In this paper, we first derive some results by using the Gelfand spectrum and spectrum in FLM algebras. Then, the characterizations of multiplicative linear mappings are also discussed in these algebras

Ali Naziri-Kordkandi

Ali Zohri

Bahman Yousefi

Fariba Ershad

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Sahand Communications in Mathematical Analysis (SCMA) Vol. 13 No. 1 (2019), 93-100http://scma.maragheh.ac.irDOI: 10.22130/scma.2017.28459A Class of New Results in FLM AlgebrasAli Naziri-Kordkandi1∗, Ali Zohri2, Fariba Ershad3, and Bahman Yousefi4Abstract. In this paper, we first derive some results by using theGelfand spectrum and spectrum in FLM algebras. Then, the char-acterizations of multiplicative linear mappings are also discussed inthese algebras.1. IntroductionNon-normed topological algebras were initially introduced around theyear 1950 for the manipulation of certain classes of these algebras thatappeared naturally in mathematics and physics. Some results concerningsuch topological algebras had been published earlier in 1947 by R. Arens[6]. It was in 1952 that Arens and Michael [6] independently publishedthe first systematic study on locally m-convex algebras, which consti-tutes an important class of non-normed topological algebras. Here, wewould like to mention about the predictions made by the famous Sovietmathematician M.A. Naimark, an expert in the area of Banach algebras,in 1950 regarding the importance of non-normed algebras and the devel-opment of their related theory. During his study concerning cosmology,G. Lassner [6] realized that the theory of normed topological algebraswas insufficient for his study purposes.Ansari [1] introduced the notion of fundamental topological spaces andalgebras and proved the Cohen’s factorization theorem for these alge-bras. Fundamental locally multiplicative (FLM) topological algebraswith a property similar to the normed algebras were introduced laterby Ansari [2]. Some celebrated theorems of Banach algebras have beengeneralized for FLM algebras in the past studies [2, 3].2010 Mathematics Subject Classification. 46H05.Key words and phrases. Fundamental topological algebra, FLM algebra, Spec-trum, Multiplicative linear functional.Received: 14 February 2017, Accepted: 20 May 2017.∗ Corresponding author.9394 A. NAZIRI-KORDKANDI, A. ZOHRI, F. ERSHAD, AND B. YOUSEFIThe aim of this paper is twofold. Firstly, we derive some results byusing the Gelfand spectrum and spectrum in FLM algebras. Secondly,we discuss the problem of how to recognize a multiplicative linear map-ping in these algebras. This paper is divided into the following sections.In Section 2, we have gathered a collection of definitions and symbols,and in Section 3, we derive some results by using the Gelfand spectrumand spectrum in FLM algebras. In Section 4, the characterizations ofmultiplicative linear mappings are discussed in these algebras.2. Definitions and SymbolsIn this section, we present a collection of definitions and symbols,which are also listed in the references.Definition 2.1. A topological linear space A is said to be fundamentalone if there exists b > 1 such that for every sequence {xn} of A, theconvergence of bn(xn − xn−1) to zero in A implies that {xn} is Cauchy.Definition 2.2. A fundamental topological algebra is an algebra whoseunderlying topological linear space is fundamental.Definition 2.3. A fundamental topological algebra is said to be locallymultiplicative, if there exists a neighbourhood U0 of zero such that forevery neighbourhood V of zero, the sufficiently large powers of U0 lie