The topological centers and factorization properties of module actions and ∗−involution algebras
aa r X i v : . [ m a t h . F A ] O c t THE TOPOLOGICAL CENTERS AND FACTORIZATIONPROPERTIES OF MODULE ACTIONS AND ∗ −
IN V OLU T ION
ALGEBRAS
KAZEM HAGHNEJAD AZAR
Abstract.
For Banach left and right module actions, we extend some propo-sitions from Lau and ¨
Ulger into general situations and we establish the rela-tionships between topological centers of module actions. We also introduce thenew concepts as Lw ∗ w -property and Rw ∗ w -property for Banach A − bimoduleB and we obtain some conclusions in the topological center of module actionsand Arens regularity of Banach algebras. we also study some factorization prop-erties of left module actions and we find some relations of them and topologicalcenters of module actions. For Banach algebra A , we extend the definition of ∗ − involution algebra into Banach A − bimodule B with some results in thefactorizations of B ∗ . We have some applications in group algebras. As is well-known [1], the second dual A ∗∗ of A endowed with the either Arens mul-tiplications is a Banach algebra. The constructions of the two Arens multiplicationsin A ∗∗ lead us to definition of topological centers for A ∗∗ with respect both Arensmultiplications. The topological centers of Banach algebras, module actions and ap-plications of them were introduced and discussed in [6, 8, 13, 14, 15, 16, 17, 21, 22],and they have attracted by some attentions.Now we introduce some notations and definitions that we used throughout this paper.Let A be a Banach algebra. We say that a net ( e α ) α ∈ I in A is a left approximateidentity (= LAI ) [resp. right approximate identity (=
RAI )] if, for each a ∈ A , e α a −→ a [resp. ae α −→ a ]. For a ∈ A and a ′ ∈ A ∗ , we denote by a ′ a and aa ′ respectively, the functionals on A ∗ defined by h a ′ a, b i = h a ′ , ab i = a ′ ( ab ) and h aa ′ , b i = h a ′ , ba i = a ′ ( ba ) for all b ∈ A . The Banach algebra A is embedded in itssecond dual via the identification h a, a ′ i - h a ′ , a i for every a ∈ A and a ′ ∈ A ∗ . Wedenote the set { a ′ a : a ∈ A and a ′ ∈ A ∗ } and { aa ′ : a ∈ A and a ′ ∈ A ∗ } by A ∗ A and AA ∗ , respectively, clearly these two sets are subsets of A ∗ . Let A has a BAI .If the equality A ∗ A = A ∗ , ( AA ∗ = A ∗ ) holds, then we say that A ∗ factors on theleft (right). If both equalities A ∗ A = AA ∗ = A ∗ hold, then we say that A ∗ factorson both sides. Let X, Y, Z be normed spaces and m : X × Y → Z be a boundedbilinear mapping. Arens in [1] offers two natural extensions m ∗∗∗ and m t ∗∗∗ t of m from X ∗∗ × Y ∗∗ into Z ∗∗ as following:1. m ∗ : Z ∗ × X → Y ∗ , given by h m ∗ ( z ′ , x ) , y i = h z ′ , m ( x, y ) i where x ∈ X , y ∈ Y , Mathematics Subject Classification.
Key words and phrases.
