DYNAMICAL SYSTEMS ON HILBERT MODULES OVER LOCALLY C -ALGEBRAS
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1 Kragujevac Journal of Mathematics Volume 42(2) (2018), Pages DYNAMICAL SYSTMS ON HILBRT MODULS OVR LOCALLY C -ALGBRAS L. NARANJANI 1, M. HASSANI 1, AND M. AMYARI 1 Abstract. Let A be a locally C -algebra and S(A) be the family of continuous C -seminorms and let be a Hilbert A-module. We prove that every dynamical system of unitary operators on defines a dynamical system of automorphisms on the compact operators on and show that under certain conditions, the converse is true. We define a generalized derivation on and prove that if is a full Hilbert A-module and δ : is a bounded generalized derivation, then δ p : p p is a generalized derivation on the Hilbert module p over the C -algebra A p for each p S(A). 1. Introduction Locally C -algebras are generalizations of C -algebras. Instead of having a single C -norm, we have a given directed family of C -seminorms, which gives a topology. Recall that a C -seminorm on a topological -algebra A is a seminorm p such that p(ab) p(a)p(b) and p(aa ) = p(a) 2 for all a, b A. A locally C -algebra is a complete Hausdorff complex topological -algebra A, whose topology is determined by its continuous C -seminorms in the sense that the net {a γ } converges to zero if and only if the net {p(a γ )} γ converges to 0, for every continuous C -seminorm p on A. For example any C -algebra is a locally C -algebra and any closed subalgebra of a locally C -algebra is a locally C -algebra. The notion of locally C -algebra was first introduced by Inoue [6] and studied more by Phillips and Fragoulopoulou [3, 10]. See also the book of Joita [7]. Hilbert modules are essentially objects, which behave similar to Hilbert spaces by allowing the inner product to take values in a locally C -algebra rather than the field Key words and phrases. Locally C -algebra, dynamical system, generalized derivation Mathematics Subject Classification. Primary: 46L08. Secondary: 46L57, 46D60. Received: December 02, Accepted: March 04,
2 240 L. NARANJANI, M. HASSANI, AND M. AMYARI of complex numbers. They play an important role in the modern theory of operator algebras, in noncommutative geometry and in quantum groups, see [5]. The paper is organized as follows. In Section 2 we recall some facts about Hilbert module over locally C -algebras [7]. In Section 3 we extend results about dynamical system of unitary operators for Hilbert C -module from [4] in the context of Hilbert modules over locally C -algebra. In Section 4 we investigate generalized derivation on Hilbert modules over a locally C -algebra. 2. Preliminaries Let A be a locally C -algebra and S(A) be the set of all continuous C -seminorms on A. For p S(A), the quotient -algebra A p = A/N p, where N p = {a A : p(a) = 0} is a C -algebra with respect to the C -norm p induced by p (i.e. a p p = p(a) for each a A, where a p = a + N p ). The canonical map from A onto A p is denoted by π p and π p (a) = a p for all a in A. For p, q S(A) with p q, the surjective canonical map π A pq : A p A q is defined by π A pq (π A p (a)) = π A q (a) for all a A. The {A p : πpq, A p, q S(A), p q} is an inverse system of C -algebras and lim p A p is a locally C -algebra which is identical with A. Suppose that A is a locally C -algebra. A right A-module, equipped with an A-valued inner product, : A satisfying the following conditions for all x, y, a A, α, β C: (i) x, x 0, x, x = 0 if and only if x = 0; (ii) x, ya = x, y a; (iii) x, y = y, x ; (iv) x, αy + βz = α x, y + β x, z ; is called a pre Hilbert A-module. If is complete