inV . Such an algebra is known an FLM algebra.Definition 2.4. Let (A, dA) be a metrizable topological algebra. Wesay that A is a sub-multiplicative metrizable topological algebra ifdA(0, xy) ≤ dA(0, x)dA(0, y),for all x, y ∈ A. For abbreviation, we denote dA(0, x) by DA(x) for anyx ∈ A .Definition 2.5. Let A be an algebra with unit element. The set of allinvertible elements of A is denoted by Inv(A).Definition 2.6. Let A be an algebra with the unit element e. Thespectrum of x ∈ A is denoted by sp(x) and defined as:sp(x) = {λ ∈ C : λe− x is not invertible}.Definition 2.7. Let A be a complex algebra with unit element.i) Define the Gelfand spectrum of A as ϕA := Hom(A,C) with thetopology of pointwise convergence on A. Note that 0 ̸∈ ϕA be-cause we require homomorphisms to respect the unit elements.A CLASS OF NEW RESULTS IN FLM ALGEBRAS 95ii) Each element a ∈ A gives rise to a continuous function â : ϕA →C by â(φ) := φ(a). The function â is called the Gelfand trans-form of a. The map a → â : A → C(ϕA), which is a homomor-phism of unital algebras, is called the Gelfand homomorphismof the algebra A.3. Some New Results by Using the Gelfand Spectrum andSpectrum in FLM AlgebraSpectral theory and the Gelfand spectrum have already been studiedin Banach algebras. Now we would like to give similar results for FLMalgebras.Theorem 3.1 (The Gelfand homomorphism). Let A be a commutativecomplete metrizable FLM algebra with the unit element e and ϕA be theGelfand spectrum of A. Then the following statements hold:i) Every element a ∈ A satisfiessp(x) = {φ(a) : φ ∈ ϕA} = â(φA);ii) The Gelfand spectrum ϕA is a compact Hausdorff space.Proof. The proof follows easily from [3, 5.5] and [3, 5.6]. □Theorem 3.2. Let A be a commutative complete metrizable FLM alge-bra with the unit element e and ϕA be the Gelfand spectrum of A. Thenthe following statements hold:i) sp(x+ y) ⊆ sp(x) + sp(y);ii) sp(xy) ⊆ sp(x)sp(y).Proof. By the preceding theorem, we havesp(x+ y) = Im((x+ y)̂) = Im(x̂+ ŷ) ⊆ Im(x̂) + Im(ŷ) = sp(x) + sp(y).The analogous calculation holds for the product xy. □The fact that exp(0) = 1 and exp(−a) exp(a) = 1 and for all x, y ∈ Awhen xy = yx, exp(x) exp(y) = exp(x+ y), have been considered in [3,5.1]. These are of course some well-known results of Banach algebras.Now, we would like to prove the following theorem for commutativecomplete metrizable FLM algebras.Theorem 3.3. Let A be a commutative complete metrizable FLM al-gebra with the unit element eA and ϕA be the Gelfand spectrum of A .Suppose thatexp(a) = exp(b), â = b̂, a, b ∈ A,then a = b.96 A. NAZIRI-KORDKANDI, A. ZOHRI, F. ERSHAD, AND B. YOUSEFIProof. By [3, 5.5], and Definition 2.7, we haveâ(ϕA) = {φ(a) : φ ∈ ϕA } = sp(a).Set c = b− a or c+ a = b. Then we haveexp(a) exp(c) = exp(a+ c) = exp(b),hence, exp(c) = eA and soc(eA +∞∑k=1ck(k + 1)!)= 0.By hypothesis, since c ∈ rad(A), it follows from [2, 4.1] that(eA +∞∑k=1ck(k + 1)!)∈ Inv(A),and so c = 0. Thus, a = b. □The following theorem is another application of J. Zemanek’s theoremfor complete metrizable FLM algebras [9, 3.4].Theorem 3.4. Let A be a semisimple complete metrizable FLM algebrawith sub-multiplicative meter dA. If x , y ∈ A do not commute, thenthere exists a ∈ A such thatsp(a− xy) ̸= sp(a− yx).Proof. Let sp(a−xy) = sp(a−yx) for all a ∈ A. Writing a+xy insteadof a, it follows thatsp(a) = sp(a+ xy − yx), ∀a ∈ A,by [9, 3.4], xy − yx ∈ rad(B). Since A is semisimple, we conclude thatxy − yx = 0 or xy = yx, which is a contradiction. □4. Multiplicative Linear MappingsA characterization of multiplicative linear functionals in a Banachalgebras