Arens regularity, bilinear mappings, Topological center, Second dual,Module action, factorization, ∗ − involution algebra. z ′ ∈ Z ∗ ,2. m ∗∗ : Y ∗∗ × Z ∗ → X ∗ , given by h m ∗∗ ( y ′′ , z ′ ) , x i = h y ′′ , m ∗ ( z ′ , x ) i where x ∈ X , y ′′ ∈ Y ∗∗ , z ′ ∈ Z ∗ ,3. m ∗∗∗ : X ∗∗ × Y ∗∗ → Z ∗∗ , given by h m ∗∗∗ ( x ′′ , y ′′ ) , z ′ i = h x ′′ , m ∗∗ ( y ′′ , z ′ ) i where x ′′ ∈ X ∗∗ , y ′′ ∈ Y ∗∗ , z ′ ∈ Z ∗ .The mapping m ∗∗∗ is the unique extension of m such that x ′′ → m ∗∗∗ ( x ′′ , y ′′ ) from X ∗∗ into Z ∗∗ is weak ∗ − to − weak ∗ continuous for every y ′′ ∈ Y ∗∗ , but the mapping y ′′ → m ∗∗∗ ( x ′′ , y ′′ ) is not in general weak ∗ − to − weak ∗ continuous from Y ∗∗ into Z ∗∗ unless x ′′ ∈ X . Hence the first topological center of m may be defined as following Z ( m ) = { x ′′ ∈ X ∗∗ : y ′′ → m ∗∗∗ ( x ′′ , y ′′ ) is weak ∗ − to − weak ∗ − continuous } . Let now m t : Y × X → Z be the transpose of m defined by m t ( y, x ) = m ( x, y ) forevery x ∈ X and y ∈ Y . Then m t is a continuous bilinear map from Y × X to Z , andso it may be extended as above to m t ∗∗∗ : Y ∗∗ × X ∗∗ → Z ∗∗ . The mapping m t ∗∗∗ t : X ∗∗ × Y ∗∗ → Z ∗∗ in general is not equal to m ∗∗∗ , see [1], if m ∗∗∗ = m t ∗∗∗ t , then m is called Arens regular. The mapping y ′′ → m t ∗∗∗ t ( x ′′ , y ′′ ) is weak ∗ − to − weak ∗ continuous for every y ′′ ∈ Y ∗∗ , but the mapping x ′′ → m t ∗∗∗ t ( x ′′ , y ′′ ) from X ∗∗ into Z ∗∗ is not in general weak ∗ − to − weak ∗ continuous for every y ′′ ∈ Y ∗∗ . So we definethe second topological center of m as Z ( m ) = { y ′′ ∈ Y ∗∗ : x ′′ → m t ∗∗∗ t ( x ′′ , y ′′ ) is weak ∗ − to − weak ∗ − continuous } . It is clear that m is Arens regular if and only if Z ( m ) = X ∗∗ or Z ( m ) = Y ∗∗ . Arensregularity of m is equivalent to the followinglim i lim j h z ′ , m ( x i , y j ) i = lim j lim i h z ′ , m ( x i , y j ) i , whenever both limits exist for all bounded sequences ( x i ) i ⊆ X , ( y i ) i ⊆ Y and z ′ ∈ Z ∗ , see [6, 18].The regularity of a normed algebra A is defined to be the regularity of its algebramultiplication when considered as a bilinear mapping. Let a ′′ and b ′′ be elementsof A ∗∗ , the second dual of A . By Goldstin , s Theorem [6, P.424-425], there are nets( a α ) α and ( b β ) β in A such that a ′′ = weak ∗ − lim α a α and b ′′ = weak ∗ − lim β b β . Soit is easy to see that for all a ′ ∈ A ∗ ,lim α lim β h a ′ , m ( a α , b β ) i = h a ′′ b ′′ , a ′ i and lim β lim α h a ′ , m ( a α , b β ) i = h a ′′ ob ′′ , a ′ i , where a ′′ b ′′ and a ′′ ob ′′ are the first and second Arens products of A ∗∗ , respectively,see [6, 14, 18].The mapping m is left strongly Arens irregular if Z ( m ) = X and m is right stronglyArens irregular if Z ( m ) = Y .This paper is organized as follows. a) In section two, for a Banach A − bimodule , we have(1) a ′′ ∈ Z B ∗∗ ( A ∗∗ ) if and only if π ∗∗∗∗ ℓ ( b ′ , a ′′ ) ∈ B ∗ for all b ′ ∈ B ∗ .(2) F ∈ Z B ∗∗ (( A ∗ A ) ∗ ) if and only if π ∗∗∗∗ ℓ ( g, F ) ∈ B ∗ for all g ∈ B ∗ .(3) G ∈ Z ( A ∗ A ) ∗ ( B ∗∗ ) if and only if π ∗∗∗∗ r ( g, G ) ∈ A ∗ A for all g ∈ B ∗ . (4) Let B has a BAI ( e α ) α ⊆ A such that e α w ∗ → e ′′ . Then if Z te ∗∗ ( B ∗∗ ) = B ∗∗ [resp. Z e ∗∗ ( B ∗∗ ) = B ∗∗ ] and B ∗ factors on the left [resp. right], but not onthe right [resp. left], then Z B ∗∗ ( A ∗∗ ) = Z tB ∗∗ ( A ∗∗ ).(5) B ∗ A ⊆ wap ℓ ( B ) if and only if AA ∗∗ ⊆ Z B ∗∗ ( A ∗∗ ).(6) Let b ′ ∈ B ∗ . Then b ′ ∈ wap ℓ ( B ) if and only if the adjoint of the mapping π ∗ ℓ ( b ′ , ) : A → B ∗ is weak ∗ − to − weak continuous.Then ( α ) ⇒ ( β ) ⇒ ( γ ).If we take T ∈ LM ( A, B ) and if B has a sequential BAI and it is
