Hilbert modules over locally C -algebras ∗ Yu. I. Zhuraev, F. Sharipov ∗ 1 0 0 2 Abstract n a In the present paper the notion of a Hilbert module over a locally C∗-algebra is dis- J cussed and some results are obtained on this matter. In particular, we give a detailed 4 proof of the known result that the set of adjointable endomorphisms of such modules is 2 itself a locally C∗-algebra. ] A Introduction O . h t General theory of Hilbert modules over an arbitrary C∗-algebra was constructed in the a m papers of W. Paschke [20] and M. Rieffel [24] as a natural generalization of the Hilbert spaces [ theory. This generalization arises if a C∗-algebra takes place of the field of scalars C. Theory of Hilbert modules is applied to many fields of mathematics, in particular, to theory of C∗- 3 v algebras [25, 21], to theory of vector bundles [3], to theory of index of elliptic operators [18, 26], 3 to K-theory [19, 9], to KK-theory of G. Kasparov [10], to theory of quantum groups and 5 0 unbounded operators [28], to some physical problems [16, 14] etc. There is also a number of 1 papers dedicated to theory of Hilbert C∗-modules proper (see, for example, [17, 6, 7, 13]). 1 0 Other topological -algebras, for example, locally C∗-algebras and some group algebras ∗ 0 can be met in applications together with C∗-algebras. By analogy with C∗-algebras locally / h C∗-algebras are applied to relativistic quantum mechanics [2, 4]. Therefore it seems useful to t a develop theory of Hilbert modules over other topological -algebras as well. As we know the m ∗ paper [15] of A. Mallios was the first one in this direction. In that paper finitely generated : v modules equipped with inner products over some topological -algebras were considered in Xi connection with Hermitian K-theory, the standard Hilbert mo∗dule l2(A) over a locally C∗- r algebra A was introduced and the index theory for elliptic operators over a locally C∗-algebra a was constructed. Inthepresent paperthenotionofaHilbertmoduleover alocallyC∗-algebraisdiscussed and some results are obtained on this matter. In particular, it is proved that the set of adjointable endomorphisms of such modules is itself a locally C∗-algebra. After the first version of this paper appeared we were informed that our main result was already known to specialists [22, 23, 27], but we think that our detailed proof may still be of interest. ∗ The work is partially supported by the INTAS grant 96-1099. 1 1 Locally C -algebras ∗ We start with some information from [8, 1, 5] on locally C∗-algebras. Definition 1.1. A complex algebra A is called a LMC-algebra if it is a separable lo- cally convex space with respect to some family of seminorms P satisfying the following α α∈∆ { } condition: (1) P (ab) P (a)P (b) for all a,b A. α α α ≤ ∈ An involutive LMC-algebra A is called a LMC -algebra if the following condition holds: ∗ (2) P (a∗) = P (a) for all a A and α ∆. α α ∈ ∈ A complete LMC -algebra A is called a locally C∗-algebra (LC∗-algebra) if ∗ (3) P (a∗a) = (P (a))2 for all a A and α ∆. α α ∈ ∈ A family of C∗-seminorms is the family of seminorms P satisfying condition (3) of α α∈∆ { } Definition 1.1. Let