ON BOUNDED MODULE MAPS BETWEEN HILBERT MODULES OVER LOCALLY C∗-ALGEBRAS
M. JOIT¸ A
Abstract. LetAbe a locally C∗-algebra and letE be a Hilbert A-module. We show that the algebra BA(E) of all boundedA-module maps on E is a locally m-convex algebra which is algebraically and topologically isomorphic toLM(KA(E)), the algebra of all left multipliers ofKA(E), whereKA(E) is the locallyC∗-algebra of all ”compact“
A-module maps onE. Also we show thatb(BA(E)), the algebra of all bounded elements inBA(E), is a Banach algebra which is isometrically isomorphic toBb(A)(b(E)).
1. Introduction
A locallyC∗-algebra is a complete Hausdorff complex topological∗-algebraAwhose topology is determined by its continuousC∗-seminorms in the sense that the net{ai}i converges to 0 if and only if the net{p(ai)}i converges to 0 for every continuousC∗-seminormponA.In fact a locallyC∗-algebra is an inverse limit ofC∗-algebras.
Hilbert modules over locallyC∗-algebras generalize the notion of Hilbert C∗-modules by allowing the inner product to take values in a locallyC∗-algebra. In [9], Phillips showed that many results about multipliers of a C∗-algebra are valid for multipliers of a locallyC∗-algebra. Thus, he proved thatM(A), the multiplier algebra of a locallyC∗-algebraA,is a locallyC∗-algebra in the topology of seminorm [9, Theorem 3.14]. In this note we show that any left multiplier of a locallyC∗-algebra Ais automatically continuous (Proposition3.4) andLM(A), the algebra of left multipliers ofA, is a complete locallym-convex algebra in the topology of seminorm (Theorem3.5).
Also, Phillips shows that ifE is a Hilbert module over a locallyC∗-algebraA, then the locallyC∗-algebraLA(E)
Received May 16, 2003.
2000Mathematics Subject Classification. Primary 46L08, 46L05, 46A13.
Key words and phrases. Hilbert modules over locallyC∗-algebras, bounded module maps, locallym-convex algebras.
of all adjointable maps onE is isomorphic toM(KA(E)), whereKA(E) is the locallyC∗-algebra of all ”compact“
A-module maps on E [9, Theorem 4.2]. This result is a generalization of Theorem 1 of [5] for Hilbert module over locally C∗-algebras. We show that the locally m-convex algebraBA(E) of all bounded A-module maps is isomorphic to LM(KA(E) (Theorem3.6). This result generalizes Theorem 1.5 of [6] in the context of Hilbert modules over locallyC∗-algebras. Finally we prove that ifE andF are Hilbert modules over a locallyC∗-algebra A,thenb(BA(E, F)), the set of all bounded elements inBA(E, F), is a Banach space in the normk·k∞which is isometrically isomorphic toBb(A)(b(E), b(F)), the Banach space of all boundedb(A)-module maps from b(E) to b(F) (Theorem3.7). In particular,b(BA(E)) is a Banach algebra which is isometrically isomorphic toBb(A)(b(E)) andb(LA(E)) is aC∗-algebra which is isomorphic toLb(A)(b(E)).
2. Preliminaries
IfA is a locally C∗-algebra and S(A) is the set of all continuousC∗-seminorms onA, then for each p∈S(A), Ap=A/ker(p) is a C∗-algebra in the norm induced bypandA= lim
p←Ap (see, for example, [9]). The canonical map fromAontoAp, p∈S(A) is denoted byπpand the image ofain Aunder πp byap.The connecting maps of the inverse system{Ap}p∈S(A) are denoted byπpq,q, p∈S(A), withp≥q.
Now we recall some facts about Hilbert modules over locallyC∗ -algebras from [9].
Definition 2.1. A pre-Hilbert A-module is a complex vector space E which is also a right A-module, compatible with the complex algebra structure, equipped with an A-valued inner product h·,·i : E×E → A which isC- andA-linear in its second variable and satisfies the following relations:
(i) hx, yi∗=hy, xi for everyx, y∈E;
(ii) hx, xi ≥0 for everyx∈E;
(iii) hx, xi= 0 if and only ifx= 0.
We say thatE is a Hilbert A-module ifE is complete with respect to the topology determined by the family of seminormspE(x) =p
p(hx, xi),x∈E,p∈S(A).
Given a HilbertA-moduleE, for eachp∈S(A),NpE = ker(pE) is a closed submodule of E andEp =E/NpE is a HilbertAp-module with (x+NpE)πp(a) =xa+NpEand
x+NpE, y+NpE
=πp(hx, yi).The canonical map fromE ontoEp is denoted byσpE, and the image ofxinE underσEp byxp,p∈S(A).