W SC , then( α ), ( β ) and ( γ ) are equivalent. b) In section three, for a Banach A − bimodule B , we define Lef t − weak ∗ − to − weak property [= Rw ∗ w − property] and Right − weak ∗ − to − weak property [= Rw ∗ w − property] for Banach algebra A and we show that(1) If A ∗∗ = a A ∗∗ [resp. A ∗∗ = A ∗∗ a ] for some a ∈ A and a has Rw ∗ w − property [resp. Lw ∗ w − property], then Z B ∗∗ ( A ∗∗ ) = A ∗∗ .(2) If B ∗∗ = a B ∗∗ [resp. B ∗∗ = B ∗∗ a ] for some a ∈ A and a has Rw ∗ w − property [resp. Lw ∗ w − property] with respect to B , then Z A ∗∗ ( B ∗∗ ) = B ∗∗ .(3) If B ∗ factors on the left [resp. right] with respect to A and A has Rw ∗ w − property [resp. Lw ∗ w − property], then Z B ∗∗ ( A ∗∗ ) = A ∗∗ .(4) If B ∗ factors on the left [resp. right] with respect to A and A has Rw ∗ w − property [resp. Lw ∗ w − property] with respect B , then Z A ∗∗ ( B ∗∗ ) = B ∗∗ .(5) If a ∈ A has Rw ∗ w − property with respect to B , then a A ∗∗ ⊆ Z B ∗∗ ( A ∗∗ )and a B ∗ ⊆ wap ℓ ( B ).(6) Assume that AB ∗ ⊆ wap ℓ B . If B ∗ strong factors on the left [resp. right],then A has Lw ∗ w − property [resp. Rw ∗ w − property ] with respect to B .(7) Assume that AB ∗ ⊆ wap ℓ B . If B ∗ strong factors on the left [resp. right],then A has Lw ∗ w − property [resp. Rw ∗ w − property ] with respect to B . c) In section four, for a left Banach A − module B , we study some relationshipsbetween factorization properties of left module action and topological centers of it.(1) Let ( e α ) α ⊆ A be a LBAI for B . Then the following assertions hold.i) For each b ′ ∈ B ∗ , π ∗ ℓ ( b ′ , e α ) w ∗ → b ′ .ii) B ∗ factors on the left with respect to A if and only if B ∗∗ has a W ∗ LBAI ( e α ) α ⊆ A .iii) B ∗∗ has a W ∗ LBAI ( e α ) α ⊆ A if and only if B ∗∗ has a left unit element e ′′ ∈ A ∗∗ such that e α w ∗ → e ′′ .(2) Suppose that b ′ ∈ wap ℓ ( B ). Let a ′′ ∈ A ∗∗ and ( a α ) α ⊆ A such that a α w ∗ → a ′′ in A ∗∗ . Then we have π ∗ ℓ ( b ′ , a α ) w → π ∗∗∗∗ ℓ ( b ′ , a ′′ ) . (3) Let B ∗ factors on the left with respect to A . If AA ∗∗ ⊆ Z B ∗∗ ( A ∗∗ ), then Z B ∗∗ ( A ∗∗ ) = A ∗∗ .(4) Let B ∗∗ has a LBAI with respect to A ∗∗ . Then B ∗∗ has a left unit withrespect to A ∗∗ .(5) Let B has a LBAI with respect to A . Then we have the following assertions.i) B ∗ factors on the left with respect to A if and only if for each b ′ ∈ B ∗ , we have π ∗ ℓ ( b ′ , e α ) w → b ′ in B ∗ .ii) B factors on the left with respect to A if and only if for each b ∈ B , wehave π ∗ ℓ ( b, e α ) w → b in B .(6) Let A has a LBAI ( e α ) α ⊆ A such that e α w ∗ → e ′′ in A ∗∗ where e ′′ is a leftunit for A ∗∗ . Suppose that Z te ′′ ( B ∗∗ ) = B ∗∗ . Then, B factors on the rightwith respect to A if and only if e ′′ is a left unit for B ∗∗ . d) In section five, for a Banach A − bimodule B we study ∗− involution algebra on B ∗∗ with respect to first Arens product with some results in the factorization in B ∗ , thatis, suppose that ( e α ) α ⊆ A is a BAI for B . Then if B ∗∗ is a Banach ∗ − involution algebra as A ∗∗ -module, then B ∗∗ is unital as A ∗∗ -module and B ∗ factors on the bothside.
