JOINT TOPOLOGICAL ZERO DIVISORS FOR A REAL BANACH ALGEBRA H. S. Mehta 1, R.D.Mehta 2, A. N. Roghelia 3

Transcription

1 Mathematics Today Vol.30 June & December 2014; Published in June 2015) ISSN 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, B.V.M. Engineering College, Vallabh Vidyanagar, India, 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.

2 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

3 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 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.

4 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.

5 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, [2] V. Muller, Spectral Theory of Linear Operators, Birkhasuser Verlag, Basel Boston Berlin, [3] Z. Slodkowski, On ideals consisting of joint topological divisors of zero, Studia Math., 48, 1973), pp [4] A. Wawrzynczyk, On ideals consisting of topological zero divisors, Studia Math., 142 8), 2000), pp [5] W. Zelazko, On a certain class of non-removable ideals in Banach Algebras, Studia Math., 44, 1972), pp