was given by Gleason-Kahane-Zelazko [4]. A version of theGleason-Kahane-Zelazko theorem is also proved for FLM algebras [3,5.5]. Now we weaken the assumption of this theorem and prove it forsub-multiplicative FLM algebras without linearity. Before giving themain theorem, we need the following lemma.Lemma 4.1. Let A be a complete metrizable FLM algebra with the unitelement e and φ be an R-linear on A such that φ(x) ∈ sp(x) for eachx ∈ A, then φ is C-linear.Proof. The proof of this lemma follows by the same reasoning as in [7,2.1]. □A CLASS OF NEW RESULTS IN FLM ALGEBRAS 97Theorem 4.2. Let A be a complete metrizable FLM algebra with thesub-multiplicative meter dA and the unit element e. Let f : A → Csatisfy f(0) = 0 and f(x) − (y) ∈ sp(x − y) for each x, y ∈ A. Then fis multiplicative and linear.Proof. First we may assume that A is separable. Suppose that f has adifferential at a point a ∈ A. We havef(a+ rx)− f(a)r∈ sp(a+ rx− a)r= sp(x), r ∈ R, r ̸= 0, x ∈ A.So,(Df)a(x) = limr→0f(a+ rx)− f(a)r∈ sp(x).Thus, by Lemma 4.1, the differential is C-linear. On the other hand, by[9], the following relation holds for spectral radiusr(x) ≤ DA(x).Hence, we have|f(x)− f(y)| ∈ |sp(x− y)| ≤ DA(x− y).Thus, f is a Lipschitz function. By [7, 2.3] and [7, 2.4], we obtain that fis an entire function. As in [7, 1.2], we define the function fa,b : C → Cby fa,b(z) = f(az + b) , for a, b ∈ A. Therefore, fa,b is Lipschitzand entire. So, it is affine. By the same reasoning as in [6, 1.2], and theGleason-Kahane-Zelazko theorem for FLM algebras [3, 5.5], we concludethat f is linear and multiplicative, when A is separable. Now we considerthe general case. Let a, b ∈ A. Clearly [e,a,b] (the subalgebra of Agenerated by e,a,b in A) is the FLM subalgebra of A. The function f inTheorem 4.2, restricted to the subalgebra [e,a,b] of A satisfies conditionsof the theorem. As [e,a,b] is separable, from the preceding part of theproof, it follows that f∣∣[e,a,b] is multiplicative and linear. Since a and bis arbitrary, we deduce that f is multiplicative and linear in the wholeof A. □The next results show the applications of the Gleason-Kahane-Zelazkotheorem.Theorem 4.3. Let A be a complete metrizable FLM algebra with thesub-multiplicative meter dA and the unit element e and let B be a semisim-ple commutative complete metrizable FLM algebra with the unit elemente′. Suppose that T is a map from A into B such thatsp(T (x) + T (y)) ⊆ sp(x+ y), ∀x, y ∈ A.Then T is linear and multiplicative.98 A. NAZIRI-KORDKANDI, A. ZOHRI, F. ERSHAD, AND B. YOUSEFIProof. Since B is semisimple and sp(T (y)+T (−y)) ⊆ sp(y+(−y)) = {0}for each y ∈ A, we obtain that T (−y) = −T (y) for each y ∈ A. Let φbe an element in ϕB . Then φoT is a function from A into C such thatφoT (0) = 0, and by the Gleason-Kahane-Zelazko theorem in [3, 5.5], wehaveφoT (x)− φoT (y) = φ (T (x)− T (y)) ∈ sp (T (x)− T (y)= sp (T (x) + T (−y)) ⊆ sp(x− y).Thus, φoT is linear and multiplicative by Theorem 4.2. Now we haveφ(T (xy)− T (xy)) = φoT (xy)− φoT (x)φoT (y) = 0.Since B is semisimple and φ ∈ ϕB is arbitrary, it follows thatT (xy) = T (x)T (y).