On locally pseudoconvex square algebras

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. 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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