case of W"-algebra-valued kernels and constructed reproducing kernel Hilbert W"-modules. On the other hand, Masani et al. treated the theory for B(X,X")-valued kernels (Masani [19], Miamee and Salehi [21], Chatterji [41], cf. Suciu and Valusescu [28, 29]). In this paper we extend all the above results to the case of Banach C"-modules and give general dilation theorems. In order to prove them, we use the method of reproducing kernel Hilbert C*-modules. More precisely, let B be a C"-algebra and W be a Banach B-module. We first construct reproducing kernel Hilbert B-modules for M(W, W')-valued positive definite kernels. Using these results, we obtain dilation theorems for MÅqW, Vti')-valued positive definite mappings defined on unital *- semigroups. As applications, we give "unitary" dilations for contractions in M(X)(not in ut(X)), where X is a Hilber.t B-module. 2. Preliminaries Let B be a C"-algebra and Wbe a right B-module. We denote the module action ofB on Wby (x, b).x¥b(xE W, beB). We assume that any module treated below has a vector space structure over the complex numbers C compatible with that of B in the sense that A(x ¥ b) == (Zx) ¥b = x' (ab) (x E W, bG B, ZE C) . Let Z be a right B-module. A mapping T: W--År Z is called a module map if T satisfies T(x ' b) = (Tx) ' b (x E W, bE B).

DEFiNmoN 2.1. A right B-module VVis called a Banach B-module if Wis a Banach space with the norm 1fi ¥ 11w satisfying

llX'bllwr ÅqÅ~ llxliwrilbll (xEW, bEB)•

Let W, Z be Banach B-modules. We denote by B(W, Z) the set of bounded linear operators of Winto Z, and write B(W) for B(W, W). Let M(W, Z) be the set of bounded linear module maps of J?Vinto Z, and M(J2V) for M(W, W). Since B itself is a Banach B-module, W' = M(W, B) becomes a Banach B-module as follows:

(T¥ b) (x) = b" Tx ; (AT) (x) - ATx ; HTIIwt = 11Tliw,B (the operator norm of T); (TE W', bEB, xE W, ZE C).

DEFiNmoN 2.2. (i) A right B-module X is said to be a pre-Hilbert B-module if X is equipped with a conjugate bilinear map Åq•, •År: X Å~ X.B(called a B-valued inner product on X) satisfying the following conditions:

(I) Åqx, xÅr År. O, and Åqx, xÅr ==O only if x=O;

(II) Åqx, yÅr = Åqy, xÅr";

(III) Åqx•b, yÅr = Åqx, yÅrb; (x, yEX, bEB).

(ii) A pre-Hilbert B-module X is called a Hilbert B-module if X is complete with respect to the norm Hxll = llÅqx, xÅrll'i2 (xEX)(cf. Paschke [25]). Hilbert B-modules are Banach B-modules (cf. Paschke [25]). In the sequel we use symbols W, Z for Banach B-modules, and X, Y for Hilbert B-modules. The C"-algebra B is also a Hilbert B-module with Åqa, bÅr == b'a(a, bEB). We denote by QI(X, Y) the set of operators TEB(X, Y) for which there exists an operaotor T"EB(Y, X) such that ÅqTx, yÅr = Åqx, T*yÅr for every xEX, yEY It is easy to see that S}I (X, Y) c M(X, Y), in particular S}I (X) c M(X), where QI (X) = E}I(X, X). In general E}l(X) does not equal to M(X) (Paschke [25]). It is shown that E}l(X) is a unital C"-algebra (Loynes [17], Paschke [25]) Reproducing Kernels in Modules 3

For any xEX, Åí=Åq•,xÅr belongs to X'. Let Xl be the set of such Åí(xEX), then Xi is a right B-submodule of X' and the mapping x.Åí(xEX) is a one to one linear module map such that ll x ll. = 11 Åí 11 ., (x EX). In general f l X' (Paschke [25] ). Now let S be a set and Åë(S, W) be the set of W-valued mappings on S. Let F(S, W) be the set of mappingsfE ÅqP(S, JÅrV) such thatf(s) = O except for a finite number of elements sES. It is clear that (P(S, W) and F(S, VV) are right B-modules with (f'b)(s) ==f(s)¥b(sES, bEB). Similarly let 45(S, W') be the set of W'-valued mappings on S. A mapping K:SÅ~S--ÅrM(W, W') is called a kernel.

DEFiNmoN 2.3. A kernel K: S Å~ S.M(W, W') is said to be positive definite (PD) if for any fEF(S, VV),

2(K(s, t)f(s))(f(t)) År. o.

s,t

REMARK 2.4. For any ,PD kernel K, we have ((K(s,t)x)(y))* = (K(t, s) y) (x) (s, tES, x, yE W).

Proof. Its; t, then the above equality is easily seen to be true. On the other hand, we have the desired conclusion for s=:t by the followings:

4((K(s, s)x) (y)) - (K(s, s) (x + y))(x + y) - (K(s, s) (x - y))(x - y) - i{(K(s, s) (x + iy))(x + iy) - (K(s, s) (x - iy))(x - iy)},

and (K(s, s)z)(z) År- O for x, y, zEX.

