Publicacions Ma em`a iques, Vol 39 (1995), 89–93.
ON LOCALLY PSEUDOCONVEX
SQUARE ALGEBRAS
A hippainen Jo ma
Abs ac
Le Abe an algeb a o e he field o complex numbe s wi h a
(Hausdo ff) opology gi en by a amily Q={qλ|λ∈Λ}o squa e
p ese ing λ-homogeneous semino ms ( λ∈(0,1]). We shall
show ha (A, T (Q)) is a locally m-con ex algeb a. Fu he mo e
we shall show ha Ais commu a i e.
In oduc ion. Le Abe a locally pseudocon ex algeb a o e he
field o complex numbe s. Le Q={qλ|λ∈Λ}be a amily o λ-
homogeneous semino ms defining a Hausdo ff opology on A. Fo each
λ∈Λ he numbe λ∈(0,1] is fixed. By λ-homogeniousi y we mean
ha qλ(αx)=|α| λqλ(x) o all x∈Aand α∈C. We shall say ha he
semino m qλis submul iplica i e i qλ(xy)≤qλ(x)qλ(y) o all xand y
in A. I e e y qλ∈Qis submul iplica i e, hen (A, T(Q)) is called a
locally m-pseudocon ex algeb a. I each qλ∈Qis squa e p ese ing in
o he wo ds i qλ(x2)=qλ(x)2 o all x∈Aand λ∈Λ we shall say ha
(A, T(Q)) is a squa e algeb a. No e ha locally pseudocon ex algeb as
include as a special case be e known locally con ex algeb as. Namely
o locally con ex algeb as we ha e λ= 1 o e e y λ∈Λ. Fo p op-
e ies o commu a i e locally con ex squa e algeb as see [1], [4]o [14].
Commu a i e locally pseudocon ex squa e and s a algeb as ha e been
s udied in [2]. I is known ha a commu a i e locally con ex squa e
algeb a is au oma ically locally m-con ex. See [1] and [5] and [16]. I
was claimed in [2] ha he co esponding esul is alid also o locally
pseudocon ex algeb as. In his pape we shall show ha indeed his
claim is ue and e en he assump ion o commu a i i y is supe fluous.
Main esul s. I is a -homogeneous submul iplica i e no m on
a complex associa e algeb a A, hen (A, ) is called a locally bounded
algeb a (mo e p ecisely a -no med algeb a). See [18]. Fi s we shall
p o e a locally bounded e sion o Theo em o [8] and Co olla y 16.8 o
[7]. See also [3], [12], [13] and [16].
90 A. Jo ma
Lemma 1. Suppose ha (A, )is a -Banach algeb a o which he e
is a cons an K>0such ha
(1) x2≤Kx2 o all x∈A.
Then A is commu a i e.
P oo : Le xbe a gi en elemen o A. Fu he mo e, le Bbe a maximal
commu a i e subalgeb a o Aincluding he elemen x. Then also (B,)
is a -Banach algeb a. By Theo ems 3.3, 4.4 and 4.8 o [18]weha e
sB(y) = limn→∞ yn1
n o all y∈B. (He e sBs ands o he spec al
adius o yin B.) I ollows om (1) ha he e is some cons an M:=
M(K)>0 such ha y≤MsB(y) o all y∈B. Since Bis a maximal
commu a i e subalgeb a o Awe ha e sB(y)=sA(y) o all y∈B. (See
o ex. [17, p. 46].) Since he abo e men ioned xis in Bwe can see ha
we ha e x1
≤M1
sA(x) and since xwas chosen a bi a ily we can see
ha his same holds o all x∈A. Bu now we ha e
sA(xy)≤xy1
≤x1
y1
≤M2
sA(x)sA(y) o all xand y∈A.
By Theo em 1 o [11] i ollows ha A/ Rad Ais commu a i e. (Rad A
s ands o he Jacobson adical o A). Bu i ollows om (1) ha
Rad A={0}and hus we can see ha Ais commu a i e.
No e ha he opological dual o Awas no used in [11] in p o ing
he commu a i i y o A/ Rad A.
We shall now p o e he gene aliza ion o he esul s o [5] and [6]. Fo
a semino m qon an algeb a Adeno e by Nq=ke q={x∈A|q(x)=0}.
Theo em 1. Le qbe a -homogeneous squa e p ese ing semino m
on a (complex associa i e) algeb a A. Then qis submul iplica i e, Nqis
an ideal o A,q1
is a semino m on A, and he quo ien algeb a A/Nqis
commu a i e.
P oo : Define he “Jo dan p oduc ” ◦o Aby x◦y=1
2(xy +yx)x,
y∈A.Now4(x◦y)=(x+y)2−(x−y)2 o all xand yin A. Thus, i
xand y∈A hen
4 q(x◦y)=q(4(x◦y))
=q((x+y)2−(x−y)2)≤q((x+y)2)+q((x−y)2)
=(q(x+y))2+(q(x−y))2≤(q(x)+q(y))2+(q(x)+q(y))2
=2(q(x)+q(y))2.
Pseudocon ex squa e algeb as 91
So we ha e q(x◦y)≤2
4 (q(x)+q(y))2 o all xand yin A. Le x,
y∈Aand >0 be a bi a y. Deno e by α=q(x)+and β=q(y)+.
Then
α−1β−1q(x◦y)=q((α−1
x)◦(β−1
y)) ≤2
4 (q(α−1
x)+q(β−1
y))2
=2
4 (α−1q(x)+β−1q(y))2≤2
4 (1+1)
2=8
4 .
