On locally pseudoconvex square algebras

Publicacions Matem`atiques, Vol 39 (1995), 89–93. ON LOCALLY PSEUDOCONVEX SQUARE ALGEBRAS Arhippainen Jorma Abstract Let Abe an algebra over the field of complex numbers with a (Hausdorff) topology given by a family Q={qλ|λ∈Λ}of square preserving rλ-homogeneous seminorms (rλ∈(0,1]). We shall show that (A, T (Q)) is a locally m-convex algebra. Furthermore we shall show that Ais commutative. Introduction. Let Abe a locally pseudoconvex algebra over the field of complex numbers. Let Q={qλ|λ∈Λ}be a family of rλhomogeneous seminorms defining a Hausdorff topology on A. For each λ∈Λ the number rλ∈(0,1] is fixed. By rλ-homogeniousity we mean that qλ(αx)=|α|rλqλ(x) for all x∈Aand α∈C. We shall say that the seminorm qλis submultiplicative if qλ(xy)≤qλ(x)qλ(y) for all xand y in A. If every qλ∈Qis submultiplicative, then (A, T(Q)) is called a locally m-pseudoconvex algebra. If each qλ∈Qis square preserving in other words if qλ(x2)=qλ(x)2for all x∈Aand λ∈Λ we shall say that (A, T(Q)) is a square algebra. Note that locally pseudoconvex algebras include as a special case better known locally convex algebras. Namely for locally convex algebras we have rλ= 1 for every λ∈Λ. For properties of commutative locally convex square algebras see [1], [4]or[14]. Commutative locally pseudoconvex square and star algebras have been studied in [2]. It is known that a commutative locally convex square algebra is automatically locally m-convex. See [1] and [5] and [16]. It was claimed in [2] that the corresponding result is valid also for locally pseudoconvex algebras. In this paper we shall show that indeed this claim is true and even the assumption of commutativity is superfluous. Main results. If is a r-homogeneous submultiplicative norm on a complex associatve algebra A, then (A, ) is called a locally bounded algebra (more precisely a r-normed algebra). See [18]. First we shall prove a locally bounded version of Theorem of [8] and Corollary 16.8 of [7]. See also [3], [12], [13] and [16]. 90 A. Jorma Lemma 1. Suppose that (A, )is a r-Banach algebra for which there is a constant K>0such that (1) x2≤Kx2for all x∈A. Then A is commutative. Proof: Let xbe a given element of A. Furthermore, let Bbe a maximal commutative subalgebra of Aincluding the element x. Then also (B,) is a r-Banach algebra. By Theorems 3.3, 4.4 and 4.8 of [18]wehave sB(y)r= limn→∞ yn1 nfor all y∈B. (Here sBstands for the spectral radius of yin B.) It follows from (1) that there is some constant M:= M(K)>0 such that y≤MsB(y)rfor all y∈B. Since Bis a maximal commutative subalgebra of Awe have sB(y)=sA(y) for all y∈B. (See for ex. [17, p. 46].) Since the above mentioned xis in Bwe can see that we have x1 r≤M1 rsA(x) and since xwas chosen arbitrarily we can see that this same holds for all x∈A. But now we have sA(xy)≤xy1 r≤x1 ry1 r≤M2 rsA(x)sA(y) for all xand y∈A. By Theorem 1 of [11] it follows that A/ Rad Ais commutative. (Rad A stands for the Jacobson radical of A). But it follows from (1) that Rad A={0}and thus we can see that Ais commutative. Note that the topological dual of Awas not used in [11] in proving the commutativity of A/ Rad A. We shall now prove the generalization of the results of [5] and [6]. For a seminorm qon an algebra Adenote by