 Advances in Pure Mathematics, 2013, 3, 660-665 Published Online November 2013 (http://www.scirp.org/journal/apm) http://dx.doi.org/10.4236/apm.2013.38088 Open Access APM A Real p-Homogeneous Seminorm with Square Property Is Submultiplicative Mohammed El Azhari Department of Mathematics, Ecole Normale Supérieure, Rabat, Morocco Email: mohammed.elazhari@yahoo.fr Received October 10, 2013; revised November 10, 2013; accepted November 17, 2013 Copyright © 2013 Mohammed El Azhari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We give a functional representation theorem for a class of real p-Banach algebras. This theorem is used to show that every p-homogeneous seminorm with square property on a real associative algebra is submultiplicative. Keywords: Functional Representation; p-Homogeneous Seminorm; Square Property; Submultiplicative 1. Introduction J. Arhippainen  has obtained the following result: Theorem 1 of . Let q be a p-homogeneous semi- norm with square property on a complex associative algebra A. The n 1) Ker(q) is an ideal of A; 2) The quotient algebra A/Ker(q) is commutative; 3) q is submultiplicative; 4) 1pq is a submultiplicative seminorm on A. This result is a positive answer to a problem posed in  and considered in [3-5]. The proofs of (3) and (4) depend on (2) which is obtained by using a locally bounded version of the Hirschfeld-Zelazko Theorem [1, Lemma 1]. This method can not be used in a real algebra; if q is the usual norm defined on the real algebra H of quaternions, Ker and H/Ker is non- commutative, then the assertion (2) does not hold in the real case.  0qHqThe purpose of this paper is to provide a real algebra analogue of the above Arhippainen Theorem, and this im-proves the result in . Our method is based on a func-tional representation theorem which we will establish; it is an extens ion of the Abel-Jar osz Theore m [7 , Theore m 1] to real p-Banach algebras. We also give a functional rep-resentation theorem for a class of complex p-Banach a lg e-bras. As a consequence, we obtain the main result in . 2. Preliminaries Let A be an associative algebra over the field K = R or C. Let 0,1p, a map .: 0,A is a p-homo- geneous seminorm if for in A and ,ab in K, aba b and paa. Moreover, if 0a imply that 0a, . is called a p-homo- geneous norm. A 1-homogeneous seminorm (resp.norm) is called a seminorm (resp.norm). . is submultipli- cative if abab for all in A. ,ab . has the square property if 22aa for all a. If A. is a submultiplicative p-homogeneous norm on A, then ,.A is called a p-normed algebra, we denote by M(A) the set of all nonzero continuous multiplicative linear functionals on A. A complete p-normed algebra is called a p-Banach algebra. A uniform p-normed algebra is a p-normed algebra ,.A such that 22aa for all aA. Let A be a complex algebra with unit e, the spectrum of an element aA is defined by 1,Sp aCeaA where 1A is the set of all invertible elements of A. Let A be a real algebra with unit e, the spectrum of aA is defined by  221,SpasitCaseteA . Let A be an algebra, the spectral radius of an element aA is defined by sup ,ra Spa. Let ,.A be a p-normed algebra, the limit 1lim npnna exists for each aA, and if A is complete, we have M. EL AZHARI 6611lim npnnra a for all . A -algebra is a aAcomplex algebra with a mapping :,AAa a , such that, for in A and ,ab ,C  ,,,.aaababaaabba The map is called an involution on A. An element is said to be hermitian if . The set of all hermitian elements of A is denoted by H(A).  aAaa3. A Functional Representation Theorem for a Class of Real p-Banach Algebras We will need the following result due to B. Aupetit and J. Zemanek ([9,10]) , th eir alg ebr aic approach works for real p-Banach algebras. Theorem 3.1. Let A be a real p-Banach algebra with unit. If there is a positive constant  such that for all in  rab rarb,abA, then for every irreducible representation  of A on a real linear space , the algebra EA is isomorphic (algebrai- cally) to its commutant in the algebra of all linear transformations on . LEELet A be a real p-Banach algebra with unit such that 1pamra for some positive constant and all maA. Let XA be the set of all nonzero multi- plicative linear functionals from A into the noncom- mutative algebra H of quaternions. For aA, we consider the map  :,JaXAHJaxxa for all xXA. We endow XA with the weakest topology such that all the functions ,Jaa