2. The topological centers of module actions
Let B be a Banach A − bimodule , and let π ℓ : A × B → B and π r : B × A → B. be the left and right module actions of A on B . Then B ∗∗ is a Banach A ∗∗ − bimodule with module actions π ∗∗∗ ℓ : A ∗∗ × B ∗∗ → B ∗∗ and π ∗∗∗ r : B ∗∗ × A ∗∗ → B ∗∗ . Similarly, B ∗∗ is a Banach A ∗∗ − bimodule with module actions π t ∗∗∗ tℓ : A ∗∗ × B ∗∗ → B ∗∗ and π t ∗∗∗ tr : B ∗∗ × A ∗∗ → B ∗∗ . We may therefore define the topological centers of the right and left module actionsof A on B as follows: Z A ∗∗ ( B ∗∗ ) = Z ( π r ) = { b ′′ ∈ B ∗∗ : the map a ′′ → π ∗∗∗ r ( b ′′ , a ′′ ) : A ∗∗ → B ∗∗ is weak ∗ − to − weak ∗ continuous } Z B ∗∗ ( A ∗∗ ) = Z ( π ℓ ) = { a ′′ ∈ A ∗∗ : the map b ′′ → π ∗∗∗ ℓ ( a ′′ , b ′′ ) : B ∗∗ → B ∗∗ is weak ∗ − to − weak ∗ continuous } Z tA ∗∗ ( B ∗∗ ) = Z ( π tℓ ) = { b ′′ ∈ B ∗∗ : the map a ′′ → π t ∗∗∗ ℓ ( b ′′ , a ′′ ) : A ∗∗ → B ∗∗ is weak ∗ − to − weak ∗ continuous } Z tB ∗∗ ( A ∗∗ ) = Z ( π tr ) = { a ′′ ∈ A ∗∗ : the map b ′′ → π t ∗∗∗ r ( a ′′ , b ′′ ) : B ∗∗ → B ∗∗ is weak ∗ − to − weak ∗ continuous } We note also that if B is a left(resp. right) Banach A − module and π ℓ : A × B → B (resp. π r : B × A → B ) is left (resp. right) module action of A on B , then B ∗ isa right (resp. left) Banach A − module .We write ab = π ℓ ( a, b ), ba = π r ( b, a ), π ℓ ( a a , b ) = π ℓ ( a , a b ), π r ( b, a a ) = π r ( ba , a ), π ∗ ℓ ( a b ′ , a ) = π ∗ ℓ ( b ′ , a a ), π ∗ r ( b ′ a, b ) = π ∗ r ( b ′ , ab ), for all a , a , a ∈ A , b ∈ B and b ′ ∈ B ∗ when there is no confusion. Theorem 2-1.