REMARK 2.5. Given a kernel K:SÅ~S.M(MZ, W'), we can define mappings K,: W. 45(S, W'), where sES, and Åë.:F(S, W).Åë(S, W') by K,w = K(s, ¥)w(we W), and

ÅëKf- 2 K,f(s) (fE F(S, W)) SES respectively. Then K, and ÅëK are module maps. For any sEW, wEMZ, define w,,EF(S, W) by (w,)(t) = w if t= s, or O if t 7E s. Then K,w =: ÅëKw,. Let K: S Å~ S.M(Vl(, VV') be a kernel and Ybe a B-submodule of Åë(S, W') forming a Hilbert B-module.

DEFiNiTioN 2.6. A kernel K is called a reproducing kernel (RK) of Yif K satisfies the following conditions: (i) for any sES and wEW, K.wEYand the set {K,w:sES. weW} spansadense B-submodule of Y; (ii) for any sES, wEWand yEY,

(y(s))(w) - ÅqK,w, yÅr•

In this case Yis called a reproducing kernel Hilbert B-module of K.

REMARK 2.7. (i) It is easy to observe that a RK K is PD (cf. Aronszajn [1], Masani [19], Chatterji [4]). (ii) If B=C (the complex numbers), then the denseness of the set {K,w: sES, wE MZ} in the above condition (i) is redundant (iii) Ifa RK K exists for Y, then it is unique. Conversely, a RK Hilbert B-module Y(it always exists for K if K is PD; cf. Theorem 3.1) is unique to K as a Hilbert B- module which is also a B-submodule of Åë(S, W'). (cf. Aronszajn [1], Chatterji [4]).

3. RK Hilbert B-Modules (I) THEoREM 3.1. Let K:SÅ~S.M(W, VV') be a PD kerneL Then there exists a unique RK Hilbert B-module of K.

Proof. Define a B-submodule E of Åë(S, W') by E == {h = ÅëKf: fe F(S, W)}. For any h, kEE with h= Åë.f, k = Åë.g, define Åqh, kÅr by

, Åqh, kÅr == Z(K(s, t)g(s))(f(t)). s,t

Then Åqh, kÅr'=:Åqk, hÅr, and Åq•,•År is well-defined. In fact, if k= ÅqPKg = ÅqPKg', where g, g'EF(S, JV), then

Z(K(s, t)g(s))(f(t)) s.,t = Z((ÅëKg) Åqt)) (f(t)) t - 2((cPKg')(t))(f(t)) - Z(K(s, t)g'(s))(f(t)). t s,t We show that Åq•,•År is a B-valued inner product on E. For each bEB,h= ÅqPKf,k = ÅëKgEE, we have h•b = ÅqP.(f•b) and Reproducing Kernels in Modules 5

Åqh • b, kÅr - 2 ((cP.g) (t)) (f(t) • b) t = (Z ((ÅëKg) (t)) (f(t))) • b t - Åqh, kÅr b.

Since K is PD, we obtain Åqh, hÅr År! O. Suppose that Åqh, hÅr =O with h == ÅëKfEE. Let p be any positive linear form on B. For each kEE, it follows that

lp(Åqk, hÅr)l2 ÅqÅ~ p(Åqk, kÅr)p(Åqh, hÅr)- O by Schwarz's inequality, hence Åqk, hÅr = O. Thus

O == ÅqK,w, hÅr - Z(K(u, v)f(u))(w,(v)) u,v = ((ÅëKf) (S)) (W) - (h(s))(w) for all sES, wE W, and h=O in E. Now, since E be a pre-Hilbert B-module, let Ybe the completion of E with respect to the norm 11h11 = 1Åqh, hÅr1'!2, then Ybe a Hilbert B- module (cf. Paschke [25]). In the following we observe that Ymay be considered as a B-submodule of Åë(S, W'). First we show that for every sES, the mapping K, is a bounded linear module map of Winto Y. For wEW,

ÅqK,w, K.wÅr = ÅqÅëKw,, ÅqPKw,År = (K(s, s)w) (w),

llÅqK,w, K,wÅrli xÅq llK(s, s)w11w, 11wllw Åqx 11 K(s, s) ll w,w, 11 w 11 ft, where 11K(s, s)11w,w, is the operator norm of K(s, s) as a linear operator of Winto W'. It follows that

11 K,w Ily Åqx llK(s, s) 11 W?w, 11wllw•

Now, for any seS, define a mapping J,:E. W' by J,h =h(s)(hEE); then it is easy to see that J, is a linear module map. For any wGW,

ÅqK,w, hÅr = (h(s))(w) = (J,h) (w), hence by Paschke [25]

11 ( Jsh) (W) ll `x 11 KsW II y 11 h II y ÅqÅ~ 11K(s, s) 11W?w, 11wllwBhlly, -

t- and

11 Jsh ll w' `Å~ ll K(s, s) 11 ti/ ?w, 11 h ll y•

Thus the mapping J, can be extended to a bounded linear module map (donoted also by J,) of Yinto W' and

ÅqK,w, yÅr = (J,y) (w) (yE Y).