with respect to the topology determined by the family of seminorms {p } p S(A), where p (x) = p( x, x ), x, then is called a Hilbert module over the locally C -algebra A (Hilbert A-module). If is a right A-module equipped with an A-valued inner-product,, then for each p S(A) and for all x, y we have the Cauchy-Schwarz inequality p( x, y ) 2 p( x, x )p( y, y ). Suppose is a Hilbert A-module and p belongs to S(A). Then N p = {x : p (x) = 0} is a closed submodule of and p = is a Hilbert module over C -algebra A p via the module multiplication (x + N p )π p (a) = xa + N p and the inner product x + N p, y + N p = π p ( x, y ). The canonical map from onto p is denoted by σ p and σ p (x) = x p, p S(A). For each p, q S(A), with p q, there is a canonical surjective linear map σ pq : p q such that σ pq (x p ) = x q for all x. Then { p ; A p ; σ pq : p q, p, q S(A)} is an inverse system of Hilbert C -modules in the following sense: σ pq (x p a p ) = σ pq (x p )π pq (a p ) for all x p p, a p A p ; σ pq (x p ), σ pq (y p ) = π pq ( x p, y p ) for all x p, y p p ; σ pk = σ qk σ pq, for p q k; σ pp = I p ; N p
3 DYNAMICAL SYSTMS ON HILBRT MODULS 241 and lim p p is a Hilbert A-module with ((x p ) p )((a p ) p ) = (x p a p ) p and (x p ) p, (y p ) p = ( x p, y p ) p. Moreover,lim p p can be identified by. Let A be a locally C -algebra and, F be two Hilbert A-modules, a map T : F is said to be adjointable if there exists a map T : F such that T x, y = x, T y for all x and y F. We use L(, F) for the set of all adjointable module maps. If = F we write L(). A map T : F is called a bounded A-module map if for each p S (A) there exists K p 0 such that p F (T x) K p p (x). The space of all bounded A-module maps between and F is denoted by B(, F). It is easy to see that P (T ) = sup{p F (T x) : p (x) 1} is a seminorm on B(, F). For y and x F, θ x,y : F is defined by θ x,y (z) = x y, z for each z. We have θ x,y = θ y,x. The closed subspace of L(, F) generated by {θ x,y : y, x F} is denoted by K(, F). When = F we use K() instead of K(, ). An element in K(, F) is called a compact operator from to F and K() is a two-sided -ideal of L(). In fact L() is a locally C -algebra with respect to the topology determined by the family of C -seminorms P for each p S(A) (see [7, Theorem 2.2.6]) and K() is a locally C -subalgebra of L(). A Hilbert A-module is called full if the closed linear span { x, y : x, y } denoted by,, coincides with A. For each p S (A) and θ x,y K A (, F), we have P (θ x,y ) p F (x)p (y), since for each z, p S(A) such that p (z) 1, it follows from [7, Corollary 1.2.3] that p F (θ x,y (z)) = p F (x y, z ) p F (x)p( y, z ) p F (x)p (y)p(z) p F (x)p (y). Throughout this paper, we assume that A is a locally C -algebra and, F are two Hilbert A-modules. An adjointable operator u from to F is said to be a unitary if u u = I and uu = I F, where I and I F are identity operators on and F, respectively. Definition 2.1. Let A and B be two locally C -algebras. A morphism from A to B is a continuous linear map ϕ : A B such that ϕ(ab) = ϕ(a)ϕ(b) and ϕ(a ) = (ϕ(a)) for all a, b A. Two locally C -algebras A and B are isomorphic if there is an isomorphism (bijective morphism) from A to B. Joita in [7] characterized the unitary operators on Hilbert modules over locally C -algebras by proving the following proposition. Proposition 2.1. [7, Proposition 2.5.3] Let u : F be a linear map. Then the following statements are equivalent: (i) u is a unitary operator from to F; (ii) u is surjective and ux, ux = x, x for all x ; (iii) p F (u(x)) = p (x) for all x, p S(A) and u is a surjective module homomorphism from to F.
