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UNIDAD IZTAPALAPA División de Ciencias Básicas e Ingeniería

DEPARTAMENTO DE MA TEMA TICAS

TITULO:

MULTIPLIERS IN PERFECT LOCALLY m-CONVEX ALGEBRAS

AUTORES:\ HARALAMPIDOU, LOURDES PALACIOS AND CARLOS

SIGNORET ^ \

MULTIPLIERS IN PERFECT LOCALLY m-CONVEXALGEBRAS

MARINA HARALAMPIDOU1, LOURDES PALACIOS2 AND CARLOS SIGNORET3

ABSTRACT. In this paper we describe the multiplier algebra of a perfect com-plete locally m-convex algebra with an approximate identity and wíth completeArens-Michael normed factors.

1. INTRODUCTION AND PRELIMINARIES

Multipliers are important in various áreas of mathematics where an algebra struc-ture appears (see e.g [ i ] ; for (non-normed) topological algebras cf. e.g. [!]).

The algebras considered throughout are taken over the field of complexes C.Denote by L(E) the algebra of all linear operators on an algebra E.

Deflnition 1.1. A mapping T : E —> E is called a left (right) multiplier on E ifT(xy) = T(x)y (resp. T(xy] = xT(y}} for all x,y E E ; it is called a two-sidedmultiplier on E if it is both a left and a right multiplier.

It is known that if E is a proper algebra, namely xE = {9} implies x = 6or Ex — {9} implies x = O, where O denotes the nuil element of E, then anytwo-sided multiplier on E is automatically a linear mapping [i.i, p. 20].

In the sequel, a two-sided multiplier will be called in short, a multiplier. Letus denote by Mi(E] the set of all left multipliers on E, by Mr(E] the set of allright multipliers on E and by M(E] that of all multipliers on E. Note that, bydefinition, M(E) = Mt(E) n Mr(E).

Obviously M(E] is a subalgebra of L(E) in case the algebra is proper. Thesame holds for Mr(E] and M¡(E). Now, for x E E, the operator lx on E given bylx(y) = xy,y E E, is, due to the asso.ciativity of E, a left multiplier. Similarly,we can also define the right multiplier with respect to x E E, say rx.

It is known that if E is a proper algebra, then the mapping

F : E —» M[(E) given by x >—> lx

defines an algebra monomorphism which identifies E with a subalgebra of M¡(E).Moreover, E is a left ideal of the algebra Mi(E) (see [';, p. 1933, Proposition 2.2]).A similar result is also valid for right multipliers. For multipliers, the algebra Ecan be identified with a two-sided ideal in M(E] (ibid, p. 1934, Corollary 2.3).

1991 Mathematics Subject Classification. Primary 46H05; Secondary 46H10. 46K05.Key words and phrases. Proper algebra. Arens-Michael decomposition. multiplier algebra,

perfect projective system, perfect locally m-convex algebra.1

2 MARINA HARALAMPIDOU, LOURDES PALACIOS, CARLOS SIGNORET

Deflnition 1.2. An approximate identity in a topological algebra E is a netsuch that for each x € -E we have:

(x — xej) — > O and (x — e¿x) — > O for all x & E.6 5

Note that an algebra with an approximate identity is proper. In this paper wedescribe the multiplier algebra M(E] in the case where E is a certain completelocally m-convex algebra with an approximate identity.

For the sake of completeness, we recall what we mean by the "Arens-Michaeldecomposition" (["/", p. 88, Theorem 3.1]).

Let (E, (pa)aeh) be a complete locally m -convex algebra and

pa : E — » E I ker(pa) = Ea defined by pa(x) = x + ker(pQ) = xa, a E A

the respective quotient maps. Then pa(xa) := pa(x), x E E, a £ A. defines on Ea

an algebra norm, so that Ea is a normed algebra and the morphisms pa, a 6 Aare continuous. Ea, a € A denotes the completion of Ea (with respect to pa. Ais endowed with a partial order by putting a < /3 if and only if pa(x) < pp(x) forevery x e E. Thus, ker(p^) C ker(pa) and henee the continuous (onto) morphismfa/3 : Ep — > Ea : x@ i— > fap(xp) — xa, a < ¡3 is defined. Moreover, fap is extendedto a continuous morphism fa/3 : Ep — > Ea, a < (3. Thus, ( E a , f a p ) , (Ea,fap),a, /? € A with a < (3 are projective systems of normed (resp. Banach) algebras,so that E = lim ̂ = lim Ea (Arens-Michael decomposition) within topologicalalgebra isomorphism.