us give some examples of locally C∗-algebras [8, 1, 5]. Example 1.1. Any C∗-algebra is a LC∗-algebra. Example 1.2. A closed -subalgebra of a LC∗-algebra is a LC∗-algebra. ∗ Example 1.3. Let M be a completely regular k-space ([12], p. 300) and C(M) an algebra of all continuous complex-valued functions on M. For each compact space K M put ⊂ q (f) := sup f(x) , f C(M). K | | ∈ x∈K Then the function q is a C∗-seminorm on C(M) and with respect to the family of these K seminorms C(M) is a LC∗-algebra. Example 1.4. Let Λ be a directed set of indices, H a family of Hilbert spaces such λ λ∈Λ { } that H H and λ µ ⊆ ( , ) = ( , ) · · λ · · µ|Hλ for λ µ. Here ( , ) is an inner product on H , λ Λ. Consider a locally convex space λ λ ≤ · · ∈ H := lim H = H . −→ λ λ λ λ [ The space H equipped with the topology of the inductive limit is called a locally Hilbert space. We denote by L(H) the set of all linear continuous operators T : H H such that → T = lim T , T B(H ). −→ λ λ λ ∈ λ Here B(H ) is the space of all linear bounded operators on H and (T ) is the inductive λ λ λ λ∈Λ family of the operators T B(H ). It is clear that L(H) is an algebra. Furthemore, if (T ) λ λ λ λ∈Λ ∈ is an inductive family of linear bounded operators on H ,λ Λ, then the family of adjoint λ ∈ operators (T∗) is inductive too. The map λ λ∈Λ : L(H) L(H), T T∗ = lim T∗ ∗ → 7→ −→ λ λ defines an involution on L(H). If is the operator norm on B(H ) then the function λ λ ||·|| q (T) := T , T L(H) λ λ λ || || ∈ 2 is a C∗-seminorm on L(H) for each λ Λ and L(H) is a LC∗-algebra with respect to the family ∈ of seminorms q . λ λ∈Λ { } Theorem 1.1. ([8], Theorem 5.1). Any LC∗-algebra is isomorphic to a closed -subalgebra ∗ of L(H) for some locally Hilbert space H. Let A be a LC∗-algebra with respect to a family of C∗-seminorms P . We denote by α α∈∆ { } I the kernel of the seminorm P , i.e. the set of elements a A such that P (a) = 0. It is clear α α α ∈ that I is a closed -ideal in A. Therefore the quotient space A = A/I is a normed -algebra α α α ∗ ∗ with respect to the norm a := P (a), a = a+I A . α α α α α || || ∈ It follows from Theorem 2.4 of [1] that the algebra A is complete, i.e. it is a C∗-algebra. α By e we will denote the identity element of A. Clearly P (e) = 1 for any non-zero seminorm α P . α If A is an algebra without the identity element, then by A+ we denote its unitalization. By Theorem 2.3 of [8], any seminorm P can be extended up to a C∗-seminorm P+ on A+ and A+ α α is a locally C∗-algebra with respect to the family of seminorms P+. α The spectrum of an element a of a unital LC∗-algebra A is the set Sp(a) = Sp (a) of A complex numbers z such that a z 1 is not invertible. If A is a non-unital algebra, then the − · spectrum of an element a A is its spectrum in the LC∗-algebra A+. It follows from Corollary ∈ 2.1 of [8] that Sp (a) = Sp (a ), a = a+I (1.1) A Aα α α α α∈∆ [ for each a A. An element a A is called positive (and is written a 0) if it is Hermitian, ∈ ∈ ≥ i.e. a = a∗ and one of the following equivalent (see [8], Proposition 2.1) conditions is true: (1) Sp(a) [0, ); ⊂ ∞ (2) a = b∗b for some b A; ∈ (3) a = h2 for some Hermitian h A. ∈ Besides, thesetofpositiveelementsP+(A)isaclosedconvexconeinAandP+(A) ( P+(A)) = − 0 . { } T Remark 1.1. If a A is a positive element then there exists a unique positive element ∈ h A satisfying the condition (3). This element is called a square root of a and is denoted by 1∈ a2 = √a. For elements a,b A the inequality a b (or b a) means that a b 0. ∈ ≥ ≤ − ≥ Lemma 1.1. ([8]). (a) If a,b P+(A) and a b, then P (a) P (b) for all α ∆. α α ∈ ≤ ≤ ∈ (b) If e A,b A and b e then the element b is invertible and b−1 e. ∈ ∈ ≥ ≤ (c) If elements a,b A are invertible and 0 a b then a−1 b−1. ∈ ≤ ≤ ≥ (d) If a,b,c A and a b then c∗ac c∗bc. ∈ ≤ ≤ Lemma 1.2. Let A be a unital locally C∗-algebra, a P+(A) and t be a positive number. ∈ Then for any α ∆ the following relations hold: ∈ (a) P ((e+ta)−1) 1; α ≤ (b) P (a(e+a)−1) 1; α ≤ (c) P (e a) 1 if P (a) 1. α α − ≤ ≤ 3 Proof. Since e+ta e, we obtain from (b) and (a) of Lemma 1.1 that the element e+ta ≥ is invertible and P ((e+ta)−1) P (e) = 1. α α ≤ Using e+a a and statement (d) of Lemma 1.1 for c = (e+a)−1 we obtain ≥ q a(e+a)−1 e. ≤ This yields (b). We will prove (c). Since A = A/I is a C∗-algebra with the identity element e = e+I α α α α and a = P (a) 1, we obtain that α α || || ≤ P (e a) = e a+I = e a 1. α α α α − || − || || − || ≤ The lemma is proved. We denote by As the set of all elements a A such that ∈ a s := supP (a) < . α || || ∞ λ∈∆ Then As is a -subalgebra of A and s is a norm for As. Moreover, the following theorem is ∗ ||·|| true. Theorem 1.2. ([1], Theorem 2.3). The algebra As is dense in A and is a C∗-algebra with respect to the norm s. ||·|| An approximate identity of a locally C∗-algebra A is any increasing net u of positive λ λ∈Λ { } elements such that 1) P (u ) 1 for all α ∆,λ Λ; α λ ≤ ∈ ∈ 2) lim(a au ) = lim(a u a) = 0 for any a A. λ λ λ − λ − ∈ Any locally C∗-algebra (and its closed ideal) has an approximate identity (see [8], Theorem 2.6). 2 Hilbert modules over LC -algebras ∗ Let A be a LC∗-algebra with respect to the family of C∗-seminorms P . Below we will α α∈△ { } assume that the algebra A has the unit e. Considering A+ instead of A, one can easily extend all results for the case of non-unital algebras. Definition 2.1. Let X be a right A-module. An A-valued inner product on X is a map < .,. >: X X A satisfying the following conditions: × → (1) < x,x > 0 for all x X; ≥ ∈ (2) < x,x >= 0 if and only if x = 0; (3) < x +x ,y >=< x ,y > + < x ,y > for all x ,x ,y X; 1 2 1 2 1 2 ∈ (4) < xa,y >=< x,y > a for all x,y X,a A; ∈ ∈ (5) < x,y >∗=< y,x > for all x,y X. ∈ ArightA-moduleequipped with anA-valuedinner product is called apre-Hilbert A-module. 4 Lemma 2.1. Let X be a pre-Hilbert A-module. Then for each α and for all x,y X ∈ △ ∈ the following Cauchy-Bunyakovskii inequality holds: P (< x,y >)2 P (< x,x >)P (< y,y >). (2.1) α α α ≤ Proof. Suppose x,y X, b A. Let us consider the expression ∈ ∈ < x+yb,x+yb >=< x,x > +b∗ < x,y > + < y,x > b+b∗ < y,y > b 0. (2.2) ≥ Assuming P (< y,y >) = 0, we put α 6 < x,y > b = −P (< y,y >) α in (2.2). It now follows that < y,x >< x,y > < y,x >< x,y > < y,x >< y,y >< x,y > < x,x > + 0, − P (< y,y >) − P (< y,y >) P (< y,y >)2 ≥ α α α