□Theorem 4.4. Let A be a complete metrizable FLM algebra with thesub-multiplicative meter dA and the unit element e and let B be a semisim-ple commutative complete metrizable FLM algebra with the unit elemente′, and p(u, v) = λu+µv, (λµ ̸= 0) be two-variable polynomials. Supposethat T is a map from A into B such thatsp(p(Tf, Tg)) ⊆ sp(p(f, g)), ∀f, g ∈ A.If λ+ µ ̸= 0, then T is linear and multiplicative.Proof. Let f, g ∈ A. Since λ ̸= 0, we havesp(Tf +µλTg)⊆ sp(f +µλg),so,sp(T (−µλg) +µλTg)⊆ sp(−µλg +µλg)= {0}.By taking f = −µλg, we obtainT (−µλg) = −µλTg, ∀g ∈ A.Since B is semisimple, it follows thatsp (Tf − Tg) = sp(Tf − T(µλ− λµg))= sp(Tf +µλT(−λµg))⊆ sp(f − g)A CLASS OF NEW RESULTS IN FLM ALGEBRAS 99for all f, g ∈ A. Now we prove that T (0) = 0 if λ + µ ̸= 0. By takingf = g = 0, we havesp (λT (0) + µT (0)) ⊆ sp(λ.0 + µ.0) = {0}.Hence, T (0) = 0. The result follows from the proof of Theorem 4.3. □Corollary 4.5. If λ + µ = 0, in Theorem 4.4, then T − T (0) is linearand multiplicative.Proof. We consider a map W from A into B by Wf = Tf − T (0).Clearly W is onto and we havesp(Wf −Wg) ⊆ sp(f − g), ∀f, g ∈ A.So, the result follows from the proof of Theorem 4.3. □Theorem 4.6. Let A and B be two complete metrizable FLM algebraswith unit elements eA and eB respectively, and let B be commutativeand semisimple. If T : A → B is a linear map such that T (eA) = eBand T (a) ∈ Inv(B) for a ∈ Inv(A), then T is multiplicative.Proof. Let φ ∈ ϕB and x, y ∈ A. Then we haveφoT (a) = φ (T (a)) ̸= 0, φoT (eA) = 1,this implies that kerφoT ⊆ sin g A. By the Gleason-Kahane-Zelazkotheorem for FLM algebras [3, 5.5], we conclude that φoT is multiplica-tive. The remainder of the proof follows from Theorem 4.3. □Theorem 4.7. Let A be a complete metrizable FLM algebra with theunit element eA and let B be a semisimple complete metrizable FLMalgebra with the sub-multiplicative meter dB and the unit element eB. Ifφ : A → B is surjective linear map such that for all x ∈ Asp(x) ̸= ∅, sp(x) = sp (φ(x)) ,then φ(eA) = eB.Proof. We havesp(φ(eA)) = sp(eA) = 1,and so φ(eA) = eB + q where q ∈ B is quasi-nilpotent. For all x ∈ A ,we obtain1 + sp (q + φ(x)) = sp (eB + q + φ(x))= sp (φ(eA) + φ(x))= sp (φ (eA + x))= sp (eA + x)= 1 + sp(x),100 A. NAZIRI-KORDKANDI, A. ZOHRI, F. ERSHAD, AND B. YOUSEFIhence,sp(q + φ(x)) = sp(x) = sp(φ(x)).By [9, 3.4], q ∈ rad(B). Since B is semisimple, we conclude that φ(eA) =eB. □Acknowledgment. The authors would like to thank the referees forhelpful suggestions.References1. E. Ansari-Piri, A class of factorable topological algebras, Proc. Edin. Math. Soc.,33 (1990), pp. 53-59.2. E. Ansari-Piri, Topics on fundamental topological algebras, Honam math. J., 23(2001), pp. 59-66.3. E. Ansari-Piri, The linear functionals on fundamental locally multiplicative topo-logical algebras, Turk. J. Math., 33 (2010), pp. 385-391.4. F.F. Bonsal and J. Doncan, Complete normed algebras, Springer-Verlag, Berlin1937.5. H.G. Dales, Banach algebras and automatic continuity, Clarendon Press, Oxford,2000.6. M. Fragoulopoulou, Topological algebras with involution, Elsevier, 2005.7. S. Kowalski and Z. Slodkowaski, A characterization of multiplicative linear func-tionals in Banach algebras, Studia Math, 67 (1980), pp. 215-223.8. W. Rudin, Functional analysis, McGraw- Hill, 1973.9. A. Zohri and A. Jabbari, Generalization of some properties of Banach algebras tofundamental locally multiplicative topological algebras, Turk J Math, 35 (2011),pp. 1-7.1 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, I.R. of Iran.E-mail address: ali naziri@pnu.ac.ir2 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, I.R. of Iran.E-mail address: zohri a@pnu.ac.ir3 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, I.R. of Iran.E-mail address: fershad@pnu.ac.ir4 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, I.R. of Iran.E-mail address: b yousefi@pnu.ac.ir

A class of new results in FLM algebras

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A class of new results in FLM algebras

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