Define a mapping 9[': Y. gP(S, W') by (!Uy)(s)=J,y(sES,yEY); then !U is a linear module map and (YZh)(s) =h(s) for any hEE. If we show that tU is one to one, then Y may be viewed as a B-submodule of Åë(S, W') by identifying Ywith tU(Y). So we suppose that !Uy=O for some yEY Then J.y=O for every sES. If follows that ÅqK,w, yÅr =(J,y)(w) =O for all sES, wEW Hence, for each h= ÅëKfEE, h == ZK,f(s)

Åqh, yÅr -2ÅqK,f(S), yÅr : O. s Since E is dense in Y, this implies y=O, and 9Pf is one to one. Finally we note the following. If {h.} c E converges to yEY, then for any sES, {h.(s)} converges to y(s) = J,y, hence y is just the "limit point" of {h.} in Åë(S, W') (cf. Aronszajn [1], Miamee and Salehi [21]).

REMARK 3.2. (i) (Conjecture) Let K:SÅ~S.M(W, MZ') be a PD kernel and Ybe the corr' esponding RK Hilbert B-module of K. For any yEdi(S, W'),yEYif and only if there exists a positive number c such that the kernel KY: S Å~ S --År M(W, VV') defined by KY (s, t) = cK (s, t) - y(t) (!D y(s) (s, tE S) is PD, where for v', w'E W', . we define v' (g) w'EM(w, w') by (v' (g) w')(w) = v'¥w'(w)(wEMz). (ii) We give the proof of (i) in the case where B== C. This is only a slight modification of that of Beatrous and Burbea [2, Theorem 2.2]

Proof. (=År)(This direction is in fact valid for general C"-algebra B.). Let yEÅqZ}(S, W') be contained in }I Choose cÅrO so that c År- 11ylii, and define KY as above for this c. For every fEF(S, W), 2y(s)(f(s))=2ÅqK,f(s),yÅr =Åq2K,f(s),yÅr. It follows that ss s 2(Ky(s, t)f(s))(f(t))

s,t

- c2(K(s, t)f(s))(f(t)) - 2(y(s)(f(s)))" (y(t) (f(t))) s,t s,t - c2(K(s, t)f(s))(f(t)) - (2 y(s) (f(s)))* (2 y(t) (f(t))) Reproducing Kernels in Modules 7

- c Åq2K,f(t), 2K,f(s)År - Åqy, 2K.f(s)År Åq2K,f(t), yÅr

År/ c Åq2 K,f(t), ZK,f(s)År ts - liyll st 2y Åq2 K,f(s), 2K,f(t)År

År-O, ts st where we use the inequality

Åqy, zÅr Åqz, yÅr ÅqÅ~ IIylliÅqz, zÅr for y, zEY(Paschke [25]). (Åq=) Suppose that for some c År O, the above KY is PD. If y = O in Åë(S, W'), then yE Y So we assume that for some rE S, y(r) 7E O, hence for some wE VV, (y (r)) (w) l O. Since K(s, t) = c-' {KY (s, t) + y(t) (8) y(s)}, Y is the direct sum of RK Hilbert C-modules (=Hilbert spaces) of KY and y(t) (21i)y(s) as a C-submodule of Åë(S, W')(cf. Aronszajn [1, pp. 352-353]). Thus the element

z = KIw + 2 (y(r))(w) y

= cK,w - (y(r))(w) y + 2 (y(r))(w) y

== cK,w + (y(r))(w) y is in Y Therefore the element 1 (y(r))(w) y= {z-cK.w} is in Y

For any (bounded linear) module map V: W. Y, we define V.: Y- W' by (V.y)(w) =ÅqVw, yÅr(yEY, wEM. Then V. is also a (bounded) module map.

CoRoLLARy 3.3. A kernel K: S Å~ S.M(W, W') is PD if and only if there exist a Hilbert B-module Y and a mapping V: S -¥ M(W, Y) such that K(s, t) = V(t). V(s).

Proof. (=År) Let Ybe the RK Hilbert B-module of K and V:S.M(W, Y) be defined by V(s) == K,(sES) in Theorem 3.1. Since for each sES, wEW, yEY,,

ÅqK,w, yÅr == (y(s))(w),

we have V(s). y = y(s), and

V(t). V(s)w = (K.w)(t) == K(s, t)w. (Åq=) For every fGF(S, PV), we obtain

2(K(s, t)f(s))(f(t)) s,t == 2(V(t). V(s)f(s))(f(t)) s,t = 2 Åq V(t)f(t), V(s)f(s) År s,t - Åq2 V(t)f(t), 2 V(s)f(s)År År. O. ts We call the above Vin Corollary 3.3 a decomposition of the PD kernel K. If the set {V(s)wlsES,wEW} spans a dense B-submodule of Y, V is called a minimal decomposition. By Theorem 3.1 and Corollary 3.3 the decomposition Vdefined by V(s) == K,(sES) is minimal. An element TEM(X,Y) is called an isometric embedding if ÅqTx,TyÅr = Åqx, yÅr (x, yEX). If Tis also an onto mapping, then we say that X is isomorphic to Y

CoRoLLARy 3.4. Let K:SÅ~S-.M(W, W) be a PD kernel and V:S.M(W, X) be a minimal dccomposition of K. Then for any decomposition U : S -År M(W, Y) of K, there exists a iSometric embedding TEM(X, Y) such that U(s) == TV(s)(sES). If U is also minimal, then Tbelongs to E}I(X, Y) and T" = T-'.