In ['i, p. 1934, Theorem 3.1], it is shown that, in a special case, the algebraM(E] is a subalgebra of ¿-(E), the algebra of all continuous linear operators inE; for completeness, we refer it here.

Theorem 1.3. Let (E, (pa)ae\) be a complete locally m-convex algebra with anapproximate identity {es}s^A- Suppose that each factor Ea = E/keTpa in theArens-Michael decomposition of E is complete. Then each multiplier T of E iscontinuous, viz. M(E] is a subalgebra of ¿C(E).

2. PERFECTNESS AND MULTIPLIERS IN LOCALLY TB-CONVEX ALGEBRAS

To proceed, we use the notion of a perfect projective system as it appeared in[", p. 199, Definition 2.7]. To fix notation, we repeat it.

Definition 2.1. A projective system {(Ea,fap)}a€^ of topological algebras iscalled perfect , if the restrictions to the projective limit algebra

E = limEa = {(XQ) E U Ea : faft(x^ = xa, if a < (3 E A}

of the canonical projections TTQ : HaeA ^a — > Ea, a E A, namely, the (continuousalgebra) morphisms

fa = Ka \E=l\mEa'- E —> Ea, a & A,

are onto maps. The resulting projective limit algebra E — lim Ea is then calleda perfect (topological) algebra.

MULTIPLIERS IN PERFECT LOCALLY m-CONVEX ALGEBRAS 3

Definition 2.2. In the sequel, by the term perfect locally m-convex algebra wemean a locally m-convex algebra (E, (pQ)aeA) f°r which the respective Arens-Michael projective system {(Ea, fa0)}a&A is perfect.

Every Fréchet locally m-convex algebra (E, (pn)n€ N) gives a perfect projectivesystem of normed algebras, and thus it is a perfect algebra (see [2], and [b]).

Example 2.3. Let E be a non-complete normed algebra. Take E = Ea for eacha 6 A and, for a < (3, let fap : Eg — > Ea be the identity map. Then A =limEa, the diagonal algebra, is a perfect locally m -convex algebra, but A isnon-complete.

Let E = (E, (pQ)aeA) be a complete perfect locally m-convex algebra with anapproximate identity and such that each factor Ea of its Arens-Michael decom-position (Ea, fap, o. < ¡3) is complete.

Remark 2.4. If ó is the isomorphism E — > lim.E'a given by <p(x) = (xa}a&\, then,for each a e A, pa = fa o </>. Therefore, kerpa = kerpa = ker(/a o 0).

Remark 2.5. By the hypothesis of perfectness, each fp is surjective, so each timewe have an element xp € Ep, we can choose an element u € E such that uip — xp,and consequently ua = fap(xp] = xa, whenever a < /3.

For each a < /?, we define the map hap : M(E@] — *• M(Ea) given by

which is well defined, according to the following lemma.

Lemma 2.6. Let (E, (pa)a€A) be a complete perfect locally m-convex algebra withan approximate identity (es)s^A o,nd such that each factor E a of its Arens-Michaeldecomposition is complete. Then ker/a/g is Tp-invariant for each Tp 6 M(Ep],that is, Tp(\SBT fap) C keTfap, i f c t < / 3 , and the map hap is a well-defined con-tinuous multiplicative linear mapping.

