Mathematics Today Vol.30 June & December 2014; Published in June 2015) 54-58 ISSN 0976-3228 JOINT TOPOLOGICAL ZERO DIVISORS FOR A REAL BANACH ALGEBRA H. S. Mehta 1, R.D.Mehta 2, A. N. Roghelia 3 1,2 Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, India, 388120 3 B.V.M. Engineering College, Vallabh Vidyanagar, India, 388120 1 hs mehta@spuvvn.edu, 2 vvnspu@yahoo.co.in, 3 aakarkhyati@gmail.com Acceptance Date: May 11, 2015) Abstract: The concept of joint topological zero divisors for a complex Banach algebra is studied by A. Wawrzynczyk [4] and V. Muller [2]. We define this concept for a Real Banach algebra and extend the results in [4], for a real Banach algebra. We also relate this idea with certain ideals in the real algebra. Keywords: Joint topological zero divisors, Real Banach algebra. 1. Introduction The concept of topological zero divisor TZD) is well known in a complex Banach algebra and this idea was generalized to joint topological zero divisors joint TZD) by W. Zelazko 1972) [5] and then further developed by Z. Slodkowski 1973) [3]. Many concepts of complex Banach algebras have been extended to Real Banach algebras. We study the concept of joint topological zero divisor for a real Banach algebra. A real Banach algebra is a real Banach Space in which the norm is submultiplicative. Throughout this paper we assume that A is a real commutative Banach algebra with identity e. For terminologies and notations regarding a real Banach algebra, we refer to [1]. 2. Joint topological zero divisors For a 1,..., a n in A, define d a 1,..., a n ) = inf { n i=1 a iz : z A, z = 1}. Definition 2.1. An element a A is said to be a topological zero divisor in A if d a) = 0.
Joint Topological Zero Divisors for a Real Banach Algebra 55 Definition 2.2. We say that a non empty subset S of A consists of joint topological zero divisors if d a 1,..., a n ) = 0 for every finite subset {a 1,..., a n } S. In particular, if S is an ideal then it is called an ideal consisting of joint topological zero divisors. The following theorem gives equivalent definition for a set consisting of joint topological zero divisors. Theorem 2.3. A set M A consists of joint topological zero divisors if and only if a net b ) A, such that b = 1 and lim b a = 0 for each a M. Proof. Suppose M consists of joint topological divisors of zero. Therefore by definition d a 1,..., a n ) = 0 for every finite subset {a 1,..., a n } of M. Let F M be a finite subset and k N. Then there exists an element b F,k A such that b F,k = 1 and a F b F,k a 1. Define, the order F, k) F, k ) if and only if F F and k k. With this k order the set = {F, k) : F M, k N} is a directed set. Hence, b F,k ) F,k is a net with b F,k = 1 also lim F,k b F,k a = 0 for each a M. Conversely, suppose there exists a net b ) A, such that b = 1 for all and lim b a = 0 for each a M. Let {a 1,..., a n } M be a finite set. Then, lim b a i = 0 for each i. Now, for a fixed i and some k N there exists λ i Λ such that for all > λ i we have b a i < 1 k n. b a i < 1 k Take λ = max {λ i } then for all > λ we have for each i. Hence, for some > λ we can show that n i=1 b a i < 1 k. Therefore, d a 1,..., a n ) = inf { n i=1 a ib : b A, b = 1} = 0. So, M consists of joint topological zero divisors. Notations: The set of all ideals of A consisting of joint TZD is denoted by l A) while L A) denote the set of those elements of l A) which are maximal ideals of A. i We shall need the following result to prove our main Theorem 2.5. Proposition 2.4. Let K be a compact Hausdorff space and let A be a real subalgebra of C K) with identity. Suppose for f 1,..., f k A, every function in the ideal I A f 1,..., f k ) has a zero in K. Then for every g A there exists µ C, µ = s + it such that I A f1,..., f k, g s) 2 + t 2) has a zero in K. Proof. We use method of contradiction to prove this proposition. Suppose that there exists a g A such that for every µ = s + it C we can find φ µ j A, 1 j k + 1 such that u µ = k j=1 φµ j f j + φ µ k+1 g s) 2 + t 2) nowhere vanishes on K. The functions