Proof. Define a mapping Tby

T(Z V(s)f(s)) - 2 U(s)f(s), ss whrerefis any element of F(S, W). Then the domain of Tis a dense B-submodule of X, and for every f, gEF(S, W),

ÅqT(2 V(s)f(s)), T(2 V(t)g(t))År st == Åq2 U(s)f(s), 2 u(t)g(t)År st = 2(U(s). U(t)g(t))(f(s)) s,t == 2(K(t, s)g(t))(f(s)) s,t = Z(V(s). V(t)g(t))(f(s)) s,t Reproducing Kernels in Modules 9

- Åq2 V(s)f(s), Z V(t)g(t)År. st Since Vis minimal, Tmay be extended to an isometric embedding (which is also denoted by T) of X into Y If U is also minimal, then Tbecomes an onto mapping, hence the inverse T-i of T exists and belongs to M(Y, X). For any xEX, yE Y, ÅqTx, yÅr =: ÅqTx, TT"yÅr = Åqx, T-iyÅr.

Thus T" = T-' and TEE}I(X, Y).

4. RK Hilbert B-Modules (II) In this section we consider mappings K:S Å~ S-S}I(X). We also call such K's kernels.

DEFiNiTioN 4.1. A kernel K:SÅ~S.S}I(X) is positive definite (PD) if for any fEF(S, X),

2ÅqK(s, t)f(s),f(t)År År! o. s,t It is easy to observe that a kernel K: S Å~ S. QI(X) is PD if and only if for each fEF(S, X),

2(K(s, t)f(s))(f(t)) År- O, s,t where we identify xEX with SE.Xl and consider K as a mapping of SÅ~S into M(X,X'). We have K(s,t)" =K(t,s)(s,tES). Thus we immediately obtain the following result by Theorem 3.1. Note that we make use of similar notations as in section 3 (cf. Remark 2.5). For example, for every sES, K, is the mapping of X into Åë(S,X) defined by (K,x)(t)=K(s,t)x(xEX,tES). We also consider K, as the mapping of S into QI(X).

THEoREM 4.2. Let K: S Å~ S. E}I(X) be a PD kernel. Then there exists a Hilbert B-module Ywhich is also a B-submodule of Åë(S, X) and has the following properties: (i) for any seS, xEX, K,xEY and the totality of such K,x spans a dense B- submodule of Y; (ii) for any sES, xEX, yE Y, Åqx, y(s)År= ÅqK,x, yÅr.

REMARK 4.3. We also call the above Ythe reproducing kernel (RK) Hilbert B- module of K (cf. Remark 2.7). 10 Shigeru lToH CoRoLLARy 4.4. A kernel K:SÅ~S.E}l(X) is PD if and only if there exist a Hilbert B-module Y and a mapping V:S.ut(X,Y) such that K(s,t) = V(t)* V(s)(s, tES).

Proof. (=År) Let Y be the RK Hilbert B-module of K. Then for every sE S, K, EM(X, Y), J, E M( Y, X), where J,y = y(s) (y E Y), and

Åqx, J,yÅr = ÅqK,x, yÅr (xEX, yE Y).

Thus K,' = J, and K,ES}I(X, Y). Define V: S.QI(X, Y) by V(s) = K,(sES), then we obtain K(s, t) = V(t)"V(s)(s, tES)(cf. Corollary 3.3). (Åq==) This is easily verified as in Corollary 3.3. The above Vis said to be a decomposition of K. If the set {V(s)x: sES, xEX} is a dense B-submodule of Y, then Vis called a minimal decomposition. We also have results corresponding to Corollary 3.4, but we omit the details. If A is a unital C"-algebra, then A itself is a Hilbert A-module (cf. Preliminaries) and QI(A)={f(a):aEA}, where f(a)b==ab(bEA). It is clear that IV(a)llA = Il a ll (a EA). Since E}{ (X) is a unital C*-algebra and for TE E}I (X), T År- O as an element of E}I(X) if and only if ÅqTx, xÅr År/ O for all xEX (Loynes [17], Paschke [25]), we have the following lemma (cf. Itoh [14, Lemma 2.6]).

LEMMA 4.5. Let K:SÅ~S.S}I(X) be a PD kerneL Then the kernel L:SÅ~S .ut(E}I(X)) defined by L(s, t) =:f(K(s, t))(s, tES) is PD. Using Lemma 4.5 and Theorem 4.1 we obtain the following corollary.