Proof. Take xp 6 ker/Qig. Since E has an approximate identity (es)se¿\dmultipliers over Banach algebras are continuous (see ['i, p. 20, Theorem 1.1.1]),then

fap(Tfi(xp)) = fap(Tp(\imxpes}} = faf3(limTf3(x/3es}) = limfap(Tp(xpe5}} =O O O

= \\rnf ap(xpT0(es}} = \ip.(fap(xp}fap(Tp(e&}}} = 0.o o

We claim that hap(Tp) is well-defined. For that, let a < /3, x £ E such thatxa = x'a and Tp € M(Ep); then O = xa - x'a = pa(x] - pa(x') = pa(x - x'}and henee O = (fa o fy(x - x1) = (fap o fp o <p)(x — x'}, which implies that(f/3°^)(x-x') G ker/a/3. Since ker fap is Tp -invariant, Tp((fpo<t>}(x-x'}} € ker/Q/3

too, and thereforeO = fap(Tp((f8 o é}(x - x')} = fa0(Tp(Pp(x - x'})} = fa0(T0(x0 - 4)) =

that is, fap(Tp(xp)) = fap(Tp(x'g)). This proves the claim.Moreover, hap(Tp) is actually a multiplier on Ea. For, let xa and ya be two

elements in Ea. Then

MARINA HARALAMPIDOU, LOURDES PALACIOS, CARLOS SIGNORET

[hap(Tp)} (xaya) = faD(Tf}(x0y0}} = fap(xpTp(yp)) = fa0(x0)fa0(T0(y3)) == x a ( f a p ( T 0 ( y 8 ) ) = xa [ha0(T3)} (ya]

and so, hap(Tp) is a right multiplier. In a similar way, one can prove that hag(T0)is a left multiplier.

It is easily seen that hap is a linear mapping. Moreover, hag is multiplicativo.For that, take Tg, Sp E M(Eg}. We have

[ha0(T(, o S0}}(xa} = fap((Tp o Sp)(x0)) = faf3(Tp(Sp(xp))). (2.1)

On the other hand, since the system is perfect, we can choose ui € E (equiv-alently (wQ)aeA) € lim^a) such that fap(Sp(xp)) = WQ; note that fap(up) — ua

too. Then Sp(xp) — uip E ker fap.But, since ker fap is Tp -invariant, we have Tp(Sp(xp) — u>p) 6 ker /Q/g, and thus

)). Besides,

O

The last, in connection with (2.1) gives the multiplicativity of /IQ^.Next, we prove that hap is continuous. Since /Qjg : E1/? — > £^Q is a continuous

mapping between normed algebras, there exists a constant K > O such thatP<*(fap(yp)} < K Pp(y0) for each y^ E E3. In particular,

< K p^Tpfo)) foreach x3 E Ep. (2.2)

Taking the supremum on the right hand of (2.2) and since M(E¡}} is a Banachalgebra (see [ ;> , p. 20, Theorem 1.1.1]), we get

A: ̂ (7>(x^)) < /r sup {p^(x?)}} <K\\T^\\ (2.3)

for every xp E E/3 with Pp(xp) < 1, and where IH^ is the norm in the multiplieralgebra M(Ep). Since fap(T@(xp}) = [hap(Tp)](xa) whenever a < ¡3 (henee

pa(xa) < Pp(xp)), then pa([ha0(Tp)](xa)) < K \\Tg\\p for every xa E Ea with

pa(xa) ^ 1 by (2.3). Taking now the supremum in this latter relation, we haveP'a([ha0(Tp)}(xa)) < K \\Tft\\p. Thus ||^CZ»||a < K \\T8\\,, namely,

each hap is continuous. D

So far, we have the family of topological algebras M(Ea) and the family ofmultiplicative continuous linear mappings hap : M(Ep) —> M(Ea), a < ¡3 in A.Actually, they form a projective system. In fact, if a < (3 < 7, then fap o fp^ =fQ-y, and therefore

(ha^}\Xa) = /a7(T7(x7)) - (faf} o f^)(T^)) =xa) =

for each xa E Ea. That is, /iQ7(T7) = (/ia,3°/i/37)(T7) for each T7 € M(E~,), whichimplies that /iQ7 = hap o hp^; it is clear that

h-aa = Id.M(Ea)-

MULTIPLIERS IN PERFECT LOCALLY m-CONVEX ALGEBRAS

Thus, we have the projective system of Banach algebras {(M(Ea}, ^a/3and we can take its inverse limit, limM(Ea).

