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Theorem 2.2. [9, Lemma 1] If a ∈ Λ then the mapping fa : S → sp(Sa), defined on C(Λ), is a bounded linear functional and ||fa|| = τ(a). Theorem 2.3. [9, Theorem 1] Each bounded linear functional on C(Λ) is of the form fa for some a ∈ τ(Λ). The correspondence a ↔ fa is an isometric isomorphism between τ(Λ) and C(Λ)∗. Also, τ(Λ) is a Banach algebra. For more details, we refer to [8, 9, 1, 2, 5].
3. Results ∗ ∗ Let L be a Hilbert H -module over a proper H -algebra Λ and let B1(L) is the unit ball in L. We construct a topology on W ′ such that the unit ball in L′ is compact with respect to this topology. Define Λ′-weak∗ topology on L′ with the base
′ ′ ′ ′ ′ L 1 n 1 n = w ∈ L | sp(Sj(w (xj) − w (xj))) < δ, j = 1, .., n , w0,x ,..,x ,S ,..,S ,δ 0 ′ ′ for w0 ∈ L , xj ∈ L, Sj ∈ C(Λ), and δ > 0. SHORT TITLE 3
The main result of this paper, the compactness of the unit ball, will be proven in Theorem 3.2 and its corollary. Before that, we state and prove a useful lemma. Lemma 3.1. Let Λ be a H∗-algebra and let S ∈ C(Λ). Then the operator Sa : Λ → Λ, Sa(x)= S(ax) belongs to C(Λ). Proof. From
Sa(λx)= S(aλx)= λS(ax)= λSa(x), Sa(xy)= S(axy)= S(ax)y = Sa(x)y and ||Sa(x)|| = ||S(ax)|| ≤ ||S|| · ||ax|| ≤ ||S|| · ||a|| · ||x||, it follows that Sa belongs to R(Λ). If Lan converges to S, then from
||(Lana − Sa)(x)|| = ||(Lana(x) − Sa(x)|| = ||anax − S(ax)||
= ||Lan (ax) − S(ax)|| ≤ ||Lan − S|| · ||ax||
≤ ||Lan − S|| · ||a|| · ||x||, it follows that Lana converges to Sa. Thus Sa ∈ C(Λ). Theorem 3.2. The set ′ ′ ′ K = w ∈ L | ||w (x)|| ≤ 1 for all x ∈ B1(L) is compact in Λ′-weak∗ topology.
Proof. The neighborhood B1(L) is absorbing because for each x ∈ L, x 6= 0, there exists a number γ(x) = ||x|| > 0 such that x ∈ γ(x)B1(L). For all ′ ′ ′ w ∈ K, it holds w (x) ≤ 1, x ∈ B1(L), hence |w (x)| ≤ ||x||, x ∈ L. According to Banach-Alaoglu theorem and Theorem 2.3, the set Dx = {a ∈ Λ|||a|| ≤ γ(x)} is compact in weak∗-topology on τ(Λ) given by semi- ∗ norms pS(·) = |sp(S(·))|, S ∈ C(Λ). Let τ1 be the product weak -topology on P = Qx Dx, the cartesian product of all Dx, one for each x ∈ L. Since ∗ each Dx is weak -compact, it follows, from Tychonoff’s theorem, that P is τ1-compact. From definition of P we have P = {f : L → τ(Λ)| ||f(x)|| ≤ ||x|| for all x ∈ L}. The set P can contain Λ-nonlinear functionals. It is clear that K ⊂ L′ ∩ P . It follows that K inherits two topologies: one from L′ (its Λ′-weak∗ topology, to which the conclusion of the theorem refers) and the other, τ1, from P . We will see that these two topologies coincide on K, and that K is a closed subset of P . ′ ∗ We now prove that topologies τ1 and Λ -weak coincide on K. Fix some ′ w0 ∈ K. Then ′ ′ ′ ′ W = { w ∈ L | sp(Sj(w (xj))) − sp(Sj(w (xj))) < δ, 1 0 x1,x2, ..., xn ∈ L, S1, ..., Sn ∈ C(Λ)} 4 F. AUTHOR and
W2 = {f ∈ P | |sp(Sj(f(xj))) − sp(Sj(Λ0(xj)))| < δ,
