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.