whence, using item (a) of Lemma 1.1, we obtain 2 < y,x >< x,y > < y,x >< y,y >< x,y > P P < x,x > + . α α Pα(< y,y >) ! ≤ Pα(< y,y >)2 ! Therefore, 2P (< x,y >)2 P (< y,x >)P (< y,y >)P (< x,y >) α α α α P (< x,x >)+ α P (< y,y >) ≤ P (< y,y >)2 α α P (< x,y >)2 α = P (< x,x >)+ . α P (< y,y >) α This implies inequality (2.1). If P (< x,x >) = 0, then it is true by the same reason. α 6 Let now P (< x,x >) = P (< y,y >) = 0. Putting b = < x,y > in (2.2), we get α α − < x,x > + < y,x >< y,y >< x,y > 2 < y,x >< x,y >, ≥ whence 2P (< x,y >)2 = 2P (< y,x >< x,y >) P (< x,x >)+ α α α ≤ +P (< y,x >)P (< y,y >)P (< x,y >) = 0. α α α Thus P (< x,y >) = 0 and inequality (2.1) is true in this case too. The lemma is proved. α Lemma 2.2. Let X be a pre-Hilbert A-module with respect to an inner product < .,. >. Put P¯ (x) := P (< x,x >)21 (2.3) α α for any α ∆. Then the function P¯ is a seminorm on X and the following conditions hold: α ∈ 5 (1) P¯ (xa) P¯ (x)P (a) for all x X, a A; α α α (2) if P¯ (x)≤= 0 for all α ∆, then∈x = 0;∈ α ∈ (3) P¯ (x) = sup P (< x,y >) for all x X, α ∆. α P¯α(y)≤1 α ∈ ∈ Proof. Under the axioms of seminorm we have ¯ 1 P (x) = P (< x,x >)2 0 α α ≥ and P¯ (λx) = P (< λx,λx >) = P (λ¯ < x,x > λ) α α α q q = P ( λ 2 < x,x >) = λ 2P (< x,x >) = λ P¯ (x) α α α | | | | | | for all x X and λ qC. q ∈ ∈ Using the Cauchy-Bunyakovskii inequality one has P¯ (x+y) = P (< x+y,x+y >) α α q = P (< x,x > + < y,x > + < x,y > + < y,y >) α q P (< x,x >)+2P (< x,y >)+P (< y,y >) α α α ≤ q P¯ (x)2 +2P¯ (x)P¯ (y)+P¯ (y)2 = P¯ (x)+P¯ (y) α α α α α α ≤ for all x,y X. Thus Pq¯ is a seminorm on X. α ∈ Let us prove (1). For all a A and x X we derive ∈ ∈ P¯ (xa) = P (< xa,xa >) = P (a∗ < x,x > a) P (a∗)P (< x,x >)P (a) = P (a)P¯ (x). α α α α α α α α ≤ q q q Suppose x X and P¯ (x) = 0 for all α . Then P (< x,x >) = 0 for all α . α α ∈ ∈ △ ∈ △ Therefore < x,x >= 0 and, consequently, x = 0. Thus (2) is true. Equality (3) follows easy from the Cauchy-Bunyakovskii inequality. The lemma is proved. Lemma 2.2 implies that X is a separable locally convex space with respect to the family of seminorms P¯ : α . α { ∈ △} Definition 2.2. Let X bea pre-Hilbert A-module equipped with theinner product< .,. >. If X is acomplete locallyconvex spacewith respect tothefamilyofseminorms P¯ defined α α∈△ { } by (2.3), then it is called a Hilbert A-module. Example 2.1. Any closed right ideal I of a locally C∗-algebra A equipped with the inner product < a,b >= a∗b is a Hilbert A-module. Example 2.2 ([15]). Let l2(A) be the set of all sequences x = (xn)n∈N of elements from a locally C∗-algebra A such that the series ∞ x∗x i i i=1 X is convergent in A. Then l (A) is a right Hilbert A-module with respect to the pointwise 2 operations and the inner product ∞ < x,y >= x∗y . i i i=1 X 6 Let A be a locally C∗-algebra and X a right Hilbert A-module equipped with the inner product < .,. >. We will denote by Xs the set of all x X such that < x,x > As. It is ∈ ∈ verified immediately that Xs is As-module. Lemma 2.3. For all x X and α ∆ one has: a) P¯ (x(e+√< x,x >)−∈1) 1; ∈ α b) lim P¯ (x x(e+t√< x≤,x >)−1) = 0. α t→+0 − Proof. a) Let x X be an arbitrary element. Then