For eachp, q∈S(A) withp≥qthere is a canonical surjective linear mapσEpq:Ep →Eq such thatσpqE(xp) = xq, x ∈E. Then{Ep;Ap;σpqE, p≥q, p, q ∈ S(A)} is an inverse system of HilbertC∗-modules in the following sense:
• σEpq(xpap) =σEpq(xp)πpq(ap) for everyxp∈Epand for every ap∈Ap;
•
σEpq(xp), σEpq(yp)
=πpq(hxp, ypi)for everyxp, yp∈Ep;
• σEqr◦σEpq=σprE,p≥q≥r;
• σEpp=idEp; and lim
p←Ep is a HilbertA-module with ((xp)p) ((ap)p) = (xpap)pandh(xp)p,(yp)pi= (hxp, ypi)p. Moreover, lim
p←Ep can be identified withE.
We recall that an elementainArespectivelyxin E is bounded if
kak∞= sup{p(a);p∈S(A)}<∞
respectively
kxk∞= sup{pE(x);p∈S(A)}<∞
The set of all bounded elements inArespectively inEwill be denoted byb(A) respectivelyb(E). We know that b(A) is a C∗-algebra in theC∗-normk·k∞, andb(E) is a Hilbertb(A)-module.
3. Bounded modules maps
LetAbe a locallyC∗-algebra and letE andF be two HilbertA-modules. AnA-module mapT :E→F is said to be bounded if for eachp∈S(A), there is Kp >0 such that pF(T x)≤KppE(x) for all x∈E. The set of all boundedA-module maps fromE toF is denoted byBA(E, F) and we writeBA(E) forBA(E, E).
Clearly, for eachp∈S(A), the mappedefined by
ep(T) = sup{pF(T x);x∈E andpE(x)≤1}, T ∈BA(E, F) is a seminorm onBA(E, F).
Proposition 3.1. LetA be a locally C∗-algebra and letE andF be two HilbertA-modules. Then we have:
1. BA(E, F)with the topology determined by the family of seminorms{p}e p∈S(A) is a complete locally convex space.
2. BA(E)with the topology determined by the family of seminorms{p}e p∈S(A)is a complete locallym-convex algebra.
Proof. (1): Letp, q∈S(A) withp≥qand letS∈BAp(Ep, Fp). Since σpqF S σpE(x)
, σpqF S σpE(x)
= πpq
S σpE(x)
, S σEp(x)
≤ kSkpπpq
σpE(x), σpE(x)
cf. [7, 2.8]
= kSkp
σqE(x), σqE(x)
for allx∈E, wherek·kpis the norm onBAp(Ep, Fp), we can define (πpq)∗(S) :Eq →Fq by (πpq)∗(S) σEq(x)
= σpqF S σpE(x)
. It is easy to see that (πpq)∗(S) is a bounded Aq-module map from Eq to Fq. Thus we have obtained a map (πpq)∗ fromBAp(Ep, Fp) to BAq(Eq, Fq). Also it is easy to see that{BAp(Ep, Fp); (πpq)∗,p≥q, p, q∈S(A)} is an inverse system of Banach spaces.
We will show that the locally convex spacesBA(E, F) and lim
p←BAp(Ep, Fp) are isomorphic.
Let p ∈ S(A) and let T ∈ BA(E, F). Since T(NpE) ⊆ NpF there is a unique linear map Tp : Ep → Fp
such that σpF ◦T =Tp◦σEp. Moreover, Tp is a bounded Ap -module map. Thus we can define a map (πp)∗ : BA(E, F)→BAp(Ep, Fp) by (πp)∗(T) =Tp, whereσpF ◦T =Tp◦σEp. Clearly (πp)∗ is a continuous linear map and (πpq)∗◦(πp)∗ = (πq)∗ for all p, q ∈ S(A) with p ≥ q. Therefore we can define a map Φ from BA(E, F) to lim
p←BAp(Ep, Fp) by Φ(T) = (πp)∗(T)
p. It is not difficult to check that Φ is linear and kΦ(T)kp = p(T)e for all T ∈ BA(E, F). To show that Φ is surjective, let (Tp)p ∈ lim
p←BAp(Ep, Fp). Define T : E → F by T(x) = Tp σpE(x)
p. Since σpqF Tp σEp (x)
= (πpq)∗(Tp) σEq (x)
= Tq σEq (x)
for all p, q ∈ S(A) with p≥q,T is well-defined. It is not difficult to check thatT is a boundedA-module map and Φ(T) = (Tp)p. Hence Φ is surjective.
Thus we showed that the topological spaces BA(E, F) and lim
p←BAp(Ep, Fp) are isomorphic, and since limp←BAp(Ep, Fp) is complete,BA(E, F) is complete.
(2): It is not difficult to check that pe is a submultiplicative seminorm on BA(E) for all p ∈ S(A) and {Bp(Ep); (πpq)∗, p ≥ q, p, q ∈ S(A)} is an inverse system of Banach algebras. Also it is easy to check that the mapΦ frome BA(E) to lim
p←BAp(Ep) defined byΦ(Te ) = (πp)∗(T)
pis an isomorphism of topological algebras, and since lim
p←BAp(Ep) is complete, the assertion is proved.
Remark 3.2. If Ais a locally C∗-algebra andE andF are Hilbert A-modules, then the locally convex spaces BA(E, F)andlim
p←BAp(Ep, Fp)as well as the locallym-convex algebrasBA(E)andlim
p←BAp(Ep)can be identified.
A mapT fromE toF is adjointable if there is a mapT∗ fromF toE such thathT(x), yi=hx, T∗(y)ifor all xinE and for ally inF. Any adjointable map fromE intoF is a boundedA-module map (cf. [11]). The set of all adjointable maps fromE intoF is denoted byLA(E, F), and we writeLA(E) forLA(E, E). ForxinE and fory in F the mapθy,x:E →F defined byθy,x(z) = yhx, ziis adjointable. The closed subspace ofLA(E, F)
generated by {θy,x;x∈E, y∈F} is denoted by KA(E, F), and we write KA(E) for KA(E, E). It is easy to verify that (πpq)∗ LAp(Ep, Fp)
⊆ LAq(Eq, Fq) and (πpq)∗ KAp(Ep, Fp)
⊆ KAq(Eq, Fq) for all p, q ∈ S(A) with p ≥ q. Then the restriction of Φ on LA(E, F) is exactly the same map as defined in Proposition 4.7 of [9]. Therefore the restriction of Φ on LA(E, F) is an isomorphism between the locally convex spaces LA(E, F) and lim
p←LAp(Ep, Fp), and the restriction of Φ on KA(E, F) is an isomorphism between the locally convex spaces KA(E, F) and lim
p←KAp(Ep, Fp) [9, Proposition 4.7]. Also the restriction ofΦ one LA(E) is an isomorphism between the locallyC∗-algebras LA(E) and lim
p←LAp(Ep), and the restriction ofΦ one KA(E) is an isomorphism between the locallyC∗-algebras KA(E) and lim
p←KAp(Ep) [9, Theorem 4.2].
In [9, Theorem 4.2], Phillips shows that the locallyC∗-algebrasLA(E) andM(KA(E)), the multiplier algebra ofKA(E), are isomorphic. We will prove here that the locally m-convex algebrasBA(E) andLM(KA(E)), the algebra of left multipliers ofKA(E), are isomorphic.
IfAis a locallyC∗-algebra, we recall that a left multiplier ofAis a linear mapl:A→Asuch thatl(ab) =l(a)b for allaandbinA. We know that any left multiplier of aC∗-algebra is automatically continuous. We will show that this result is still valid for left multipliers of a locallyC∗-algebra. Recall that in [11], Weinder showed that the multipliers of a locallyC∗-algebra are automatically continuous.
Lemma 3.3. Let a be an element of a locallyC∗-algebra A. If 0< α <1, then there is an element uin A such thata=u|a|α, where|a|2=aa∗.
Proof. We know that for eachpinS(A), there is an elementupinAp such thatπp(a) =up|πp(a)|α. Moreover, up= lim
n πp(a) 1n+|πp(a)|2−12
|πp(a)|1−α (see, for example, [8, 1.4.6]).
To show that (up)p is a coherent sequence inAp, p ∈S(A), let p, q ∈S(A) with p≥q. Sinceπpq preserves spectral functions, we have
πpq(up) = lim
n πpq πp(a) 1
n+|πp(a)|2 −12
|πp(a)|1−α
!
= lim
n πq(a) 1
n+|πq(a)|2 −12
|πq(a)|1−α=uq.
Hence (up)p is a coherent sequence inAp, p∈S(A). LetuinAbe such thatπp(u) =up for allp∈S(A). Then, sinceπp(|a|α) =|πp(a)|αfor allp∈S(A) (see [9] or [2]), we havea=u|a|α.