Nq=kerq={x∈A|q(x)=0}. Theorem 1. Let qbe a r-homogeneous square preserving seminorm on a (complex associative) algebra A. Then qis submultiplicative, Nqis an ideal of A,q1 ris a seminorm on A, and the quotient algebra A/Nqis commutative. Proof: Define the “Jordan product” ◦of Aby x◦y=1 2(xy +yx)x, y∈A.Now4(x◦y)=(x+y)2−(x−y)2for all xand yin A. Thus, if xand y∈Athen 4rq(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. Pseudoconvex square algebras 91 So we have q(x◦y)≤2 4r(q(x)+q(y))2for all xand yin A. Let x, y∈Aand >0 be arbitrary. Denote by α=q(x)+and β=q(y)+. Then α−1β−1q(x◦y)=q((α−1 rx)◦(β−1 ry)) ≤2 4r(q(α−1 rx)+q(β−1 ry))2 =2 4r(α−1q(x)+β−1q(y))2≤2 4r(1+1) 2=8 4r. This shows that q(x◦y)≤8 4rq(x)q(y) for all xand yin A.Ifwenow define p=8 4rq, then pis a r-homogeneous seminorm on Asatisfying p(x◦y)≤p(x)p(y) for all xand yin Aand p(x)2≤8 4rp(x2) for all xin A.Forx,y∈Adenote [x, y]=xy −yx. As in the proof of Proposition 1 of [9] it can be shown that there is some constant K for which p([x, y])2≤Kp(x)2p(y)2for all xand yin A. Since xy = x◦y+1 2[x, y]x,y∈A, we can see that there is some constant Rsuch that p(xy)≤Rp(x)p(y) for all xand yin A. Thus there is some constant Mfor which q(xy)≤Mq(x)q(y) for all xand yin A. It follows from this inequality that Nqis an ideal of A. Write B:= A/Nqand let ˙q denote the induced r-homogeneous norm on B. Then .:= M˙qis a submultiplicative r-homogeneous norm on Bsatisfying x2≤Mx2 for all x∈A. Applying Lemma 1 to the completion of (B,), we can see that Bis commutative. To prove that ˙q1 ris a norm on Blet ˙sqbe the spectral norm of (B, ˙q) i.e. ˙sq(x) = limn→∞ n ˙q(xn)1 n. By Theorem 3.3 of [18]˙sqsatisfies the triangle inequality and on the otherhand we have ˙q(x)= ˙sq(x) for all x∈B(since ˙qis square preserving). By Corollary 3.5 of [18]˙q1 ris a usual (1-homogeneous) norm on B.Thusq1 ris a seminorm on A. See also [2, Lemma 9]. Note also that q1 ris submultiplicative. See [1]or[5]. Corollary 1. Suppose that (A, T(Q)) is a square algebra. Then (A, T(Q)) is a locally m-convex commutative algebra. Proof: It follows from Theorem 1 that each quotient algebra A/Nλ is commutative (Nλ=kerqλ). This implies that qλ(xy −yx) = 0 for all λ∈Λ and since we assumed that T(Q) is a Hausdorff topology this implies that Ais commutative. By Theorem 1 for each λ∈Λ, q 1 rλis a usual 1-homogeneous submultiplicative seminorm on A.ThusT(Q)is equivalent with a locally m-convex topology T(P) where P={q 1 rλ|λ∈ Λ}. Acknowledgement. The author wants to express his sincere thanks to the referee for his contribution and advice in the first draft of this paper. 92 A. Jorma References 1. J. Arhippainen, On locally convex square algebras, Funct. Approx. XXII (1993), 57–63. 2. J. Arhippainen, On functional representation of locally m-pseudoconvex algebras, submitted for a publication. 3. J. W. Baker and J. S. Pym, A remark on continuous bilinear mappings, Proc. Edingburgh Math. Soc. (2) 17 (1971), 245–248. 4. E. Beckenstein, L. Narici and C. Suffel,“Topological Algebras,” North Holland Publ. Comp., New York, 1977. 5. S. J. Bhatt and D. J. Karia, Uniqueness of the uniform norm with an application to topological algebras, Proc. Amer. Math. Soc. 116(2) (1992), 499–503. 6. S. J. 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