Proyecciones Vol. 28, No 1, pp. 21—26, Mayo 2009. Universidad Cat´olica del Norte Antofagasta - Chile
ARENS REGULARITY OF SOME BILINEAR MAPS M. ESHAGHI GORDJI ´ SEMNAN UNIVERSITY, IRAN Received : February 2008. Accepted : November 2008
Abstract Let H be a Hilbert space. we show that the following statements are equivalent: (a) B(H) is finite dimension, (b) every left Banach module action l : B(H) ×H −→ H, is Arens regular (c) every bilinear map f : B(H)∗ −→ B(H) is Arens regular. Indeed we show that a Banach space X is reflexive if and only if every bilinear map f : X ∗ × X −→ X ∗ is Arens regular. Subjclass [2000] : Primary 46H25, 16E40.
Keywords : Banach algebra, Bilinear map, Arens products
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M. Eshaghi Gordji
1. Introduction The regularity of bilinear maps on norm spaces, was introduced by Arens in 1951. Let X , Y and Z be normed spaces and let f : X × Y −→ Z be a continuous bilinear map, then f ∗ : Z ∗ × X −→ Y ∗ (the transpose of f ) is defined by hf ∗ (z ∗ , x), yi = hz ∗ , f (x, y)i
(z ∗ ∈ Z ∗ , x ∈ X , y ∈ Y ).
f ∗ is a continuous bilinear map. Set f ∗∗ = (f ∗ )∗ and f ∗∗∗ = (f ∗∗ )∗ , .... Then f ∗∗∗ : X ∗∗ × Y ∗∗ −→ ∗∗ Z is the unique extension of f such that f ∗∗∗ (·, y 00 ) : X ∗∗ −→ Z ∗∗ is weak∗ − weak ∗ continuous for every y 00 ∈ Y ∗∗ . Let f r : Y × X −→ Z defined by f r (y, x) = f (x, y), (x ∈ X, y ∈ Y ), then f r is a continuous bilinear map. f is called Arens regular whenever f ∗∗∗ = f r∗∗∗r . It is easy to show that f is Arens regular if and only if f ∗∗∗ (x00 , ·) : Y ∗∗ −→ Z ∗∗ is weak∗ − weak∗ continuous for every x00 ∈ X ∗∗ . For further details we refer the reader to [A], [E-F], [D-R-V], and [M-Y]. Let A be a Banach algebra and let π : A × A −→ A be the product of A. Then the second dual space A∗∗ of A is Banach algebra with both products π ∗∗∗ and π r∗∗∗r . A is called Arens regular if π is Arens regular. See [A], [D-H] and [F-S]. An element f ∈ A∗ is called weakly almost periodic if the maps a 7→ a.f, a 7→ f.a, A −→ A∗ are weakly compact. The collection of weakly almost periodic elements in A∗ is denoted by W ap(A∗ ). 1.Theorem. Let X be a Banach space and let l : B(X) × X −→ X defined by l(T, x) = T (x), (T ∈ B(X), x ∈ X). If l is Arens regular, then B(X)∗∗ is isomorphic to a subalgebra of B(X ∗ ). Proof: Let π : B(X) × B(X) −→ B(X) be the product of B(X). Then for every e ∈ X, e0 ∈ X ∗ , and T, S ∈ B(X), we have hl∗ (e0 , π(T, S)), ei
= he0 , l(T, S(e)i
= hl∗ (e0 , T ), l(S, e)i
= hl∗ (l∗ (e0 , T )S), ei. This means that l∗ (e0 , π(T, S)) = l∗ (L∗ (e0 , T ), S)
(1).
Arens regularity of some bilinear maps
23
By applying (1), for every e00 ∈ X ∗∗ , we have hπ ∗ (l∗∗ (e00 , e0 ), T ), Si
= hl∗∗ (e00 , e0 ), π(T, S)i = he00 , l∗ (e0 , π(T, S)i
= he00 , l∗ (L∗ (e0 , T ), S)i
= hl∗∗ (e00 , L∗ (e0 , T ), Si. Thus we have π ∗ (l∗∗ (e00 , e0 ), T ) = l∗∗ (e00 , L∗ (e0 , T )
(2).