Proposition 3.4. Any left multiplier of a locallyC∗-algebraAis automatically continuous.
Proof. Letl be a left multiplier ofA, let p∈S(A) anda∈ker(p). By Lemma 3.3, there isu∈A such that a=u|a|12, and then
p(l(a)) =p(l(u)|a|12)≤p(l(u))p(a)12
whence we conclude thatl(a)∈ker(p). Hence there is a unique linear maplp:Ap→Ap such thatπp◦l=lp◦πp. Moreover,lp is a left multiplier ofAp and so it is continuous (see, for example, [8, 3.12.2]). From these facts we
conclude thatlis continuous and the proposition is proved.
We consider on LM(A), the set of all left multipliers of A, the seminorm topology (that is the topology determined by that family of seminorms{p}e p∈S(A), where p(l) = sup{p(l(a)),e a∈Aand p(a)≤1}).
Theorem 3.5. LetA be a locally C∗-algebra. Then we have:
(1) LM(A)is a complete locallym-convex algebra.
(2) IfA= lim
λ∈Λ←Aλ and the canonical mapsπλ :A→Aλ are all surjective, then the locallym-convex algebras LM(A)and lim
λ∈Λ←LM(Aλ)are isomorphic.
Proof. To prove this theorem we use the same arguments as in the proof of Theorem 3.14 of [9].
(1): Letp, q ∈ S(A) with p≥q. Since πpq is surjective, there is a unique morphism πpq00 : A00p →A00q which extends πpq and πpq00(LM(Ap))⊆LM(Aq) (see, for example, [8, 3.7.7 and 3.12]). Then{LM(Ap); πpq00|LM(Ap), p≥q,p, q∈S(A)}is an inverse system of Banach algebras. It is not difficult to check that the map Ψ :LM(A)→ limp←LM(Ap) defined by Ψ(l) = (lp)p, whereπp◦l=lp◦πpfor allp∈S(A), is an isomorphism of locallym-convex algebras.
(2): Exactly as in the proof of Theorem 3.14 of [9] we show that the inverse systems {LM(Aλ)}λ∈Λ and {LM(Ap)}p∈S(A)have the same inverse limit and thus the assertion is proved.
The following theorem is a generalization of Theorem 1.5 of [6] in the context of Hilbert modules over locally C∗-algebras.
Theorem 3.6. Let A be a locally C∗-algebra and let E be a Hilbert A-module. Then the locally m-convex algebras BA(E)andLM(KA(E))are isomorphic.
Proof. Letp, q ∈S(A) with p ≥q. Since (πpq)∗ (θy,x) = θσpq(y),σpq(x) for allx, y ∈ Ep, and since the map σpq fromEp toEq is surjective, the morphism (πpq)∗fromKAp(Ep) toKAq(Eq) is surjective. Then according to Theorem3.5(2), the locallym-convex algebras LM(KA(E)) and lim
p←LM(KAp(Ep)) are isomorphic.
For each p ∈ S(A), the map Φp : BAp(Ep) → LM(K Ap(Ep)) defined by Φp(Tp)(Sp) = Tp◦Sp is an iso- metric isomorphism of Banach algebras [6, Theorem 1.5]. It is easy to check that (Φp)p is an inverse system of isometric isomorphisms of Banach algebras. Then lim
p←Φp is an isomorphism of locallym -convex algebras from limp←BAp(Ep) onto lim
p←LM(K Ap(Ep)) and the theorem is proved.
We say that an elementT of BA(E, F) is bounded in BA(E, F) if there is M > 0 such that p(T)e ≤M for allp∈S(A) and denote byb(BA(E, F)) the set of all bounded elements inBA(E, F). It is clear that the map
k·k∞:b(BA(E, F))→[0,∞) defined by
kTk∞= sup{p(Te );p∈S(A)}
is a norm onb(BA(E, F)).
Theorem 3.7. IfE andF are Hilbert A-modules, then b(BA(E, F)) is a Banach space in the norm k·k∞. Moreover,b(BA(E, F))is isometrically isomorphic to Bb(A)(b(E), b(F)).
Proof. LetT ∈b(BA(E, F)).Then, since
pF(T x)≤ kTk∞kxk∞
for everyx∈b(E) and for every p∈S(A), T(b(E))⊆b(F) and it is easy to see that the restrictionT|b(E) ofT onb(E) is an element in Bb(A)(b(E), b(F)). Moreover,
T|b(E)
≤ kTk∞. On the other hand, sinceb(E) is dense inE [4, Proposition 3.1], and since
T|b(E)x, T|b(E)x
≤ T|b(E)
2hx, xi for everyx∈b(E) (cf. [7, 2.8]), we havekTk∞≤
T|b(E)
. Hence kTk∞= T|b(E)
. Define Ψ :b(BA(E, F))→ Bb(A)(b(E), b(F)) by
Ψ(T) =T|b(E).