A, are continuous. The map :,,JACXAH Jaa, is a homomorphism from A into the real algebra of all continuous function s from XA into H. Theorem 3.2. If  is an irreducible representation of A, then A is isomorphic to , or RCH. Proof. Let and , we have ,ab A1nnnnaba b, then 111npnppaba b. Letting , we obtain n 2rab mrarb. Let  be an irreducible representation of A on a real linear space . By Theorem 3.1, EA is isomorphic to its commutant Q in the algebra LEy of all linear transformations on . Let 0 be a fixed nonzero element in . For EEyE, we consider 0inf, and EyaaAayBy the same proof as in [11, Lemma 6.5], .E is a p-norm on and Q is a real div ision p-normed algebra of continuous linear operators on . By , Q is isomorphic to , or EERCH. Proposition 3.3. A is semisimple and XA is a nonempty set which separates the elements of A. Proof. By the condition 1pamra for all aA, we deduce that A is semisimple. Let be a nonzero element in aA, since A is semisimple, there is an irre- ducible representation  of A such that 0a. By Theorem 3.2, there is :AHTo an isomor- phism (into). We consider the map ,:TAH is a multiplicative linear functional. Moreover, 0Ta a since 0a and  is injective. Proposition 3. 4. 1) 1pxaa for all and aA;xXA 2) An element is invertible in A if and only if aJa is invertible in ,;CXA H 3) aaASp aSpJ for all .Proof. (1): Since H is a real uniform Banach algebra under the usual norm  1.,pHAxarxa raa for all aA and .xXA (2): The direct implication is obvious. Conversely, let  be an irreducible representation of A. By Theorem 3.2, there is :AH an isomorphism (into). Since AoX and Ja is invertible, 0Jao a , then 0.a Consequently, a is invertible. (3): sit Spa iff 221aseteAIff e1,CXAH22Jaset by (2) Iff   212,JasJetJeCXA H Iff .sitSp Ja Proposition 3.5. XA is a Hausdorff compact space. Proof. Let x1, x2 in 1,2XAx x, there is an element aA such that 12xaxa, i.e. 12Jax Jax, so XA is Hausdorff. Let and aA1,paKqHqa , y. aK is compact in H. Let K be the topological product of aK for all is compact by the Tychonoff Theorem. By Proposition 3.4(1), ,KaAXA is a subset of K. It is easy to see that the topology of Open Access APM M. EL AZHARI 662 XA is the relative topology from K and that XA is closed in . Then KXA:, is compact. Theorem 3.6. The m a p ,,JAC HaJaXA is an isomorphism (into) such that 111ppsmaaaJ for all , where aA.s is the supnorm on ,A HCX . If , we have 1m1psaJa for all .aAProof. By Proposition 3.3, J is an injective homo- morphism. Let by Proposition 3.4(3), ,aA sra rJaJa since is a real uniform Banach algebra A,CXHunder the supnorm .s. Moreover, 1psJaa by Proposition 3.4(1). Then 111.ppsmara Jaa  As an application, we obtain an extension of the Kul- karni Theorem [13, Theorem 1] to real p-Banach alge- bras. Theorem 3.7. Let be an element in aA such that then belongs to the center of Sp a,Ra.A Proof. By Theorem 3.6, :,JACXASp aRAH is an isomorphism (into). Let a with Let .xXA and xast where sR and 12 3.ttitjtk  Suppose that We have 0.t222 222123xas ttttt, Then 220xa st. Consequently si tSpxaSpa with 0,t a contradiction. Then ,JaCXAR and  JaJb JbJa for all in b,A i.e. for all b in 0Jab baA. Since J is injective, for all b in 0ab ba .A 4. A Functional Representation Theorem for a Class of Complex p-Banach Algebras Let . be a submultiplicative p-homogeneous se- minorm on a complex algebra A. For ,aAa is de- fined as follows: 1inf n1piaaiA, where the infimum is taken over all decompositions of satisfying the condition 1i, 1aniaa,, .naa By [14, Theorem 1], . is a submultiplicative seminorm on ,A it is called the support seminorm of .. Also, it is shown  the following result: Theorem 2 of . Let A be a complex algebra, . a submultiplicative p-homogeneous seminorm on A, and . the support seminorm of .. Then 11lim limnnpnnnnaa  for all .aA In the proof of this theorem, Xia Dao-Xing uses the following inequality: If 1maa a and , then 1n1111!!!mmpnmnmnaaa . If the algebra is commutative, 11111!!! mmnnmmnmaaanaa , then 1111!.!!mmpnmnmnaa a This inequality is not justified in the noncommutative case; if the algebra is noncommutative, we only have 1111!!!mmnmnmnaaa . For the sequel, we will use Theorem 2 of  in the commutative case. Theorem 4.1. Let ,.A be a complex p-normed algebra such that 22ama for some positive con- stant and all .maA Then 11ppaa ma and 222pama for all aA, where . is the support seminorm of .. Proof. The