Let B ∈ B(X)∗∗ , then by (2), we have hπ ∗∗ (B, l∗∗ (e00 , e0 ), T )i
= hB, π ∗ (l∗ (e00 , e0 ), T )i
= hB, l∗∗ (e00 , L∗ (e0 , T ))i
= hl∗∗∗ (B, e00 ), L∗ (e0 , T )i
= hl∗∗ (l∗∗∗ (B, e00 ), e0 ), T i. This means that π ∗∗ (B, l∗∗ (e00 , e0 ) = l∗∗ (l∗∗∗ (B, e00 ), e0 )
(3).
Let now A ∈ B(X)∗∗ . By (3), we have hl∗∗∗ π ∗∗∗ (A, B), e00 ), e0 i = hπ ∗∗∗ (A, B), l∗∗ (e00 , e0 )i = hA, π ∗∗ (B, l∗∗ (e00 , e0 ))i
= hA, l∗∗ (l∗∗∗ (B, e00 ), e0 )i. Therefore l∗∗∗ (π ∗∗∗ (A, B), e00 ) = l∗∗∗ (A, l∗∗∗ (B, e00 )). This means that the mapping f : B(X)∗∗ −→ B(X ∗∗ ), defined by f (A) = l∗∗∗ (A, .) (A ∈ B(X)∗∗ ), is a Banach algebras homomorphism. On the other hand l is Arens regular, then f (A) : X ∗∗ −→ X ∗∗ is weak∗ − weak ∗ continuous for every A ∈ B(X)∗∗ . Let now Bw∗ (X ∗∗ ) := {U ∈ B(X ∗∗ ) | U : X ∗∗ → X ∗∗ is weak ∗ −weak∗ continuous }, and let φ : B(X ∗ ) −→ Bw∗ (X ∗∗ ) defined by φ(T ) = T ∗ (T ∈ B(X ∗ )). Then φ−1 of : B(X)∗∗ −→ B(X ∗ ) is a injective Banach algebras anti homomorphism.
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M. Eshaghi Gordji
2.Theorem. Let A be a Banach algebra and let L be a reflexive left Banach A module with module action g : A×L −→ L. Then g ∗∗ (L∗∗ ×L∗ ) ⊆ W ap(A∗ ). Proof: Let x00 ∈ L∗∗ , x0 ∈ X ∗ , and let {an } and {bn } be sequences in A, in which the limits: lim limhg ∗∗ (x00 , x0 ), am an i, m
n
and
lim limhg ∗∗ (x00 , x0 ), am an i, n
m
are both exist. Then we have lim limhg ∗∗ (x00 , x0 ), am bn i m
= limm limn hx00 , g ∗ (x0 , am bn )i
n
= limm limn hx00 , g ∗ (g ∗ (x0 , am ), bn )i
= limm limn hg ∗∗ (x00 , g ∗ (x0 , am )), bn i
00 ∗ 0 = limm limn hg ∗∗∗ (bc n , x ), g (x , am )i
00 ∗ 0 = limn limm hg ∗∗∗ (bc n , x ), g (x , am )i (L is reflexive)
= limn limm hg ∗∗ (x00 , g ∗ (x0 , am )), bn i = limn limm hx00 , g ∗ (g ∗ (x0 , am ), bn )i = limn limm hx00 , g ∗ (x0 , am bn )i
= limm limn hg ∗∗ (x00 , x0 ), am bn i. This means that g ∗∗ (x00 , x0 ) ∈ W ap(A∗ ) [Pa, Theorem 1.4.11]. 3.Corollary. Let A, L and g are as above. If g∗∗ (L∗∗ × L∗ ) = A∗ , then A is Arens regular. 4.Corollary [D]. Let X be a Banach space and let L be a reflexive left b ∗ −→ B(X)∗ by Banach B(X) module. We define the map φ : L⊗L hφ(f ⊗ µ), T i = hµ, T.f i
b ∗ , T ∈ B(X)). (f ⊗ µ ∈ L⊗L
If φ is surjective then B(X) is Arens regular. 5.Theorem. Let X be a Banach space. Then X is reflexive if and only if every bilinear map f : X ∗ × X −→ X ∗ is Arens regular. Proof: Obviously every bilinear map f : X ∗ × X −→ X ∗ is Arens regular if X is reflexive. For the converse, let 0 6= x0 ∈ X. Then by Hahn-Banach Theorem, there exists g ∈ X ∗ such that hg, x0 i = 1. Let f : X × X −→ X defined by f ((x, y)) = hg, yix. f is a bilinear map. Then f ∗ : X ∗ ×X −→ X ∗ weak∗
is Arens regular. Let now x000 ∈ X ∗∗∗ and let x00α − −−→ x00 in X ∗∗ , then weak∗
−−→ f ∗∗∗∗ (x000 , x00 ) in X ∗∗∗ . Thus for every y 00 ∈ X ∗∗ we f ∗∗∗∗ (x000 , x00α ) −
Arens regularity of some bilinear maps
25
have limhx000 , f ∗∗∗ (x00α , y00 )i = limhf ∗∗∗∗ (x000 , x00α , y 00 i α
α
= hf ∗∗∗∗ (x000 , x00 , y 00 i
= hx000 , f ∗∗∗ (x00 , y00 )i
(4).