Now, we prove a lemma extracted from the proof of Theorem 3.1 in [ , p. 1934]that will be useful in the sequel.

Lemma 2.7. Let (E, (pa)aeA) be a locally m-convex algebra with an approximateidentity (e¿)s&^ and let T E M(E}. Then, for each a 6 A, kerpQ is T-invariant;that is, T(kerpQ) C kerpa.

Proof. Take x G kerjv Since the seminorms are continuous, for e > O, thereexists an Índex 60 G A such that pa(T(x) — T(x}e¡) < e whenever ó > S0. Wehave

pa(T(x}} = pa(T(x - xe¿0 + xeSo)} = pa(T(x) - T(xeSo) + T(xeSo))< pa(T(x) - T(xeSo}) + Pa(T(xeSo)) = pa(T(x) - T(x)eSo) + pa(xT(e&0))< pa(T(x] - T(x)eSo) + pa(x)pa(T(eSo)) < e.

Since this is true for an arbitrary e > O, we conclude that pa(T(x}} = O, that is,T(x) e kerpQ. D

Now we state our main Theorem.

Theorem 2.8. Let (E, (pa)aeA.) be a complete locally m-convex algebra with anapproximate identity (e¡)s^¿\., such that the respective projective system is perfectand each factor Ea = E/kerpa in its Arens-Michael decomposition is complete.Then M(E] = lim M(Ea) within a topological algebra isomorphism.

Proof. Take T 6 M(E). Due to Lemma 2.7, T induces a well-defmed mapTa : Ea — > Ea such that Ta o pa = pa o T for each a e A, that is,Ta(xa) = Ta(Pa(x}) = Pa(T(x)) = T(x)a for each x e E.Since for xa,ya e Ea,

Ta(xaya) = Pa(T(xy}} = Pa(xT(y}} = xaT(y)a = xaTa(ya),Ta is a right multiplier. In a similar way it can be shown that it is a left multiplier,as well.

Note also that (Ta)aeA is an element of lim M(Ea). Indeed, for a < (3 andpa(x] = xa € Ea, we have

) = [ha0(T0)](xa) = faff(Tft((x^)} =- /Q/3((//3 o <t>}(T(x}}} = ((faft o /^) o

= (fa o 4>}(T(x}} = Pa(T(x}} = T a ( p a ( x } ) .Therefore /iQ/a(7» = Taifa</3.

Now we define the map$ : M(E] — * lim M(Ea] by $(T) = (TQ)aeA,

which obviously is linear. Moreover, for T, 5 € M(E] and xa 6 Ea, we have(xa) = (ToS)a(xa) = ((ToS)(x))a = (T(S(x)))a = Ta(S(x)a) =

which implies that (T o S)a = Ta o Sa, and therefore $(T o S1) = $(T) o $(5),namely, $ is multiplicative.

Next, we show that $ is one to one. For that, take T, S E M(E] such that(Ta)ae\ $(T) = $(5) = (5Q)Q£A; then TQ = 5a for each a e A. Therefore

6 MARINA HARALAMPIDOU, LOURDES PALACIOS, CARLOS SIGNORET

pa o T = pa o S for each a € A; then T = S. Moreover, <1> is an onto map. Indeed,for (Wa)a<=h € lim M(Ea] define the map

W : E -* E byV(:r) = <j)which obviously is linear. Also

W(xy) = <t>-l((Wa(xy}a}a^}= ^~a((xQWa(ya))aeA) = </)"1((xQ)aeA)</>"1((H/-a(2/a))Q6A) = xW(y),

and similarly on the other side, so W is a multiplier on E. Finally, it is clear that

We claim that $ is continuous. By [:•., p. 1934, Theorem 3.1], M(E] is asubalgebra of L(E], the algebra of all continuous linear operators on E, so thatthe topology on M(E] is the operator topology. Let us denote by

ga : M(E) -» M(Ea)the map ga(T) = Ta, which, by Lemma 2.7, is well defined and obviously linear.Let us denote by ha : lim M(Ea) — > M(Ea] the canonical continuous homomor-phism from the inverse limit to one of its factors. Note that ha o $ = ga holdsfor each a € A.