We have the following assertions.(1) Assume that B is a Banach left A − module . Then, a ′′ ∈ Z B ∗∗ ( A ∗∗ ) if andonly if π ∗∗∗∗ ℓ ( b ′ , a ′′ ) ∈ B ∗ for all b ′ ∈ B ∗ .(2) Assume that B is a Banach right A − module . Then, b ′′ ∈ Z A ∗∗ ( B ∗∗ ) if andonly if π ∗∗∗∗ r ( b ′ , b ′′ ) ∈ A ∗ for all b ′ ∈ B ∗ . Proof. (1) Let b ′′ ∈ B ∗∗ . Then, for every a ′′ ∈ Z B ∗∗ ( A ∗∗ ), we have h π ∗∗∗∗ ℓ ( b ′ , a ′′ ) , b ′′ i = h b ′ , π ∗∗∗ ℓ ( a ′′ , b ′′ ) i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = h π t ∗∗∗ tℓ ( a ′′ , b ′′ ) , b ′ i = h π t ∗∗∗ ℓ ( b ′′ , a ′′ ) , b ′ i = h b ′′ , π t ∗∗ ℓ ( a ′′ , b ′ ) i . It follow that π ∗∗∗∗ ℓ ( b ′ , a ′′ ) = π t ∗∗ ℓ ( a ′′ , b ′ ) ∈ B ∗ .Conversely, let a ′′ ∈ A ∗∗ and let π ∗∗∗∗ ℓ ( a ′′ , b ′ ) ∈ B ∗ for all b ′ ∈ B ∗ . Let( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Then for every b ′ ∈ B ∗ , we have h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = h b ′ , π ∗∗∗ ℓ ( a ′′ , b ′′ α ) i = h π ∗∗∗∗ ℓ ( b ′ , a ′′ ) , b ′′ α i = h b ′′ α , π ∗∗∗∗ ℓ ( b ′ , a ′′ ) i→ h b ′′ , π ∗∗∗∗ ℓ ( b ′ , a ′′ ) i = h π ∗∗∗∗ ℓ ( b ′ , a ′′ ) , b ′′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i . Consequently a ′′ ∈ Z B ∗∗ ( A ∗∗ ).(2) Proof is similar to (1). (cid:3) Theorem 2-2.
Assume that B is a Banach A − bimodule . Then we have the followingassertions.(1) F ∈ Z B ∗∗ (( A ∗ A ) ∗ ) if and only if π ∗∗∗∗ ℓ ( g, F ) ∈ B ∗ for all g ∈ B ∗ .(2) G ∈ Z ( A ∗ A ) ∗ ( B ∗∗ ) if and only if π ∗∗∗∗ r ( g, G ) ∈ A ∗ A for all g ∈ B ∗ . Proof. (1) Let F ∈ Z B ∗∗ (( A ∗ A ) ∗ ) and ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Then forall g ∈ B ∗ , we have h π ∗∗∗∗ ℓ ( g, F ) , b ′′ α i = h g, π ∗∗∗ ℓ ( F, b ′′ α ) i = h π ∗∗∗ ℓ ( F, b ′′ α ) , g i→ h π ∗∗∗ ℓ ( F, b ′′ ) , g i = h π ∗∗∗∗ ℓ ( g, F ) , b ′′ i . Thus, we conclude that π ∗∗∗∗ ℓ ( g, F ) ∈ ( B ∗∗ , weak ∗ ) ∗ = B ∗ .Conversely, let π ∗∗∗∗ ℓ ( g, F ) ∈ B ∗ for F ∈ ( A ∗ A ) ∗ and g ∈ B ∗ . Assume that b ′′ ∈ B ∗∗ and ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Then h π ∗∗∗ ℓ ( F, b ′′ α ) , g i = h g, π ∗∗∗ ℓ ( F, b ′′ α ) i = h π ∗∗∗∗ ℓ ( g, F ) , b ′′ α i = h b ′′ α , π ∗∗∗∗ ℓ ( g, F ) i → h b ′′ , π ∗∗∗∗ ℓ ( g, F ) i = h π ∗∗∗∗ ℓ ( g, F ) , b ′′ i = h π ∗∗∗ ℓ ( F, b ′′ ) , g i . It follow that F ∈ Z B ∗∗ (( A ∗ A ) ∗ ).