On the other hand for each y00 ∈ X ∗∗ , there exists a net (yβ ) in Y such weak∗
c0 ) : X ∗∗ −→ X ∗∗ is that ycβ − −−→ y 00 in X ∗∗ . We know that f ∗∗∗ (·, x weak∗ − weak∗ continuous, then 00 c0 ) = w∗ − lim f ∗∗∗ (yc c0 ) = w∗ − lim yc f ∗∗∗ (y 00 , x β, x β =y β
(5).
By (4) and (5), we have
c0 )i limhx000 , x00α i = limhx000 , f ∗∗∗ (x00α , x α
α
c0 )i = hx000 , f ∗∗∗ (x00 , x
= hx000 , x00 i.
This means that x000 : X ∗∗ −→ C is weak∗ − weak∗ continuous. Thus d∗ , and X ∗ is reflexive. So X is reflexive. x000 ∈ X 6.Corollary. Let H be Hilbert space. Then the following assertions are equivalent: (a) B(H) is finite dimension. (b) The mapping l : B(H) × H −→ H defined by l(T, x) = T (x), (T ∈ B(H), x ∈ H), is Arens regular. (c) Every bilinear map f : B(H)∗ × B(H) −→ B(H)∗ is Arens regular. Proof. (a) ⇔ (b): Let B(H) be finite dimension, the B(H) is reflexive. Then by Theorem 1, l is Arens regular. For the converse we know that H∼ = H ∗ as Banach spaces. Then by Theorem 1, we have B(H)∗∗ = B(H), so B(H) is finite dimension. (a) ⇔ (c): It follows from the Theorem 5. Example. Let H = l1 (N ). Then the bilinear map l : B(H) × H −→ H defined by l(T, x) = T (x), (T ∈ B(H), x ∈ H), is not Arens regular. Acknowledgements: The author would like to express his sincere thanks to referee for his invaluable comments. Also he would like to thank the Semnan University for its financial support.
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M. Eshaghi Gordji
References [A] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2(1951), pp. 839-848, (1951). [D-R-V] H. G. Dales, A. Rodriguez-Palacios and M. V. Velasco, The second transpose of a derivation, J. London Math. Soc. (2) 64, pp. 707-721, (2001). [D] M. Daws, Arens regularity of the algebra of operators on a Banach space. Bull. London Math. Soc. 36, No. 4, pp. 493—503, (2004). [D-H] J. Duncan and S. A. Hosseiniun, The second dual of Banach algebra, Proc. Roy. Soc. Edinburgh Set. A 84, pp. 309-325, (1979). [E-F] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math., Vol 181, No 3, pp. 237-254, (2007). [F-S] M. Filali and A. I. Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras. General topological algebras (Tartu, 1999), pp. 95— 124, Math. Stud. (Tartu), 1, Est. Math. Soc., Tartu, 2001. MR1853838 [M-Y] Gh. Mohajeri-Minaei, T. Yazdanpanah, Arens regularity of modules. J. Inst. Math. Comput. Sci. Math. Ser. 18, No. 3, pp. 195—197. MR2194165, (2005). [Pa] Theodore W. Palmer, Banach Algebra and The General Theory of ∗-Algebras Vol 1 Cambridge University Press,(1994). M. Eshaghi Gordji Department of Mathematics Semnan University, P. O. Box 35195-363, Semnan, Iran e-mail : maj
[email protected] ;
[email protected]