Clearly Ψ is an isometric morphism fromb(BA(E, F)) to Bb(A)(b(E), b(F)). To show that Ψ is surjective, letS
∈L(b(E), b(F)). Since
hSx, Sxi ≤ kSk2hx, xi
for allxin b(E) (cf. [7, 2.8]) andb(E) is dense inE,S can be extended to a boundedA-module mapSefromE toF. Moreover, sincep(eS)e ≤ kSkfor allp∈S(A),Se is a bounded element inBA(E, F). Hence Ψ is surjective.
Thus we showed thatb(BA(E, F)) is isometrically isomorphic toBb(A)(b(E), b(F)), and so b(BA(E, F)) is a
Banach space.
It is easy to check that an elementT in b(BA(E, F)) is adjointable if and only ifT|b(E)is adjointable.
Remark 3.8. The restriction of Ψ on b(LA(E, F)) is an isometric isomorphism from b(LA(E, F)) onto Lb(A)(b(E), b(F))).
Knowing that for eachp∈S(A), epis a submultiplicative seminorm on BA(E) andp|eLA(E)is aC∗-seminorm onLA(E), it is easy to see thatk·k∞is a submultiplicative norm on b(BA(E)) and aC∗-norm onb(LA(E)).
Corollary 3.9. LetA be a locally C∗-algebra and letE be a HilbertA-module. Then we have:
(1) b(BA(E))with the normk·k∞ is a Banach algebra which is isometrically isomorphic to Bb(A)(b(E)).
(2) b(LA(E))with the norm k·k∞ is aC∗-algebra which is isomorphic to Lb(A)(b(E)) [4, Theorem 3.3].
Proof. PuttingF=E in Theorem3.7, it is easy to verify that Ψ is an isometric isomorphism fromb(BA(E)) ontoBb(A)(b(E)) and the restriction Ψ onb(LA(E)) is an isomorphism fromb(LA(E)) ontoLb(A)(b(E)).
Remark 3.10. Let E and F be two Hilbert A-modules. In general, b(KA(E, F)) is not isomorphic to Kb(A)(b(E), b(F)).
Example. Let A = C(Z+), the ∗-algebra of all complex valued functions on Z+. It is not difficult to see that A is just
∞
Q
n=1C. Also it is not difficult to check that A with the topology determined by the family of C∗-seminorms{pn}n, wherepn((an)n) = sup{|ak|; 1≤k≤n}, is a locallyC∗-algebra, andApn can be identified with the product of the firstnfactors ofAfor eachn.
Let E =
∞
Q
n=1
Cn. We make E into a Hilbert A-module via (ξn)n(an)n = (ξnan)n and h(ξn)n,(ηn)ni = (hξn, ηnin)n, whereh·,·in denotes the usualC-inner product onCn. ClearlyE is not finitely generated as Hilbert A-module. Moreover, Epn can be identified with the product of the first nfactors of E for each n. Therefore, LApn(Epn) =KApn(Epn) for each n. This implies thatLA(E) =KA(E) [9, Example 4.9], and by Corollary3.9, b(KA(E)) is isomorphic with Lb(A)(b(E)).
Suppose thatb(KA(E)) is isomorphic withKb(A)(b(E)). Then theC∗-algebrasKb(A)(b(E)) andLb(A)(b(E)) are isomorphic. This implies thatb(E) is finitely generated as Hilbertb(A)-module [10] and soEis finitely generated as HilbertA-module, a contradiction. Thereforeb(KA(E)) is not isomorphic withKb(A)(b(E)).
Remark 3.11. IfA is a locallyC∗-algebra thenAis a HilbertA-module with ha, bi=a∗b, a, b∈Aand the locallyC∗-algebrasLA(A) andM(A), whereM(A) is the set of all multipliers ofA, are isomorphic [9]. Putting E=Ain Corollary3.9, we deduce that theC∗-algebrasM(b(A)) andb(M(A)) are isomorphic, a result obtained independently by Bhatt and J. Karia [1, Theorem 5.1] and the author [3, Theorem 2].
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M. Joit¸a, Department of Mathematics, Faculty of Chemistry, University of Bucharest, Bd. Regina Elisabeta nr. 4–12, Bucharest, Romania,e-mail:[email protected]
