ISSN1842-6298 Volume1(2006), 23 – 32
ON THE LINKING ALGEBRA OF HILBERT
MODULES AND MORITA EQUIVALENCE OF
LOCALLY
C
-ALGEBRAS
Maria Joi¸ta
Abstract. In this paper we introduce the notion of linking algebra of a Hilbert module over a locallyC -algebra and we extend in the context of locallyC -algebras a result of Brown, Green and Rie¤el [Paci…c J.,1977] which states that twoC -algebras are strongly Morita equivalent if and only if they are isomorphic with two complementary full corners of aC -algebra.
1
Introduction and preliminaries
A locallyC -algebra is a complete Hausdor¤ complex topological -algebraAwhose topology is determined by a directed family of C -seminorms in the sense that a net faigi2I converges to 0 if and only if the net fp(ai)gi2I converges to 0 for all continuousC -seminormpon A:The term of locallyC -algebra is due to Inoue [3].
Now, we recall some facts about locally C -algebras from [2], [3], [5] and [8]. For a given locally C -algebra A we denote by S(A) the set of all continuous
C -seminorms onA:For p2S(A);the quotient -algebraA=kerp;denoted byAp; wherekerp=fa2A;p(a) = 0gis a C -algebra in the C -norm induced byp: The canonical map fromAtoAp is denoted by p:Forp; q2S(A)withp q; there is a surjective canonical map pq from Ap onto Aq such that pq( p(a)) = q(a) for all
a2A:Then fAp; pqgp;q2S(A);p q is an inverse system ofC -algebras and lim
p
Ap is
a locallyC -algebra which can be identi…ed with A:
A Fréchet locallyC -algebra is a locallyC -algebra whose topology is determined by a countable family ofC -seminorms.
A morphism of locallyC -algebras is a continuous -morphism from a locallyC -algebraA to another locally C -algebra B. An isomorphism of locally C -algebras
2000 Mathematics Subject Classi…cation: 46L08, 46L05
Keywords: locallyC -algebras, strongly Morita equivalence, linking algebra.
from A toB is a bijective map :A ! B such that and 1 are morphisms of locally C -algebras.
Hilbert modules over locallyC -algebras are generalizations of HilbertC -modules by allowing the inner product to take values in a locallyC -algebra rather than in a C -algebra. Here we recall some facts about Hilbert modules over locally C
-algebras from [5] and [8].
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 anA-valued inner producth; i:E E!Awhich isC-andA-linear in its second variable and
satis…es the following relations:
i. h ; i =h ; i for every ; 2E;
ii. h ; i 0 for every 2E;
iii. h ; i= 0 if and only if = 0.
We say thatE is a HilbertA-module ifEis complete with respect to the topology determined by the family of seminormsfpEgp2S(A)wherepE( ) =
qthere is a canonical surjective morphism of vector spaces E
pq fromEpontoEq such
that E
pq( Ep( )) = Eq( ); 2 E: Then fEp;Ap; Epq; pqgp;q2S(A);p q is an inverse
system of HilbertC -modules in the following sense: E
pq( pap) = Epq( p) pq(ap); p 2 module morphisms from E into F becomes a locally convex space with topol-ogy de…ned by the family of seminorms fpeLA(E;F)gp2S(A); where peLA(E;F)(T) =
For 2Eand 2F we consider the rank one homomorphism ; fromEintoF
Two locallyC -algebrasAandB are strongly Morita equivalent if there is a full HilbertA-moduleE such that the locallyC -algebrasB andKA(E)are isomorphic [4]. In [4], we prove that the strong Morita equivalence is an equivalence relation on the set of all locallyC -algebras. Also we prove that two Fréchet locallyC -algebras
A and B are strongly Morita equivalent if and only if they are stably isomorphic. This result extends in the context of locally C -algebras a well known result of Brown, Green and Rie¤el [1, Theorem 1.2]. In this paper, we extend in the context of locallyC -algebras another result of Brown, Green and Rie¤el [1, Theorem 1.1] which states that twoC -algebrasAandB are strongly Morita equivalent if and only if there is aC -algebraCsuch thatAandBare isomorphic with two complementary full corners in C, Theorem 9. For this, we introduce the notion of linking algebra of a Hilbert moduleE over a locallyC -algebraA:Finally, using the fact that any Hilbert C -module E is non-degenerate as Hilbert module (that is, for any 2 E
there is 2 E such that = h ; i); and taking into account that the linking algebra ofE is in fact the inverse limit of the linking algebras of Ep; p2S(A);we prove that any Hilbert module E over a locally C -algebra A is non-degenerate as Hilbert module, Proposition 10.
Any locally C -algebra A is a Hilbert A -module with the inner product de-…ned by ha; bi = a b; a; b 2 A; and the locally C -algebras M(A) and LA(A) are isomorphic [8].
Let C be a locally C -algebra.
De…nition 1. A locally C -subalgebra A of C is called a corner if there is a pro-jection P 2M(C) ( that is,P =P andPP =P) such that A=PCP:
Proposition 7. Let C be a locallyC -algebra and letA be a full corner inC. Then
A and C are strongly Morita equivalent.
