9_441_gordji.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 4, 2009, (574-577) ISSN 1307-5543 – www.ejpam.com Second Duals of Measure Algebras M. Eshaghi Gordji1∗, and A. Ebadian2 1 Department of Mathematics, Semnan University, Semnan, Iran 2 Department of Mathematics, Urmia University, Urmia, Iran Abstract. In this paper we show that M(G)∗∗ determines G when G is a compact topological group. It is a new proof for theorem of Gharamani and Mcclure. 2000 Mathematics Subject Classifications: 46HXX Key Words and Phrases: Topological group, Arens product, Isomorphism The second dual space A ∗∗ of a Banach algebra A admits the Banach algebra product known as first (left) Arens product. This product extends the product of A as canonically embedded in A ∗∗. We briefly recall the definition of this product. For m, n ∈A ∗∗, their first (left) Arens product indicated by mn is given by 〈mn, f 〉 = 〈m, nf 〉 ( f ∈A ∗), where nf ∈A ∗ is defined by 〈nf , a〉 = 〈n, f a〉 (a ∈A ). ∗Corresponding author. Email addresses: madjid.eshaghi�gmail. om & madjideg�walla. om (M. Gordji),ebadian.ali�gmail. om (A. Ebadian) http://www.ejpam.com 574 c© 2009 EJPAM All rights reserved. M. Gordji and A. Ebadian / Eur. J. Pure Appl. Math, 2 (2009), (574-577) 575 (See [1] and [2]). Wendel in [6] proved that for locally compact groups G1 and G2, the group algebras L1(G1) and L1(G2) are isometrically isomorphic if and only if G1 and G2 are isomorphic in the category of topological groups. Johnson in [5] proved that the algebra M(G) determines G when G is a locally compact group. In [3] Ghahramani and Lau have proved that L1(G) ∗∗ determines G when G is a locally compact group. Ghahramani and Mcclure in [4] proved that the algebra (M(G)) ∗∗ determines G when G is a compact topological group. In this paper we define some new ideals in Banach algebras and we apply this ideals to consider a new proof to show that (M(G))∗∗ determines G when G is compact. Let A be a Banach algebra. We consider Zl(A ) := {a ∈A : A ∗∗ · b⊣ ⊆ cA}. It is easy to show that Zl(A ) is a two sided ideal ofA so it is a left ideal ofA ∗∗. Also Zl(A ) is the union of all two sided ideals ofA which are left ideals ofA ∗∗. First we prove the following lemma. Lemma 1. Let θ :A →B be an isometrically isomorphism between Banach algebras. Then θ (Zl(A )) = Zl(B). Proof. Let θ :A →B be an isometrically isomorphism between Banach algebras. Then θ ′′ is a isometrically isomorphism between Banach algebras A ∗∗ and B∗∗. Let a ∈ Zl(A ) and b′′ ∈B∗∗. Then there exists a′′ ∈A ∗∗ such that b′′ = θ ′′(a′′). Thus b′′Õθ (a) = θ ′′(a′′)Õθ (a) = θ ′′(a′′ba) =Úθ (a′′b)a ∈Øθ (A ) = cB . Then θ (Zl(A ))⊂ Zl(B). � Theorem 1. Let G be a compact group. Then Zl((M(G)) ∗∗ ) = π ′′ (Ł1(G))∗∗. Proof. Let (eα) be a bounded approximate identity of L1(G) with bound 1, and with cluster point E ∈ L1(G) ∗∗ . We denote π : L1(G) −→ M(G) the inclusion map, M. Gordji and A. Ebadian / Eur. J. Pure Appl. Math, 2 (2009), (574-577) 576 then the map m 7−→ (π ′′ (E))bm : M(G)−→ π ′′ (L1(G)∗∗) is isometric embedding. We denote this map with ΓE . Since the restriction of ΓE to L1(G) is identity map, then ΓE(m) ∈Ûπ(L1(G)) if and only if m ∈ L1(G). It is easy to show that ΓE ′′ is isometrically embedding from (M(G)∗∗) into π ′′′′ ((L1(G))∗∗∗∗). The restriction of ΓE ′′ to π ′′ (L1(G)∗∗) is identity map, then for every m′′ ∈ (M(G)∗∗), ΓE ′′ (m ′′ ) ∈ Ûπ′′(L1(G)∗∗) if and only if m ′′ ∈ Ûπ′′(L1(G)∗∗). Let now m′′ ∈ Zl((M(G) ∗∗)), then (M(G))∗∗∗∗Óm′′ ⊆Û(M(G)∗∗). Thus π ′′′′ (L1(G))∗∗∗∗Óm′′ ⊆Û(M(G)∗∗). (1) On the other hand, we have direct sum decompositions (L1(G))∗∗∗∗ =ÛL1(G)∗∗⊕Û(L1(G)∗) ⊥ (2) and (M(G))∗∗∗∗ =ÚM(G)∗∗⊕Û(M(G)∗) ⊥ . (3) So we have π ′′′′ (Û(L1(G)∗) ⊥ ) ⊆Û(M(G)∗) ⊥ . (4) Since π ′′′′ (L1(G)∗∗∗∗) is an ideal of M(G)∗∗∗∗, then by (2) and (4), we have π ′′′′ (L1(G))∗∗∗∗Óm′′ ⊆ [(Û(M(G)∗∗))∩π ′′′′ (L1(G))∗∗∗∗] = Ûπ ′′ (L1(G)∗∗). Therefore ΓE ′′(m′′) ∈ Û π ′′ (L1(G)∗∗) and m′′ ∈ π′′(L1(G)∗∗), hence, Zl(M(G) ∗∗) ⊆ π′′(L1(G)∗∗). On the other hand since G is compact then π′′(L1(G) ∗∗ ) is a two sided ideal of π′′′′(L1(G)∗∗∗∗), so Zl(π ′′(L1(G)∗∗)) = π′′(L1(G)∗∗) and Zl(π ′′(L1(G)∗∗)) is a two sided ideal of M(G)∗∗∗∗. Hence, π′′(L1(G)∗∗)⊆ Zl(M(G) ∗∗). � We now apply above theorem to show that M(G)∗∗ determines G when G is a compact topological group. It is a new proof for the main result of [4]. By Lemma 1 and Theorem 1 we have the following. REFERENCES 577 Corollary 1 (Theorem 7 of 4). . If G1 and G2 are compact groups, and if θ is an isometric isomorphism from M(G1) ∗∗ onto M(G2) ∗∗, then θ (L1(G1) ∗∗) = L1(G2) ∗∗. Since L1(G)∗∗ determines G [3] , then we have Corollary 2. If G is a compact group, then M(G)∗∗ determines G. References [1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2(1951), 839–848. [2] J. Duncan and S. A. Hosseiniun, The second dual of Banach algebra,Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 309–325. [3] F. Ghahramani and Anthony To-Ming Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, Journal of functional analysis 132 (1995) 170–191. [4] F. Ghahramani and J. P. Mcclure, The second dual algebra of the measure algebra of a compact group, Bull. London Math. Soc. 29 (1997) 223–226. [5] B. E. Johnson, Isometric isomorphisms of measure algebras, Proc. Amer. Math. Soc. 15(1964), 186–188. [6] J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952) 251–256.