for each α ∆ we have by statement ∈ ∈ (b) of Lemma 1.2 that P¯ (x(e+√< x,x >)−1)2 = P (< x(e+√< x,x >)−1,x(e+√< x,x >)−1 >) α α = P ((e+√< x,x >)−1 < x,x > (e+√< x,x >)−1) α = P (√< x,x >(e+√< x,x >)−1)2 1. α ≤ b) For each x X and each positive number t we have ∈ x x(e+t√< x,x >)−1 = (x(e+t√< x,x >) x)(e+t√< x,x >)−1 − − = tx√< x,x >(e+t√< x,x >)−1. Therefore by statement (a) of Lemma 1.2 P¯ (x x(e+t√< x,x >)−1) tP¯ (x√< x,x >). α α − ≤ This implies b). The lemma is proved. Corollary 2.1. The set Xs is dense in X. Proof. Let x X be an arbitrary element and t a positive number. Then Lemma 2.3 ∈ implies that the elements (e+√< x,x >)−1, x(e+t√< x,x >)−1 belong to Xs and lim(x(e+t√< x,x >)−1) = x t→0 in X. Thus Xs is dense in X. Theorem 2.1. Xs is a Hilbert C∗-module over the C∗-algebra As. Proof. First we will show that the restriction of the inner product < .,. > from X to Xs is an As-valued inner product on Xs. Indeed, by the Cauchy-Bunyakovskii inequality (2.1) we have P (< x,y >)2 P (< x,x >) P (< y,y >) < x,x > s < y,y > s α α α ≤ · ≤ || || || || for all x,y Xs and α ∆. Therefore, < x,y > As. ∈ ∈ ∈ Let us prove completeness of Xs with respect to the norm x s = ( < x,x >) s)12 = supP¯ (x). (2.4) α || || || || α∈△ Let x be a fundamental sequence in Xs, i.e. for any ε > 0 there exists a natural number n n ε { } such that x x s < ε if m,n > n , whence m n ε || − || P¯ (x x ) x x s < ε α m n m n − ≤ || − || 7 for all α and m,n > n . This means that x is a Cauchy sequence in X and as X is ε n ∈ △ { } complete, so we conclude that the limit x = lim x n n→∞ exists in X. It follows from the inequalities P¯ (x ) P¯ (x ) P¯ (x x ), x s x s x x s α m α n α m n m n m n − ≤ − | || || −|| || |≤ || − || (cid:12) (cid:12) (cid:12) (cid:12) that the seque(cid:12)nces P¯ (x ) an(cid:12)d x s are Cauchy sequences of numbers and therefore are α n n { } {|| || } convergent. Besides, for each α ∈ △ P¯ (x) = lim P¯ (x ) lim x s < α α n n n→∞ ≤ n→∞|| || ∞ hence x Xs. For all α and n > n we have ε ∈ ∈ △ ¯ ¯ P (x x ) = lim P (x x ) ε. α n α m n − m→∞ − ≤ Thus for n > n ε x x s = supP¯ (x x ) ε. n α n || − || − ≤ α∈△ This means that the sequence x converges to the element x with respect to the topology of n { } Xs. The theorem is proved. For any α let us put ∈ △ I = a A : P (a) = 0 , J = I As, α α α α { ∈ } ∩ I¯ = x X : < x,x > I , J¯ = I¯ Xs. α α α α { ∈ ∈ } ∩ The subset I¯ is a closed A-submodule in X, J is a closed ideal of the C∗-algebra As and J¯ α α α is a closed As-submodule in Xs. Therefore the quotient X := X/I¯ is is a normed space with α α respect to the norm ¯ ¯ ¯ x+I := inf P (x+y) = P (x), x X, α α α || || y∈I¯α ∈ and the quotient Xs := Xs/J¯ is a Banach space with respect to the norm α α x+J¯ = inf x+y s, x Xs. α || || y∈J¯α|| || ∈ Lemma 2.4. Let u be an approximate identity in the ideal J of the C∗-algebra As. λ λ∈Λ α { } Then a) lim y yu s = 0 for any y J¯ . λ α λ || − || ∈ b) x+J¯ = lim x xu s for any x Xs. α λ || || λ || − || ∈ c) a+J = lim a au s for any a As. α λ || || λ || − || ∈ 8 Proof. a) Suppose y J¯ . Then for any β ∆ we have α ∈ ∈ P¯ (y yu )2 = P (< y yu , y yu >) = P ((e u ) < y,y > (e u )) β λ β λ λ β λ λ − − − − − P (< y,y > < y,y > u ) < y,y > < y,y > u s β λ λ ≤ − ≤ || − || because P (e u ) e u s 1. Since β is arbitrary, we get β λ λ − ≤ || − || ≤ ( y yu s)2 < y,y > < y,y > u s. λ λ || − || ≤ || − || Using < y,y > J we obtain lim y yu s = 0. α λ ∈ λ || − || b) Let now x Xs and take ε > 0. By definition of infimum there exists an element y J¯ α ∈ ∈ such that ε x+y s < x+J¯ + . α || || || || 2 Item a) implies that there exists λ Λ such that for λ > λ one has 0 0 ∈ ε y yu s < . λ || − || 2 Then for all λ > λ 0 x xu s = x(e u )+y(e u ) y(e u ) s (x+y)(e u ) s + y yu s λ λ λ λ λ λ || − || || − − − − || ≤ || − || || − || ε ε (x+y) s + y yu s x+J¯ + + = x+J¯ +ε. λ α α ≤ || || || − || ≤ || || 2 2 || || Therefore for λ > λ we have 0 x+J¯ x xu s < ε. α λ || ||−|| − || (cid:12) (cid:12) (cid:12) (cid:12) Thus statement b) is true. State(cid:12)ment c) follows from b)(cid:12)for X = A. The lemma is proved. Lemma 2.5. For any x Xs the equality ∈ x+J¯ = P¯ (x) α α || || holds. Proof. Let x Xs and u approximate identity of the ideal J . Then by statements λ λ∈Λ α ∈ { } b) and c) of Lemma 2.4 we have x+J¯ 2 = lim( x xu s)2 = limsupP¯ (x xu )2 = limsupP (< x xu ,x xu >) α λ β λ β λ λ || || λ || − || λ β − λ β − − = limsupP ((e u ) < x,x > (e u )) limsupP (< x,x > < x,x > u ) β λ λ β λ λ β − − ≤ λ β − = lim < x,x > < x,x > u s = < x,x > +J = P (< x,x >) = (P¯ (x))2. λ α α α λ || − || || || The next to last equality follows from uniqueness of C∗-norm of C∗-algebra As/J (see [1]). α Thus we have proved that x+J¯ P¯ (x). α α || || ≤ 9 The inverse inequality is verified immediately: P¯ (x) = x+I¯ = inf P¯ (x+y) inf P¯ (x+y) inf x+y s = x+J¯ . α α α α α || || y∈I¯α ≤ y∈J¯α ≤ y∈J¯α|| || || || The lemma is proved. Theorem 2.2. The quotient module X = X/I¯ is a Hilbert module over the C∗-algebra α α A . α Proof. We define an action of the algebra A on X by the formula α α (x+I¯ )(a+I ) := xa+I¯ , x X, a A. α α α ∈ ∈ With respect to this action X is a right A -module. Let us define the A -valued inner product α α α in X by the following formula α < x+I¯ , y +I¯ >:=< x,y > +I , x,y X. α α α ∈ The inner product axioms are easily verified. Positive definiteness follows from equality (1.1). Further, < x+I¯ , x+I¯ > = < x,x > +I = P (< x,x >) = (P¯ (x))2 = x+I¯ 2, α α α α α α || || || || || || i.e. ¯ ¯ ¯ 1 x+I = < x+I , x+I > 2. α α α || || || || To establish the completeness of X let us consider the map α ϕ : Xs/J¯ X/I¯ = X α α α → defined by the formula ϕ(x+J¯ ) = x+I¯ , x Xs. α α ∈ It is clear that ϕ is an injective linear map. Besides, it follows from Lemma 2.5 that this map preserves the norm, ϕ(x+J¯ ) = x+I¯ = P¯ (x) = x+J¯ , x Xs. α α α α || || || || || || ∈ Consequently, the image ϕ(Xs/J¯ ) = X˜s is closed in X . Let us prove that the set X˜s is dense α α in X . α Let x X be an arbitrary element. Then Lemma 2.3 implies that for any t > 0 the element ∈ x(e+t√< x,x >)−1 belongs to Xs and limx(e+t√< x,x >)−1 = x t→0 in X. Therefore, x(e+t√< x,x >)−1 +I¯ X˜s α ∈ and lim[x(e+t√< x,x >)−1 +I¯ ] = x+I¯ α α t→0 in X . Since X˜s is closed in X , we conclude that X˜s = X . Thus the space X is complete. α α α α The theorem is proved. 10