Proof. Let P 2 M(C) such that A= PCP. Then CP is a full Hilbert A -module with the action of Aon CP de…ned by (cP; a)!caand the inner product de…ned byhcP; dPi=Pc dP. Moreover, for eachp2S(C);the HilbertAp -modules(CP)p and Cp
00
p(P) are unitarily equivalent. Then the locally C -algebras KA(CP) and
lim strongly Morita equivalent [1, Theorem 1.1]. Moreover, Cp
00
is a morphism ofC -algebras. Indeed, from
k p( p(cPd))kK
Ap(Cp 00p(P)) = p(cP); p(d P) KAp(Cp 00p(P))
= supfk p(cPdeP)kCp;e2C;k p(eP)kCp 1g
for all c; d2E;we conclude that p is continuous, and since
It is not di¢cult to check that ( p)p is an inverse system of isomorphisms of
C -algebras. Let = lim
p
p: Then is an isomorphism of locally C -algebras
from lim
Cp and C, we conclude that the locally C -algebras
KA(CP) and C are isomorphic. Therefore the locally C -algebras A and C are strongly Morita equivalent.
From Proposition7 and taking into account that the strong Morita equivalence is an equivalence relation on the set of all locallyC -algebras we obtain the following corollary.
Corollary 8. Let C be a locally C -algebra. If A andB are two full corners inC, thenA and B are strongly Morita equivalent.
Let E be a HilbertA-module.
The direct sumA Eof the HilbertA-modulesAandEis a HilbertA-module with the action ofA onA E de…ned by
(A E; A)•(a ; b)!(a )b=ab b2A E
and the inner product de…ned by
(A E; A E)•(a ; b )! ha ; b i=a b+h ; i 2A:
Moreover, for eachp2S(A);the HilbertAp-modules(A E)p and Ap Ep can be identi…ed. Then the locally C -algebras LA(A E) and lim
p
LAp(Ap Ep) can be
Let a2A, 2E, 2E andT 2KA(E). The map
La; ; ;T :A E !A E
de…ned by
La; ; ;T(b ) = (ab+h ; i) ( b+T( ))
is an element in LA(A E). Moreover, (La; ; ;T) = La ; ; ;T : The locally C -subalgebra ofLA(A E) generated by
fLa; ; ;T;a2A; 2E; 2E; T 2KA(E)g
is denoted byL(E) and it is called the linking algebra ofE:
By Lemma III 3.2 in [7], we have
L(E) = lim
p
( p) (L(E));
where( p) (L(E))means the closure of the vector space( p) (L(E))inLAp(Ap
Ep):Let p2S(A):From
( p) (La; ; ;T) =L p(a); pE( ); Ep( );( p) (T)
for all a; b2A; for all ; ; 2E; and taking into account that L(Ep); the linking algebra of Ep;is generated by
fL p(a); E
p( ); Ep( );( p) (T);a2A; 2E; 2E; T 2KA(E)g
since p(A) =Ap; Ep(E) =Ep;and( p) (K(E)) =K(Ep);we conclude that
L(E) = lim
p
L(Ep):
Moreover, since L(Ep) = KAp(Ap Ep) and the locally C -algebras KA(A E)
and lim
p
KAp(Ap Ep) can be identi…ed, the linking algebra of E coincides with
KA(A E):
The following theorem is a generalization of Theorem 1.1 in [1].
Theorem 9. Two locally C -algebras A and B are strongly Morita equivalent if and only if there is a locally C -algebra C with two complementary full corners isomorphic with A respectively B:
1. iAp : Ap ! Cp de…ned by iAp(ap) = Lap;0;0;0 is an isometric morphism of
inverse system of isometric morphisms of C -algebras as well as iKAp(Ep)
p. Let
Therefore the locally C -algebrasA and KA(E) are isomorphic with two com-plementary corners inC:Moreover, these corners are full since
and
CQC = lim
p
p(CQC) = lim
p
CpQpCp = lim
p
Cp =C.
Thus we showed that the locally C -algebras A and B are isomorphic with two complementary full corners in C:
The converse implication is proved by Corollary 8.
It is well known that a Hilbert module E over a locally C -algebra A is non-degenerate as topological A -module in the sense that the linear space generated by f a; 2 E; a 2 Ag is dense in E (see, for example, [5]). Also it is well known that a Hilbert C -moduleE is non-degenerate as Hilbert module, that is, for any
2 E there is 2 E such that = h ; i: By analogy with the case of Hilbert
C -modules, we prove that any Hilbert module E over a locally C -algebra A is non-degenerate as Hilbert module.
Proposition 10. Let E be a HilbertA -module. Then for each 2E there is 2E
such that = h ; i .
Proof. Let T = L0; ; ;0 2 L(E): Clearly, T = T : As in the case of Hilbert C -modules, consider the function f :R ! R de…ned by f(t) = t13 . Then, for each p2S(A);there is p 2Ep such that
f L0; E
p( ); Ep( );0 =L0; p; p;0
(see, for example, [9, Proposition 2.31]) and moreover, E
p ( ) = p p; p :But, by
functional calculus (see, for example, [8]),
( p) (f(T)) =f ( p) (T) =f L0; E
p( ); Ep( );0
for all p2S(A);and then
L0; E
pq( p); Epq( p);0 = ( pq) L0; p; p;0
= ( pq) f L0; E
p( ); Ep( );0
= ( pq) ( p) (f(L0; ; ;0))
= ( q) (f(L0; ; ;0))
= f L0; E
q( ); Eq( );0
= L0; q; q;0
for all p; q2S(A)with p q: This implies that forp; q2S(A) withp q we have
E
for all 2 E:From this fact, and taking into account that E
q (E) = Eq;we obtain
E
pq( p) = q: Therefore ( p)p 2 lim
p
Ep: Let 2 E such that Ep( ) = p for all
p2S(A):From
E
p ( h ; i) = Ep ( ) Ep ( ) Ep ( ); Ep ( )
= Ep ( ) p p; p = 0
for all p2S(A); wededuce that = h ; i and the proposition is proved.
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