56 H. S. Mehta, R.D.Mehta & A. N. Roghelia φ µ j can be chosen so that uµ > 1. Let be a closed disk in complex plane centered at 0 and g K). Claim 1: There exists a collection of function ψ µ j A and D > 0 such that ψ µ and k j=1 ψµ j f j + ψ µ k+1 g s) 2 + t 2) > 1 for every µ, µ=s+it. k+1 D For every fixed µ there exists r µ) > 0 such that for λ = s 1 + it 1 satisfying λ µ < r µ) we have k j=1 φµ j f j + φ µ k+1 g s1 ) 2 + t1) 2 = k j=1 φµ j f j + φ µ k+1 g s) 2 + t 2) +φ µ k+1 2 s s 1) g + s 2 1 s 2 ) + t 2 1 t 2 )) = u µ + φ µ k+1 2 s s 1) g + s 2 1 s 2 ) + t 2 1 t 2 )) > 1 But is compact, hence there exists a finite set {µ i } m i=1 and corresponding finite collection of functions { } φ µ i j such that for every µ, µ = s + it satisfying µ µ i < r µ i ) we have k j f j + φ µ i k+1 g s) 2 + t 2) > 1. j=1 φµ i Now, for a given µ let i 0 = min {i : µ µ i < r µ i )}. Also, define ψ µ j = φµ i 0 j take D = max { φ µ i } k+1 : 1 i m we have ψ µ k+1 D and k j=1 ψµ j f j + ψ µ k+1 g s) 2 + t 2) > 1. and Take r = 1 D and let {D ν i, r)} l i=1 be a finite cover of, where ν i, ν i = s i + it i. Define h i = u ν i = k j=1 ψν i j f j + ψ ν i k+1 g si ) 2 + t 2 i ) then hi > 1 for 1 i l, also the inverse function w i = h 1 i satisfy w i < 1. Let B be the smallest closed unital real subalgebra of C K) which contains A and the functions w i, 1 i l. Claim 2: The closed ideal M generated in B by the functions f j, 1 j k, is proper. Let F p) be a finite subset of N l. N j = n j1,..., n jl ) F p) and b Nj A. Define p = k j=1 f j N j b Nj w n j 1 1...w n j l l. The set of functions of the form p is dense in the ideal M. Let L = max ji n ji. We obtain q = h L 1...h L l p = k f j b Nj h L nj1 1...h L n j l l I A f 1,..., f k ) j=1 N j so q x) = 0 for some x K. Hence q is not invertible in B and hence p is not invertible. Thus, ideal M contains a dense subset of non invertible elements. Thus, it is proper.
Joint Topological Zero Divisors for a Real Banach Algebra 57 Every proper ideal in a commutative Banach algebra contained in some maximal ideal M. Now for real Banach algebra ker 1 M) consists of { φ, φ } for some φ Car B) Set of all nonzero homomorphisms from B to C). We take any one of them then we have φ M) = 0. Then for any g for which φ g) = s 0 + it 0 we have the function g s 0 ) 2 + t 2 0 belongs to kernel of φ. This implies that ν 0 := φ g). There exists i, 1 i l such that ν 0 ν i < r. Take u = k j=1 ψ ν i j f j + ψ ν i k+1 g s0 ) 2 ) + t 2 0 = h i + ψ ν i k+1 2 si s 0 ) g + ) )) s 2 0 s 2 i + t 2 0 t 2 i Also, we have uw i 1 < 1. Hence, uw i is invertible in B, so u is also invertible. This is a contradiction because u belongs to ideal generated by the k+1-tuple f 1,..., f k, g s) 2 +t 2 which is proper. So, it belongs to ker φ. We call the following result the projection property of the family of ideals consisting of TZD. Theorem 2.5. Let J A be an ideal consisting of TZD. Let a 1,..., a k J. For every c A, there exists λ C, λ = s+it such that the ideal generated by a 1,..., a k, c s) 2 +t 2 consists of TZD. Proof. Let χ : A C L A)) be Banach algebra homomorphism defined by χ a) = â /LA). Take A = χ A) and let f i = χ a i ), g = χ c). The functions f i, 1 i k, satisfy the assumptions of Proposition 2.4. Hence, there exists a λ = s + it C such that the Gelfand transform of an arbitrary elements of I A f1,..., f k, g s) 2 + t 2) vanishes at some point in L A). Hence, I A a1,..., a k, c s) 2 + t 2) consists of TZD. Acknowledgement: This research is supported by the SAP programme to the Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar by UGC. We are thankful to the referee for some suggestions.
58 H. S. Mehta, R.D.Mehta & A. N. Roghelia References [1] S. H. Kulkarni & B.V. Limaye, Real Function Algebras, Marcel Dekker, Inc. New York, 1992. [2] V. Muller, Spectral Theory of Linear Operators, Birkhasuser Verlag, Basel Boston Berlin, 2007. [3] Z. Slodkowski, On ideals consisting of joint topological divisors of zero, Studia Math., 48, 1973), pp. 83-88. [4] A. Wawrzynczyk, On ideals consisting of topological zero divisors, Studia Math., 142 8), 2000), pp. 245-251. [5] W. Zelazko, On a certain class of non-removable ideals in Banach Algebras, Studia Math., 44, 1972), pp. 87-92.