completion of B,.A is a p-Banach algebra such that 2bmb2 for all b, it is com- mutative by [1, Lemma 1], so BA is commutative. By induction, 2122nnnam a for all a and , A1nthen 1lim nnnama for all By the com- .aAmutative version of [14, Theorem 2], we have Open Access APM M. EL AZHARI 663111111limlimnppnpnnppnnaa mamama for all . From the above inequalities, aA212222pppaa mama . Corollary 4.2. Let ,.A be a complex uniform p-normed algebra. Then 1paa for all .aATheorem 4.3. Let ,.A be a complex p-Banach algebra with unit such that 22ama for some posi- tive constant and all . Then the Gelfand map is an isomorphism (into) such that mMaA:GA CA2111ppppsma maGaaa  for all where ,aA.s is the supnorm on .CMA Proof. A is commutative by [1, Le mma 1]. By Theore m 4.1, 11ppaa ma for all , then aA,.A is a complex commutative Banach algebra with unit. Clearly  ,. ,.MAMAMA is a nonempty compact space. As in the proof of Theorem 4.1, we have  11111limsup ,.npnnpppsam amfafMmGa maA Let from the above inequalities, ,aA211pp 1ppsma maGaaa . Corollary 4.4. Let ,.A be a complex uniform p-Banach algebra with unit. Then the Gelfand map is an isomorphism (into) such that :GA CMA1psaa Ga for all . aATheorem 4.5. Let ,.A be a complex p-normed -algebra with unit such that 1) 22ama for some positive constant and all m;aA2) Every element in HA has a real spectrum in the completion of .BA Then the involution is continuous on A and the Gelfand map :GB MBC is a -isomorphism such that 211pppsmb Gbb for all in b.BProof. By Theorem 4.3, it remains to show that the involution * is continuous on A, Gb Gb for all ,bB and G is surjective. Let ,hHA,BSphf hfM BR by (2). Let ,aA we have with 1ahih212,hhHA. Let ,fMB  121 212 12fafh ihfhifhfhifhfh ihfa   since 1fh and 2fh are real. Then GaGa for all .aA By Theorem 4.3,  211ppspssma GaGaGaa for all ,aA then 2ama for all .aA Con- sequently, the involution  is continuous on A and can be extended to a continuous involution on which we will also denote by B. Let there exists a sequence ,bBnna in A such that n. Since the in- volution on B and the Gelfand map abCMB:GB are continuous, we have nGa Gb and ,nGa Gb then .Gb Gb By the Stone-Weierstrass Theorem, we deduce that is surjective. GAs a consequence, we obtain the main result in . Corollary 4.6. Let A be a complex uniform p- normed -algebra with unit such that every element in HA has a real spectrum in the completion of B.A then is a commutative -algebra. B C5. The Main Result Theorem 5.1. Let A be a real associative algebra. Every p-homogeneous seminorm with square pro- qperty on A is submultiplicative and 1pq is a sub- multiplicative seminor m on .A Proof. By , there exists a positive con stant such that mqabmqaqb for all . ,abAKerq is an ideal of ,A the norm . on the quotient algebra KerAq defined by qKeraqa is a p-norm with square property . Define Ker Keraqmaq for all .aA Let ,,ab A Open Access APM M. EL AZHARI 664    2Ker KerKer KerKerKer ,ab qmab qma qb qaqbq   then Ker, .Aq is a real p-normed algebra. Let ,aA 2222121Ker KerKerKerKeraqmaqma qmmaqma q i.e.  22KerKer .aqma q The completion of BKer, .Aq satisfies also the property 22bmb for all and by induc- ,bBtion 212 2nnnbm b for all and then bB1,npbmrb for all We consider two cases: .bBB is unital: By section 3, XB: is a nonempty com- pact space and the map ,JBCXBH is an iso- morphism (into). By Proposition 3.4(3), rb rJb for all . Let bB,bB pppsbmrb mrJbmJb since is a real uniform Banach algebra under the supnorm ,CXB H.s. Then 1psbmb Jb for all KerbA q, so . is submultiplicative and 1.p is a submultiplicative norm. Consequently, is qsubmultiplicative and 1pq is a submultiplicative semi- norm. B is not unital: Let 1 be the algebra obtained from by adjoining the unit. By the same proof of [15, Lemma 2] which works for real p-Banach algebras, there exists a p-norm on such that BBBN11) is a real p-Banach algebra with unit; 1,BN2)  113pBNbmr b for all ; 1bB3) and N. are equivalent o n .BBy section 3, 1XB1: is a nonempty compact space and the map 1,JBCXBHbB is an isomorphism (into). Let ,  1ppBBbmrbmrb by (3) pmr J b by Proposition 3.4(3) psmJb by the square property of the supnorm. Then 1psbmb Jb for all Ker ,bA q so . is submultiplicative and 1.p is a submultipli- cative norm. Consequently, is submultiplicative and q1pq is a submultiplicative seminorm. 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