4 242 L. NARANJANI, M. HASSANI, AND M. AMYARI Remark 2.1. If ϕ : A A is an automorphism of locally C -algebras, then ϕ p : A p A p is a well-defined automorphism of C -algebras for each p S(A). Thus p(ϕ(a)) = (ϕ(a)) p p = ϕ p (a p ) p = a p p = p(a) for each a A and p S(A). 3. Dynamical Systems on Hilbert modules In this section we generalized the definitions of dynamical systems on Hilbert modules over locally C -algebras. Definition 3.1. Let be a Hilbert A-module and U() be the set of all unitary operators on. A mapping α : R U(), t α t is said to be a one-parameter group of unitaries if for each t, s R (i) α 0 = I; (ii) α t+s = α t α s. We say that α is a strongly continuous one-parameter group (C 0 -group) of unitaries if, in addition, limα t (x) = x in. In this case, α is called a dynamical system of unitary operators on. { The infinitesimal generator} of α is the mapping δ : D(δ), where D(δ) = α t (x) x α t (x) x x : lim exists and δ(x) = lim for each x D(δ). The t t above limit is taken in the topology on. Remark 3.1. Let A be a locally C -algebra and Aut(A) be the set of all automorphisms on A, then, similar to Definition 3.1, we can define a dynamical system of automorphisms on A. In the following theorem, we show that every dynamical system of unitary operators on Hilbert A-module, defines a dynamical system of automorphisms on locally C - algebra K(). Theorem 3.1. Let α be a dynamical system of unitary operators on and u : R Aut(K()) defined by u t (T ) = α t T α t for each T K(), then u is a dynamical system of automorphism on K(). Proof. Obviously u 0 = I and u t+s = u t u s. It is enough to show that lim u t (T ) = T for each T K(). Put S = θ x,y for some x, y. Then P (u t (S) S) = P (α t Sα t S) = P (θ αt(x),α t(y) θ x,y ) = P (θ αt(x),α t(y) θ x,y θ x,αt(y) + θ x,αt(y)) = P (θ αt(x) x,α t(y) + θ x,αt(y) y) P (θ αt(x) x,α t(y)) + P (θ x,αt(y) y) p (α t (x) x)p (α t (y)) + p (x)p (α t (y) y).
5 DYNAMICAL SYSTMS ON HILBRT MODULS 243 Since α t is a unitary operator, by Proposition 2.1 we have p (α t (y)) = p (y), so the right of above inequality tends to zero. We know that T = lim T n, where each T n is t of the form T n = k n i=1 λ n i θ x n i,y n i, where λn i C, x n i, y n i. By continuity of seminorns, lim P (u t (T ) T ) = lim P (α t T α t T ) = 0. Hence limu t (T ) = T. The converse of Theorem 3.1 is not true in general, we want to show that under some mild conditions on a dynamical system α of automorphism on K(), there is a dynamical system u of unitary operators on such that α t (T ) = u t T u t for each T K(). Theorem 3.2. Let α be a dynamical system of automorphisms on K(). If there is x such that x, x = 1 and α t (θ x,x ) = θ x,x for each t R, then there is a dynamical system u of unitary operators on such that α t (T ) = u t T u t for each T K(). Proof. For each T K(), x with x, x = 1, let us define u t : by u t (T x) = α t (T )x. Then Therefore, p (T x) = p (T x 1) = p (T x x, x ) = p (T (x x, x )) = p (T θ x,x (x)) P (T θ x,x )p (x) P (T θ x,x ) = P (θ T x,x ) p (T x)p (x) p (T x). p (T x) = P (θ T x,x ) = P (α t (θ T x,x )) (α t is an automorphism and by Remark 2.1) = P (α t (T θ x,x )) = P (α t (T )α t (θ x,x )) = P (α t (T )θ x,x ) = P (θ αt(t )x,x) = p (α t (T )x) = p (u t (T x)). Thus, p (T x) = p (u t (T x)) for each x with x, x = 1. Let y be an arbitrary element in. Then y = y 1 = y x, x = θ y,x (x). Put T 0 = θ y,x. Then T 0 K() and y = T 0 x. Now put z = αt 1 (T 0 )x and let T = αt 1 (T 0 ). Then u t (T x) = α t (T )x = α t (αt 1 (T 0 ))x = α t (α t (T 0 ))x = α t α t (T 0 )x = α 0 (T 0 )x = T 0 x = y. Hence u t is onto. Since p (T 0 x) = p (u t (T 0 x)) or p (y) = p (u t (y)) for each y, so by Proposition 2.1, u t is unitary and u t = u 1 t. The equations α t (u t (T x)) = α t (α t (T )x) = α t (α t (T ))x = α 0 (T )x = T x and u t (α t (T )x) = T x show that (u t ) 1 (T x) = α t (T )x. Let z, y, then there exist T 0, S 0 K() such that z = T 0 x and y = S 0 x. Hence u s u t (z), y = u s u t (T 0 x), S 0 x = u t (T 0 x), (u s ) (S 0 x) = α t (T 0 )x, α s (S 0 )x = (α s (S 0 )) α t (T 0 )x, x = α s (S 0)α t (T 0 )x, x = α s (S 0α t+s (T 0 ))x, x = u s (S 0α t+s (T 0 )x), x = S 0α t+s (T 0 )x, (u s ) θ x,x (x) (x = x1 = x x, x = θ x,x (x))