Since $ is continuous if and only if, for each a £ A, ha o $ is continuous (see[; , p. 89, the proof of Theorem 3.1]), we have to prove that ga is continuous(for each a € A). Let us recall that the topology of M(E) can be given bythe set of seminorms (pa,a E A) defined as pa(T] = sup pa(T(x}) for each

M*)<iT 6 M(E] . Further, the topology of M(Ea) can be given by the norm || • ||Q definedas ||5||Q = sup p(S(x}} for each S £ M(Ea), where, as usual, p is the induced

p(x)<l

norm in Ea given by p(xa) = p(x + ker pa] = pa(x). The topology of lim M(Ea)can be defined by the local base consisting of neighborhoods V = HILi 'la1(^Qi)>where Vaí is a basic neighborhood in M(Eai).

Let Si > O be given and let Va. = {56 M(Eaí] : \\S\\, < £í} andUaí = {T e M(E) : pa.(T] < a}. We claim that

TeU^^T^eVa.. (2.4)

Indeed,pai(T] <£>*=> sup Paí(T(x)) < e,

»««<(*)< i<=> sup p x Q . < £i <=» sup

p(xQi)<l p(xa.)<l

<=> Tai e Vai.Now, let Va be a basic neighborhood of O in M(Ea), say

Va = {Se M(Ea) : \\S\\ < e}.Put E7« = 5- l(Va). Then

This implies the continuity of ga for each a 6 A. Henee í> is continuous.Finally, we show that $ is an open map. Let V = 0"=: ̂ «¿H^aJ be a basic

neighborhood of O in M(E] . Take T € V; then T € h~^(Vai) foV alH = 1, . . . , n.Therefore Ta. = h^(T) e Vai and, due to (2.4), T E Ua*. Then *(T) 6 U = (Z7a),

MULTIPLIERS IN PERFECT LOCALLY m-CONVEX ALGEBRAS 7

where Ua — Uai for a = a\, a2,. .., an and Ua = M(Ea] otherwise. This provesthat $ is an open map, and the proof is complete. D

ACKNOWLEDGMENTS

We want to thank Professor Mohamed Oudadess for the suggestion of Example2.3 and for his fruitful comments.

We also want to thank the referees for their useful sugestions.

REFERENCES[1] G.F. Bachelis and J.W. McCoy, Left centralizers of an H*-algebra, Proc. Amer. Math. Soc.

' 43(1)(1974), 106-110.[21 M. Haralampidou, The Krull nature of locally C* -algebras. Function Spaces (Edwardsville.

IL, 2002), 195-200, Contemp. Math., 328, Amer. Math. Soc., Providence, RI, 2003.[3] M. Haralampidou, L. Palacios, C. Signoret, Multipliers in locally convex ''-algebras. Rocky

' Mountain J. Math, 43, No. 6, 2013, 1931-1940.[4] T. Husain, Multipliers of topological algebras, Dissertationes Math. (Rozprawy Mat.),

285(1989), 40 pp.[5] G. Kó'the, Topological Vector Spaces, I. Springer-Verlag, Berlín, 1969.[6] R. Larsen, The multiplier problem, Lectures Notes in Math. No. 105, Springer-Verlag, Berlin,

1969.[7] A. Mallios, Topological Algebras. Selected Tapies, North-Holland, Amsterdam, 1986.

1 DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ATHENS, PANEPISTIMIOUPOLIS, ATHENS15784, GREECE.

E-mail address: mharalam@math.uoa.gr;

2 UNIVERSIDAD AUTÓNOMA METROPOLITANA IZTAPALAPA, MÉXICO CITY 09340, MÉX-ICO;

E-mail address: pafaSxanum.uam.mx;

3 UNIVERSIDAD AUTÓNOMA METROPOLITANA IZTAPALAPA, MÉXICO CITY 09340, MÉX-ICO;

E-mail address: casi@xanum.uam.mx