CoRoLARy 4.6. Let K: S Å~ S. QI(X) be a PD kernel. Then there exists a S}I(X)- submodule of Åë(S, E}I(X)) which forms a Hilbert E}I(X)-module and satisfies the following conditions : (i) for each seS, aEE}I(X), K.aEYand the set {K,a: sES, aEE}I(X)} spans a dense QI(X)-submodule of Y; (ii) for each sES, aEQI(X), yE Y,

a*y(s) =: Åqy, K,aÅr,

in particular, y(s) = Åqy, K,År.

5. Boundedness Condition In the sequel let T be a unital *-semigroup, that is, r is a semigroup with the unit 6 and the involution * such that 4"" == e and (4n)" = ny"4" (4, nET). It is easy to see that Åí" =6. Each group G is a unital *-semigroup with E=e (the identity of G) and g" =, g-i(gEG)¥ Reproducing Kernels in Modules 11

DEFiNmoN 5.1. (i) For any mapping ip:T.M(W, W'), the mapping Kip:TÅ~ r .M(W, W') defined by Kip(4, ny) == ip(ny"4)(4, nET) is called the kernel induced by ip. (ii) A mapping ip: r.M(VV, Jli') is called PD if the kernel Kip induced by ip is PD. We also adopt similar definitions for mappings ip:T.E}I(X). Such notices are also applied to the following definitions.

DEFiNmoN 5.2. (i) A PD kernel K:rxI".M(W, W) is said to satisfy boundedness condition (BC) if there exists a function c: r. [o, oo) such that for any ctEI',fEF(I-', VV),

2(K(ct4, ctn)f(4))(f(q)) 4,n

ÅqÅ~ c(ct)Z(K(4, q)f(4))(f(q))• 4,n (ii) A PD mapping ip:T.M(W, W') is said to satisfy BC if the PD kernel Kip induced by ip satisfies BC. If r is a group, any PD mapping ip:T.M(W, W') satisfies BC with c(4) =- 1(eEr). We have the following characterization of BC concerning PD kernels which is an extension of Lemma 3.1 in [14]. The proof is a slight modification of that given in [14].

LEMMA 5.3. Let K:rxI".M(W, W') be a PD kernel satisfying K(ct4, ctn) = K(ct"ct4, q)(ct, 4, nEr). Then the following conditions are equivalent: (i) K satisfies BC; (ii) there exists a function d: T. [o, co) such that for every ct, CEI", wE W,

(K(ct4, ct4)w)(w) ÅqÅ~ d(ct)(K(e, 6)w)(w);

(iii) there exist a function s: T. [o, co) and a constant C År O such that for every C, nel", s(4q) Å~Åq s(4)s(n) and 11K(e, q)il.,.' Åqx C s(4)s(n)

Proof. (i) =År (ii) This is obvious. (ii) =År (iii) Define a function 't: r. [o, co) by t(4) = inf {d(4):d satisfies condition (ii)}, then it is easy to observe that t also satisfies condition (ii) and t(4n) Åq. t(4)t(ny) for any 4, nEr. Let p be any positive linear form on B. Since K is PD, for any f, gEF(I", W), it follows that

lp(2 (K(4, n)f(4))(g(ny)))12 4,n

Å~Åq p(2 (K(4, n)f(4))(f(ny)))p(2 (K(4, ny)g(C))(g(q)))• 4,n 4,n 12 Shigeru lToH In particular, for each u, vEW, 4, n,EL we have 1p((K(4, ny)u)(v))l2 Åqx p((K(4, 4)u) (u))p((K(n, ny)v) (v)).

Denoting b= (K(4, ny)u)(v)EB, we obtain

p(b"b)2 == p(b"(K(4, n)u)(v))2 = p((K(4, n) (u ¥ b))(v))2

Åqx p((K(4, 4) (u ' b))(u ' b))p((K(ny, n)v) (v))

Åqx t(4)p((K(6, e) (u • b))(u • b))t(n)p((K(6, Åí)v)(v))

ÅqÅ~ t(4) 11 p IHI K(6, Åí) llw,rv, 11 u il ft 11 b ll2t(n) ll p 11 II K(6, s) 11 w,w, ll v H ft, hence p(b"b) Åqx, t(4)'i2t(ny)'i2HpII llK(s, e) llw,w, IIu11wHvllw11bll•

Since p is arbitrary, we can take the supremum in p with 11pll ÅqÅ~1 (cf. Dixmier [6]). This implies

ll b"b ii = Il b ll2 Åq- ,, t(4)i'2t(n)'i2 ll K(Åí, Åí) 11 w,w, ll u liw ll v llw ll b 11 and

ll b 11 Åq., t(4)'/2t(n)'/2 ii K(e, Åí) llw,w, ll u ll rv "v (1wi

Therefore we obtain

ilK(4, q) 11w,w, Åqx t(C)'i2t(n)ii2 llK(e, Åí) ilw,.,.