6 244 L. NARANJANI, M. HASSANI, AND M. AMYARI = S 0α t+s (T 0 )x, α s (θ x,x )x = S 0α t+s (T 0 )x, x = u t+s (T 0 x), S 0 (x) = u t+s (z), y, whence u t+s = u t u s. Also u 0 (y) = u 0 (S 0 x) = α 0 (S 0 )x = S 0 x = y, so u 0 = I. Hence p ((u t )y y) = p ((u t )(T x) T x) = p ((α t (T ) T )x) p(α t (T ) T )p (x). Therefore, limu t (y) = y, so u is a dynamical system of unitary on. Also u t T u t (z) = u t T u t (T 0 x) = u t T (α t ) 1 (T 0 x) = α t (T αt 1 (T 0 ))x = α t (T )(α t αt 1 (T 0 )x) = α t (T )T 0 x = α t (T )z so α t (T ) = u t T u t. Theorem 3.3. Let α be an automorphism on K() such that α(θ x,x ) = θ y,y, where x and y, y = 1. Then, there is a unitary operator u in K() such that α(t ) = ut u for each T K(). Proof. For each T K() we define u(t x) = α(t )y. Then, by the some reasoning as in the proof of Theorem 3.2, we have p (u(t x)) = p (α(t )y) = p(θ α(t )y,y ) = p(α(t )θ y,y ) = p(α(t )α(θ x,x )) = p(α(t θ x,x ) = p(α(θ T x,x )) = p(θ T x,x ) = p (T x). Also, u is onto since for each z there exists T 0 K() such that z = T 0 x. One can see u(α 1 (T 0 )y) = α(α 1 T 0 )y = T 0 y = z. So u is well-defined, onto and p (u(t x)) = p (T x). Hence, by Proposition 2.1, u is a unitary operator. Let S K() and x be arbitrary, then ut u (Sx) = ut α 1 (S)x = α(t α 1 (S))x = α(t )Sx, which implies that ut u = α(t ). 4. Generalized derivations on Hilbert modules Let A be an algebra, a linear mapping d : D(d) A A, where D(d) is a dense subalgebra of A is called a derivation if d(ab) = d(a)b + ad(b) for each a, b D(d). We introduce the notion of a generalized derivation on Hilbert modules over locally C -algebras. This definition is similar to that of a generalized derivation on Hilbert C -modules introduced in [1]. Definition 4.1. Let be a full Hilbert A-module. A linear map δ : D(δ), where D(δ) is a dense subspace of, is called a generalized derivation if there exists a mapping d : D(d) A A, where D(d) is a dense subalgebra of A such that D(δ) is an algebraic left D(d)-module and δ(xa) = δ(x)a + xd(a) for each x D(δ) and a D(d). Recall that if is a full Hilbert A-module and a A such that xa = 0 for each x, then a = 0 (see [8, Remark 2.1]).
7 DYNAMICAL SYSTMS ON HILBRT MODULS 245 Proposition 4.1. Let A be a locally C -algebra and be a full Hilbert A module and δ : be a bounded generalized derivation. Then δ p : p p defined by δ p (x+n p ) = δ(x)+n p is a generalized derivation for each p S(A). Conversely, if δ p is a generalized derivation for each p S(A) then there is a generalized derivation on. Proof. Let δ : be a bounded generalized derivation. There then exists a mapping d : D(d) A A such that δ(xa) = δ(x)a + xd(a) for all x and a D(d). For any a, b A and x we have δ(xab) = δ(x)(ab) + xd(ab). Also, δ(xab) = δ((xa)b) = δ(xa)b + (xa)d(b) = (δ(x)a + xd(a))b + (xa)d(b) = δ(x)(ab) + xd(a)b + xad(b). So xd(ab) = xd(a)b + xad(b) or x(d(ab) (d(a)b + ad(b))) = 0 for all x. Since is full, we have d(ab) = ad(b) + d(a)b. It means that d is a derivation. Similarly, we can show that d is linear. For each p S(A) consider the mapping d p : D(d p ) A p A p, defined by d p (a + N p ) = d(a) + N p. We show that d p is well-defined. Indeed, if a N p, 4 then by [2] there exist elements b 1, b 2, b 3, b 4 N p such that a = i k b 2 k and p(b k ) = 0 for k = 1, 2, 3, 4 and ( ( 4 )) p(d(a)) = p d i k b 2 k ( 4 ) = p i k d(b 2 k) ( 4 ) = p i k (b k d(b k ) + d(b k )b k ) 4 p(b k )p(d(b k )) + p(d(b k ))p(b k ) = 0. Therefore a N p implies that p(d(a)) = 0. Now, if a + N p = a + N p, then a a N p so p(d(a a )) = 0 thus d(a) + N p = d(a ) + N p. It means that (d(a)) p = (d(a )) p. So d p is well defined. Obviously, the mapping d p is a derivation. Also, δ p (x p a p ) = δ p ((xa) p ) = δ(xa) + N p = (δ(x)a + xd(a)) + N p = (δ(x)a + N p ) + (xd(a) + N p ) = δ p (x p )a p + x p d p (a p ), hence δ p is a generalized derivation. Now suppose that δ p is a generalized derivation for each p S(A), then there exists a mapping d p : A p A p such that δ p is d p derivation.