This shows ha q(x◦y)≤8
4 q(x)q(y) o all xand yin A.I wenow
define p=8
4 q, hen pis a -homogeneous semino m on Asa is ying
p(x◦y)≤p(x)p(y) o all xand yin Aand p(x)2≤8
4 p(x2) o all
xin A.Fo x,y∈Adeno e [x, y]=xy −yx. As in he p oo o
P oposi ion 1 o [9] i can be shown ha he e is some cons an K
o which p([x, y])2≤Kp(x)2p(y)2 o all xand yin A. Since xy =
x◦y+1
2[x, y]x,y∈A, we can see ha he e is some cons an Rsuch
ha p(xy)≤Rp(x)p(y) o all xand yin A. Thus he e is some cons an
M o which q(xy)≤Mq(x)q(y) o all xand yin A. I ollows om
his inequali y ha Nqis an ideal o A. W i e B:= A/Nqand le ˙q
deno e he induced -homogeneous no m on B. Then .:= M˙qis a
submul iplica i e -homogeneous no m on Bsa is ying x2≤Mx2
o all x∈A. Applying Lemma 1 o he comple ion o (B,), we can
see ha Bis commu a i e. To p o e ha ˙q1
is a no m on Ble ˙sqbe he
spec al no m o (B, ˙q) i.e. ˙sq(x) = limn→∞
n
˙q(xn)1
n. By Theo em 3.3
o [18]˙sqsa isfies he iangle inequali y and on he o he hand we ha e
˙q(x)= ˙sq(x) o all x∈B(since ˙qis squa e p ese ing). By Co olla y 3.5
o [18]˙q1
is a usual (1-homogeneous) no m on B.Thusq1
is a semino m
on A. See also [2, Lemma 9]. No e also ha q1
is submul iplica i e. See
[1]o [5].
Co olla y 1. Suppose ha (A, T(Q)) is a squa e algeb a. Then
(A, T(Q)) is a locally m-con ex commu a i e algeb a.
P oo : I ollows om Theo em 1 ha each quo ien algeb a A/Nλ
is commu a i e (Nλ=ke qλ). This implies ha qλ(xy −yx) = 0 o
all λ∈Λ and since we assumed ha T(Q) is a Hausdo ff opology his
implies ha Ais commu a i e. By Theo em 1 o each λ∈Λ, q
1
λis a
usual 1-homogeneous submul iplica i e semino m on A.ThusT(Q)is
equi alen wi h a locally m-con ex opology T(P) whe e P={q
1
λ|λ∈
Λ}.
Acknowledgemen . The au ho wan s o exp ess his since e hanks
o he e e ee o his con ibu ion and ad ice in he fi s d a o his
pape .
92 A. Jo ma
Re e ences
1. J. A hippainen, On locally con ex squa e algeb as, Func . App ox.
XXII (1993), 57–63.
2. J. A hippainen, On unc ional ep esen a ion o locally m-pseudo-
con ex algeb as, submi ed o a publica ion.
3. J. W. Bake and J. S. Pym, A ema k on con inuous bilinea
mappings, P oc. Edingbu gh Ma h. Soc. (2) 17 (1971), 245–248.
4. E. Beckens ein, L. Na ici and C. Su el,“Topological Alge-
b as,” No h Holland Publ. Comp., New Yo k, 1977.
5. S. J. Bha and D. J. Ka ia, Uniqueness o he uni o m no m
wi h an applica ion o opological algeb as, P oc. Ame . Ma h. Soc.
116(2) (1992), 499–503.
6. S. J. Bha , A semino m wi h squa e p ope y on a Banach algeb a
is submul iplica i e, P oc. Ame . Ma h. Soc. 117 (1993), 435–438.
7. F. F. Bonsall and J. Duncan,“Comple e No med Algeb as,”
Sp inge Ve lag, Be lin, 1973.
8. R. A. Hi sch eld and W. Zelazko, On spec al no m Banach
algeb as, Bull. Acad. Polon. Sci. Se . Sci. Ma h. As onom. Phys.
16 (1968), 195–199.
9. M. Cab e a, A. Mo eno and A. Rod ´
ıguez, On he beha iou
o Jo dan-algeb a no ms on asso ia i e algeb as, o appea in S udia
Ma h.
10. A. Inoue, Locally C∗-algeb as, Mem. Sci. Kyushu Uni . (Se A)
25 (1971), 197–235.
11. A. Kokk, Almos commu a i i y o spec ally bounded algeb as,
Ac a Comm. Uni . Ta uensis 960 (1993), 29–40.
12. M. Oudadess, -sa u a ed uni o mly A-con ex algeb as, Ma h.
Japonica 35(4) (1990), 615–620.
13. C. Le Page, Su quelques condi ions en ainan la commu a i i e
dans les algeb es de Banach, C.R. Acad. Sci. Pa is Se A-B 265
(1967), 235–237.
14. A. Mallios,“Topological Algeb as. Selec ed Topics,” Else ie Sci-
ence Publ. Comp., New Yo k, 1986.
15. C. Ricka ,“Gene al Theo y o Banach Algeb as,” Robe E.
Publ. Comp., Hun ing on New Yo k, 1960.
16. Z. Sebes yen, E e y C∗-semino m is au oma ically submul iplica-
i e, Pe . Ma h. Hung. 10 (1979), 1–8.
Pseudocon ex squa e algeb as 93
17. B. Yood,“Banach Algeb a-an in oduc ion,” Ca le on-O awa
Ma h. Lec u e No e Se ies 9, O awa, 1988.
18. W. Zelazko, Selec ed opics in opological algeb as, Lec u e No es
Se . Ma h.-Aa hus Uni . 31 (1971).
Depa men o Ma hema ics
Uni e si y o Oulu
P.O.Box 400
FIN 90571 Oulu
FINLAND
P ime a e si´o ebuda el 29 d’Ab il de 1994,
da e a e si´o ebuda el 9 de No emb e de 1994