Define a function s: r. [o, oo) by s(4) = t(oi!2

and let C== llK(6, s)ilw,w,. Then

11K(4, ny)llw,w, Åq-•c Cs(C)s(ny)•

(iii)=År(i) Choose anyfEF(r, W), ctEI". Let 6= ct"ct, and define gEF(r, W) by g(64) ==f(4) iff(4) ; O, othetwise g(4) = O. Note that if 64, = ¥¥¥ = 64. = ny, then we define

g(q) n== 2f(4i). Since K is PD, for every positive linear form p on B, we have i=1 O ÅqÅ~ (p(21(K(ct4, ctn)f(4))(f(n))))2 4,n = (p(Z (K(fi4, ny)f(4))(f(n))))2 4,n Reproducing Kernels in Modules 13

= (p(2(K(4, n)g(4))(f(ny))))2 4,n ÅqÅ~ p(Z (K(e, q)g(4))(g(q)))p(2 (K(4, q)f(4))(f(ny))) 4,n C,n = p(2 (K(624, q)f(4))(fn)))p(2 (K(4, ny)f(4))(f(ny))¥ e,n 4,n Write

q(6) - p(2 (K(64, n)f(4))(f(ny))) 4,n and r = q(6) ; then q(62")2 ÅqÅ~ ,q(62"")(n = O,1.2,..,).

By induction we obtain q(6) Åqx r'-2-n nq(62 -n )2 (n= 1,2,•••).

On the other hand, s(62 ) Åq. nn s(6)2 (n = 1,2,•••), and

O Åqx q(62 ) Åqx ll p nn" Cs (6)2 2 s(4)s(n) 11 f(4) li va II fny) H w• e,n Hence

liminf q(B2 n -n )2 Åq. s(6), n and q(6) Å~Åq rs(6).

Difine c:T.[o, oo) by c(4)=s(4'4). Then for this c,K satisfies BC

CoRoLLARy 5,4. Let K: TÅ~ I".M(W, W') be a PD kernel such that for every ct, 4, nET, K(ct4, ctny) == K(ct"ct4, ny) and supllK(4, ny)11w,w, Åq oo. Then K satisfies BC C,n It is easy to observe that we may let s(Åí) =1 in Lemma 5.3, hence we have the following corollary.

CoRoLLARy 5.5. Let ip:I".M(W, W') be a PD mapping. Then the following 14 Shigeru lToH conditions are equivalent: (i) ip satisfies BC; (ii) there exists a function d: I'. [o, co) such that for any ct, 4Er, wE W,

(ip(4"ct"ct4)w)(w) ÅqÅ~ d(ct)(ip(4"C)w)(w);

(iii) there exist a function s: r. [o, oo) and a constant C År O such that for any C, nET,

II ip(4)11w,w' Åqx Cs(4) and s(4n) ÅqÅ~ s(4)s(q).

6. Dilation Theorems A mapping n: I'. !I(X) is called a representation of T on X if for every g, nyEL n(C)" = n(4"), z(4n) = n(4)n(q) and n(Åí) - 1. (the identity mapping on X).

THEoREM 6.1. Let ip:T-ÅrM(J)V, W') be PD and satisfies BC. Let Ybe the RK Hilbert B-module of the kernel Kip induced by ip. Then there exist a representation n: r-År QI(Y) of T on Yand bounded linear module maps r: W. Yand q: Y. W' with the following properties: (i) q5(4)=qn(C)r(4Er) and the set {z(6)rw:wEW, CEI'} spans a dense B- submodule of Y (minimality condition); (ii) for each yE Y, ct, 6ET, (n(ct)y)(4) = y(ct"4).

Proof. Notations are the same as in Theorem 3.1. (i) For any ctEI', hEE with h=2K$f(4)(fEF(I', J?V)), define c n(ct)h = ZK8ef(C)¥ 4 Then z(ct) is a linear module map of E into E and

Åqz(ct)h, z(ct)hÅr - 2(Kip(cte, ctn)f(4))(f(ny)) 4,ny Å~Åq c(ct)2(Kip(4, ny)f(4))(f(n)) 4,n = c(ct) Åqh, hÅr.

Thus n(ct) may be extended to a bunded linear module map (also denoted by n(ct)) of Y into Y. For each kEE with k==2K$g(4)(gEF(T, w)), e Åqz(ct)h, kÅr - 2(Kip(4, ctop)g(g))(f(ny)) C,n Reproducing Kernels in Modules 15

- 2(Ke(ct*C, ny)g(4))(f(n)) 4,n == Åqh, n(ct')kÅr.

This implies that z(ct)" ==n(ct") and z(ct)EE}I(]Y). It is easy to see that n(ct)n(6) =n(ct6)(ct,6Er) and n(e)=lx. Hence n is a representation of r on Y. Denote r = K,ip and q = J,ip (This is just the J, corresponding to Kip (cf. the proof of Theorem 3.1)). Then we have qr = Kip(s, 6) and for every 4ET, wE W, z(4)rw = z(4)K,ipw = K2w, thus {Y, n} satisfies mimimality condition and qz(4)rw - J,ip Kew = Kip (4, 4)w

=: ip(4)w.