8 246 L. NARANJANI, M. HASSANI, AND M. AMYARI Now, if we define δ : lim p p lim p p by δ((a p ) p ) = (δ p (a p )) p and d : lim p A p lim p A p by d((a P ) P ) = (d p (a P )) p, then δ is d generalized derivation. Indeed, which is stated. δ(x)a + xd(a) = (δ p (x p )) p (a p ) p + (x p ) p (d p (a p )) p = (δ p (x p )a p ) p (x p d p (a p )) p = (δ p (x p )a p + x p d p (a p )) p = (δ p (x p a p )) p = δ(xa), Proposition 4.2. Suppose that is a full Hilbert A-module, α is a dynamical system of unitaries on and δ is the infinitesimal generator of α such that D(δ) is a dense subspace of. Then there exists a derivation d : D(d) A A such that δ(xa) = δ(x)a + xd(a) for all a D(d), x D(δ). Proof. Since α is a dynamical system of unitaries on, the mapping α t : is a unitary for each t R. It follows from [8, Corollary 3.6] there is an isomorphism of locally C -algebras α t : A A such that α t( x, y ) = α t (x), α t (y) and α t (xa) α t (x)α t(a), α t (xa) α t (x)α t(a) =α t( xa, xa ) α t( xa, x )α t(a) α t(a )α t( x, xa ) + α t(a )α t( x, x )α t(a) = 0. Whence α t (xa) = α t (x)α t(a). In addition, p (xα t(a) xa) = p (xα t(a) α t (x)α t(a) + α t (x)α t(a) xa) p (xα t(a) α t (x)α t(a) + p (α t (x)α t(a) xa) p(α t(a))p (α t (x) x) + p (α t (x)α t(a) xa). Hence limxα t(a) xa = 0 for each x. Thus, limα t(a) = a for each a A. Therefore α : R Aut(A) is a dynamical system of automorphisms on A. The rest of proof is similar to [1, Theorem 4.3] and we remove it. Acknowledgements. The authors would like to express their sincere thanks to the referees for their helpful comments. References [1] Gh. Abbaspour Tabadkan, M. S. Moslehian and A. Niknam, Dynamical systems on Hilbert C -modules, Bull. Iranian Math. Soc. 31(1) (2005), [2] R. Becker, Derivations on LMC -algebras, Math. Nachr. 155 (1992), [3] M. Fragoulopoulou, Topology Algebras with Invilution, North Holland, Amsterdam, [4] M. Hassani, A. Niknam, On C 0 -group of linear operator, J. Sci. Islam. Repub. Iran 15(2) (2004), [5] J. M. Gracia-Bondia and J. C. Varilly and H. Figueroa, lements of non-commutative Geometry, Birkhauser, 2000.
9 DYNAMICAL SYSTMS ON HILBRT MODULS 247 [6] A. Inoue, Locally C -algebras, Mem. Fac. Sci. Kyushu Univ. A 25 (1972), [7] M. Joita, Hilbert Modules over Locally C -Algebras, University of Bucharest Press, [8] M. Joita, A note about full Hilbert modules over Frechet locally C -algebras, Novi Sad J. Math. 37(1) (2007), [9] M. S. Moslehian, On full Hilbert C -modules, Bull. Malays. Math. Sci. Soc. 24(1) (2001), [10] N. C. Phillips, Inverse limits of C -algebras, J. Operator Theory 19(1) (1988), mail address: lnaranjani@yahoo.com 1 Department of mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran -mail address: hassani@mshdiau.ac.ir -mail address: amyari@mshdiau.ac.ir
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