x1,x2, ..., xn ∈ L, S1, ..., Sn ∈ C(Λ)} ′ ∗ ′ are local bases in (L , Λ-weak ) and (P,τ1), respectively. From K ⊂ L ∩ P we have K ∩ W1 = K ∩ W2, so topologies coincide on K. ′ Suppose w0 is in the τ1-closure of K. If S is from C(Λ), then, from Lemma 3.1, operator S(a·): Λ → Λ belongs to C(Λ) for all a ∈ Λ. For any a, b ∈ Λ, S ∈ C(Λ), x,y ∈ L, ε > 0, there is Λ-linear w′ ∈ K from ′ ′ ′ ′ Vxa+yb,w0,S,ε ∩ Vx,w0,S(a·),ε ∩ Vy,w0,S(b·),ε (τ1-neighborhood of w0). Therefore, it holds sp(S[a(w′(x) − w′ (x))]) < ε, sp(S[b(w′(y) − w′ (y))]) < ε, 0 0 sp(S[(w′(xa + yb) − w′ (xa + yb))]) < ε. 0 We have ′ ′ ′ |sp(S(w0(xa + yb) − w0(x)a − w0(y)b))| ′ ′ ′ ′ = |sp (S[(w0(xa + yb) − w (xa + yb)) − (w0(x)a − w (x)a) ′ ′ − (w0(y)b − w (y)b)] )| ′ ′ ′ ′ = |sp( S(w0(xa + yb) − w (xa + yb)) − S[(w0(x) − w (x))a] ′ ′ − S[(w0(y) − w (y))b] )| ≤ sp(S(w′ (xa + yb) − w′(xa + yb))) + sp(S[(w′ (x) − w′(x))a]) 0 0 + sp(S[(w′ (y) − w′(y))b]) 0 = sp(S(w′ (xa + yb) − w′(xa + yb))) + sp(aS[(w′ (x) − w′(x))]) 0 0 + sp(bS[(w′ (y) − w′(y))]) 0 ≤ ε + ε + ε = 3ε. Since ε> 0 was arbitrary, we see that ′ ′ ′ sp(S(w0(xa + yb))) = sp(S(w0(x)a)+ S(w0(y)b)) for all S ∈ C(Λ), i.e. ′ ′ ′ sp(S(w0(xa + yb) − w0(x)a − w0(y)b)) = 0
′ ′ ′ for all S ∈ C(Λ). For S = Lw0(xa+yb)−w0(x)a−w0(y)b) we have ′ ′ ′ ∗ ′ ′ ′ sp((w0(xa + yb) − w0(x)a − w0(y)b)) (w0(xa + yb) − w0(x)a − w0(y)b)) = 0. ′ ′ ′ ′ Hence w0(xa + yb) = w0(x)a + w0(y)b for all x,y ∈ L, a, b ∈ Λ, so w0 is Λ-linear. ′ ′ Let x ∈ B1(L). For arbitrary ε> 0 and S ∈ CΛ there is w ∈ B1(L ) such ′ ′ that |sp(S(w0(x))) − sp(S(w (x)))| < ε. Hence ′ ′ ′ |sp(S(w0(x)))| < |sp(S(w (x)))| + ε ≤ ||S|| · ||w (x)|| + ε ≤ ||S|| + ε. SHORT TITLE 5
Next, from Theorem 2.2 we have
′ ′ ′ ′ ||w0(x)|| = τ(w0(x)) = ||fw0(x)|| = sup |sp(S(w0(x)))| S∈C(Λ), ||S||=1 ≤ sup ||S|| + ε ≤ 1+ ε S∈C(Λ), ||S||=1 ′ for arbitrary ε> 0. Thus ||w0(x)|| ≤ 1. ′ ′ ′ We have proven that w0 ∈ B1(L ), and that B1(L ) is a closed subset of P . ′ ′ ∗ Since P is τ1-compact, B1(L ) is a closed subset of P , and τ1 and Λ -weak ′ ′ ′ ∗ topology coincide on B1(L ), we have that B1(L ) is Λ -weak compact. ∗ Corollary 3.3. The unit ball B1(L) in L is compact in Λ-weak topology with the base
Wx0,y1,...,yn,S1,...,Sn,δ = {x ∈ L| |sp(Sj([x,yj] − [x0,yj]))| < δ, j = 1, ..., n} , for x0,yj ∈ L, Sj ∈ C(Λ),δ> 0. Proof. For each w ∈ L′ there is y ∈ L such that w′(x) = [y,x] for all x ∈ L (Theorem 2.1), so from Theorem 3.2 it follows that the unit ball B1(L) in L is compact in given topology. Acknowledgement. The author was supported in part by the Ministry of education and science, Republic of Serbia, Grant 174034. References [1] W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), 364–386. ∗ [2] D. Baki´cand B. Guljaˇs, Operators on Hilbert H -modules, J. Operator theory 46 (2001), 123–137. [3] M. Cabrera, J. Martinez and A. Rodr´ıguez, Hilbert modules revisited: Orthonormal bases and Hilbert-Schmidt operators, Glasgow Math. J 37 (1995), 45–54. [4] G. R. Giellis, A characterization of Hilbert modules, Proc. Amer. Math. Soc. 36 (1972), 440–442. [5] D. Iliˇsevi´c, On redundance of one of the axioms of generalized normed space, Glasnik Matematiˇcki 37 (2002), 135–141. [6] L. Moln´ar, Modular bases in a Hilbert A-module, Czechoslovak Math. J. 42 (1992), 649–656. ∗ [7] W. L. Paschke, Inner product modules over B -algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. [8] P. P. Saworotnow, A generalized Hilbert space, Duke Math. J. 35 (1968), 191–197. ∗ [9] P. P. Saworotnow, Trace class and centralizers of an H -algebra, Proceedings of the American Mathematical Society 26, no. 1 (Sep., 1970), 101–104. [10] J. F. Smith, The structure of Hilbert modules, J. London Math. Soc. 8 (1975), 741– 749.
(Zlatko Lazovi´c) Faculty of Mathematics, University of Belgrade, 11 000 Belgrade, Serbia E-mail address: [email protected]