(ii) For any ct, 4ET, wEW, yEY, it follows that

((n(ct)y) (C)) (w) - ÅqKew, n(ct)yÅr

== Åqn(ct")K$w, yÅr

= ÅqKadi*eW, YÅr = (Y(ct"4))(W), thus (z(ct)y)(4) - y(ct"4)¥

REMARK 6.2. (i) In Theorem 6.1, {Y, n} is unique up to an isomorphism (in the frame of Hilbert B-modules) if Ysatisfies minimality condition (cf. Corollary 3.4). We call {Y, n} the minimal dilation of {X, ip}. This remark and the following remarks (ii), (iii), (iv) and (v) are also extended to Corollaries 6.3 and 6.4. (ii) Suppose that for some ct, 6, 7EI-', ip(4ctn) == ip(46ny) + ip(47q)(4, nET). Then we have z(ct) = z(6) + n(7). It ris a *-algebra and a PD ip: T.M(W, W') is linear, then the representation n: r.S}I(Y) is also linear. (iii) Assume that T is endowed with a topology and for any w, zEW, 4, qEL {(ip(4ctny)w)(z)} converges to (ip(46ny)w)(z) whenever {ct} converges to 6 with sup {c(ct)1ctEr} Åq oo. Then for any u, vE Y, {Åqn(ct)u, vÅr} converges to Åqn(6)u, vÅr. This in turn implies that for each yE Y, wE W, {(y(ct))(w)} converges to .(y(6))(w) because

(y(ct))(w) - Åqn(ct)Kgw, yÅr.

(iv) When r is a group, every n(6) is unitary, i.e., n(4)"n(e) = n(4)n(4)" = ly. In this case {Y, z} is called the minimal unitary dilation of {X, ip}. (v) If I-' is a topological group and for each w, zEW, the mapping (ip(¥)w)(z):r .B is (norm) continuous, then for each yE Y, n(¥)y: T. Yis continuous, hence y: T . W' is (norm) continuous. Proof of (v). For any yEY,- 16 Shigeru lToH ll z(ct)y - n(6)y lii

= 112Åqy, yÅr - Åqn(6-'ct)y, yÅr - Åqy, n(6-ict)yÅr Il.

Thus, by (iii) above {n(ct)y} converges to n(6)y in Y whenever {ct} converges to 6. Moreover, for any wEW,

li (y(ct))(w) - (y(6))(w) II = ll Åqn(ct)K,diw, yÅr - Åqn(6)K,ipw, yÅr 11

= II ÅqK,ip w, z(ct - ')y - n(6 - ')yÅr II .

ÅqÅ~ 11 K,ipw11yz(ct-i)y - z(6-')y iiy

by Paschke [25]. Since ll Kgwl12y -- 11ÅqK,ipw, KgwÅrII - IK (K3 w) (Åí)) (w) ll - II (K di (e, s) w) (w) 11

- 11 (Åë(6) w) (w) 11

Å~Åq 11 ip(E) 11w,w, 11vv 11ftr,

we obtain

11 (y(ct))(w) - (y(6))(w) il

ÅqÅ~ 11 ip (6) ll ilti' ?w, ll w ll w 11 z(ct - ')y - n(6 - ')y ll y

and

lly(ct) - y(/il) 11 Jv, xÅq ll q5(e) 11 lti'?w, lln(ct-')y - z(6-')y11y.

Therefore, if {ct} converges to 6, then {y(ct)} converges to y(6) in W'. The following two results are montioned in [14], but we also describe them by the method employed in this paper for the sake of convenience.

CoRoLLARy 6.3. Let ip: T-År S}((X) be PD and satisfy BC, and Ybe the RK Hilbert B-module of the kernel Kip induced by ip. Then there exist a representation n:T -År QI(Y) of T on Yand mappings rEQI(X, Y) and qEQI(Y, X) having the following propertles : (i) ip(4) = qn(e)r(4Er) and the linear span of the set {z(4)rx: xGX, 4eI'} is dense in Y; (ii) for each yEY, ct, CEr, (n(ct)y)(4) =:y(ct*4). If ip(e)= 1., then r becomes an isometric embedding, and if we identify X with r(X), then q is a projection in S}I(Y).

Proof. Let r= K,ip and q= r" = J,ip (cf. the proof of Corollary 4.4), then it is not diflicult to observe that (i) and (ii) hold for these r and q by Theorem 6.1. If ip(e) == lx, Reproducing Kernels in Modules 17 then we can verify that r is an isometric embedding and, identifying r(X) with X, q* = q2 ,. q.

CoRoLLARy 6.4. Let gb:r.S}I(X) be PD and satisfy BC. Let LÅë:rxT r E}l(E}I(X)) be the kernel introduced by Kip (cf. Lemma 4.5) and Ybe the RK Hilbert -- E}I(X)-module of Lip. Then there exist a representation z: T. E}{(Y) of T on Yand an element zE Y satisfying the following conditions: (i) q5(4)=Åqz(C)z, zÅr(4ET) and the set {z(4)z•a:aEE}I(X), CEjr,} spans a dense S}l(X)-submodule of Y; (ii) for any yE Y, ct, 4Er, (z(ct)y)(4) =y(ct*4)¥

Proof. Let z== K,ip. Then we obtain the above results by Corollary 4.6 and Theorem 6.1.

7. Applications Unitary dilations of any TE E}I(X) with ll TII. ÅqÅ~ 1 can be obtained by the same way as in Loynes [18] and Itoh [14] (cf. Sz.-Nagy [34], Sz.-Nagy and Foias [35]). In this section we treat "unitary" dilations of every TEM(X) with 11TllxÅqÅ~ 1. Such an operator Tis called a contraction. Under appropriate conditions, X' becomes a Hilbert B""-module and an contraction TEM(X) may be extended to a contraction Ti belonging to E}l(X')(Paschke [25], Zettl [39]), where B"* is the second dual of B. Hence, we can give unitary dilations of such T's in the frame of Hilbert B""- modules. But we consider dilations of such T's in the frame of Hilbert B-modules.

THEoREM 7.1. For any contraction TEM(X), there exist a Hilbert B-module Y, a unitary operator UEE}I(Y), an isometric embedding rGM(X, Y) and an operator qEM( Y, X') such thatAthe set {U"rx ln = O, Å} 1, Å} 2, ...,xEX} spans a dense B- submodule of Yand T"x =qU"rx(n = o,1,2,...,xEX).

Proof. Let Z be the set of integers. Define a mapping ip:Z.M(X,X') by (ip(n)x) (y) = Åqy, T"xÅr(n År O), or Åqy, xÅr (n = O), or ÅqT-"y, xÅr(n Åq O). Since ÅqTx, TxÅr Åqx Åqx, xÅr(xEX) by Paschke [25], ip is PD, i.e., for anyfEF(Z, X),

2 (ip(m - n)f(m))(f(n)) -År o m,n (cf. Loynes [18], Nagy-Foias [35]). By Theorem 6.1, there exist a Hilbert B-module Y, operators rEM(X, Y) and qEM(Y, X'), and a representation n:Z.S}I(Y) such that ip(n) == qz(n)r(nEZ) and the linear span of the set {n(n)rx:nEZ, xEX} is dense in Y As in the proof of Theorem 6.1, we have 18 Shigeru lToH Åqrx, rzÅr = ÅqK8x, KipozÅr

- (K8(o, o)z) (x)

- (Åë(O)z) (x) - Åqx, zÅr for every x,zEZ; hence r is an isometric embedding of X Xnto Y Denote U == n(1). Then U",= z(n) (nE Z) and for any n År O, xE X, 7Mx = ip (n)x : qn (n)r = qU"rx. Let {T,} be a set of operators T,eM(X)(t År- O) such that for any s,tÅr/ O, T,T, = T,+t and T. = lx. Such a set {T,} is called a 1-parameter semigroup of operators in M(X). {T,} is called weakly continuous if for each x, yEX, {ÅqT,x, yÅr} converges to ÅqT,x, yÅr as {t} converges to s. {T,} is called strongly continuous if for every xeX, {T,x} converges to T,x (in X) as {t} converges to s. A set of unitary operators {U,} c ut(X), where t varies in the real numbers R, is called a 1-parameter group of unitaries on X if for any s, tER, U,U,= U,+,, U,=Ix. Walso adopt similar definitions as above for this {U,}.

THEoREM 7.2. Let {T,} be a weakly continuous 1-parameter semigroup of contractions in M(X). Then there exist a Hilbert B-module Y, a strongly continuous 1- parameter group of unitaries {U,}, an isometric embedding rEM(X, Y) and an operator qAEM(Y, X') such that the set {U,rx: teR, xEX} spans a dense B-submodule of Yand T,x = qU,rx (t )-?År O, xEX).

Proof. Define ip:R.M(X, X') by (ip(t)x)(y) = Åqy, T,xÅr(tÅrO), or Åqy, xÅr(t= O), or ÅqT.,y, xÅr(t Åq O). Then, as in the proof of Sz.-Nagy [34] (cf. Loynes [18]), for any fEF(R, X),

Z(ip(s - t)f(s))(f(t)) !År o.

s,t

By Theorem 6.1 there exist a Hilbert B-module Y, operators rEM(X, Y) and qEM(Y, X'), and a representation z:R.E}I(Y) such that for each tER, ip(t) = qz(t)r and the set {n(t)rx:tER, xEX} spans a dense B-submodule of Y. As in the proof of Theorem 7.1År.r is an' isometric embedding. Let U,=n(t)(tER). Then for any xGX, t År! O, T,x =qU,rx. By Remark 6.2 (v), {U,} is strongly continuous.

CoRoLLARy 7.3: Let {T,} be as in Theorem 7.2. Then {T,} is in fact strongly contmuous. Proof. Let {U,},q and r be as in Theorem 7.2 corresponding to {T,}. The'n for xEX, Reproducing Kernels in Modules -, 19

ll Ttx "'Tsx Ilx =: ll TtxAA - Tsx Ilx,

= 11qU,rx - qU,rx IIx,

ÅqÅ~ 11qlly,x, ll U,x - U,rx lly, ' where 11qlly,x, is the operator norm of q from Yinto X'. Since {U,} is strongly continuous, this implies that {T,} is strongly continuous.

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Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804