(2) Proof is similar to (1). (cid:3) In the preceding theorems, if we take B = A , we obtain some parts of Lemma 3.1from [14].An element e ′′ of A ∗∗ is said to be a mixed unit if e ′′ is a right unit for the first Arensmultiplication and a left unit for the second Arens multiplication. That is, e ′′ is amixed unit if and only if for each a ′′ ∈ A ∗∗ , a ′′ e ′′ = e ′′ oa ′′ = a ′′ . By [4, p.146], anelement e ′′ of A ∗∗ is mixed unit if and only if it is a weak ∗ cluster point of some BAI( e α ) α ∈ I in A .Let B be a Banach A − bimodule and a ′′ ∈ A ∗∗ . We define the locally topologicalcenter of the left and right module actions of a ′′ on B , respectively, as follows Z ta ′′ ( B ∗∗ ) = Z ta ′′ ( π tℓ ) = { b ′′ ∈ B ∗∗ : π t ∗∗∗ tℓ ( a ′′ , b ′′ ) = π ∗∗∗ ℓ ( a ′′ , b ′′ ) } ,Z a ′′ ( B ∗∗ ) = Z a ′′ ( π tr ) = { b ′′ ∈ B ∗∗ : π t ∗∗∗ tr ( b ′′ , a ′′ ) = π ∗∗∗ r ( b ′′ , a ′′ ) } . Thus we have \ a ′′ ∈ A ∗∗ Z ta ′′ ( B ∗∗ ) = Z tA ∗∗ ( B ∗∗ ) = Z ( π tr ) , \ a ′′ ∈ A ∗∗ Z a ′′ ( B ∗∗ ) = Z A ∗∗ ( B ∗∗ ) = Z ( π r ) . Let B be a Banach A − bimodule . We say that B is a left [resp. right] factors withrespect to A , if BA = B [resp. AB = B ]. Definition 2-3.
Let B be a Banach left A − module and e ′′ ∈ A ∗∗ be a mixed unitfor A ∗∗ . We say that e ′′ is a left mixed unit for B ∗∗ , if π ∗∗∗ ℓ ( e ′′ , b ′′ ) = π t ∗∗∗ tℓ ( e ′′ , b ′′ ) = b ′′ , for all b ′′ ∈ B ∗∗ .The definition of right mixed unit for B ∗∗ is similar. B ∗∗ has a mixed unit, if it hasequal left and right mixed unit.It is clear that if e ′′ ∈ A ∗∗ is a left (resp. right) unit for B ∗∗ and Z e ′′ ( B ∗∗ ) = B ∗∗ ,then e ′′ is left (resp. right) mixed unit for B ∗∗ . Theorem 2-4.
Let B be a Banach A − bimodule with a BAI ( e α ) α such that e α w ∗ → e ′′ . Then if Z te ′′ ( B ∗∗ ) = B ∗∗ [resp. Z e ′′ ( B ∗∗ ) = B ∗∗ ] and B ∗ factors on the left[resp. right], but not on the right [resp. left], then Z B ∗∗ ( A ∗∗ ) = Z tB ∗∗ ( A ∗∗ ). Proof.
Suppose that B ∗ factors on the left with respect to A , but not on the right.Let ( e α ) α ⊆ A be a BAI for A such that e α w ∗ → e ′′ . Thus for all b ′ ∈ B ∗ there are a ∈ A and x ′ ∈ B ∗ such that x ′ a = b ′ . Then for all b ′′ ∈ B ∗∗ we have h π ∗∗∗ ℓ ( e ′′ , b ′′ ) , b ′ i = h e ′′ , π ∗∗ ℓ ( b ′′ , b ′ ) i = lim α h π ∗∗ ℓ ( b ′′ , b ′ ) , e α i = lim α h b ′′ , π ∗ ℓ ( b ′ , e α ) i = lim α h b ′′ , π ∗ ℓ ( x ′ a, e α ) i = lim α h b ′′ , π ∗ ℓ ( x ′ , ae α ) i = lim α h π ∗∗ ℓ ( b ′′ , x ′ ) , ae α i = h π ∗∗ ℓ ( b ′′ , x ′ ) , a i = h b ′′ , b ′ i . Thus π ∗∗∗ ℓ ( e ′′ , b ′′ ) = b ′′ consequently B ∗∗ has left unit A ∗∗ − module . It follows that e ′′ ∈ Z B ∗∗ ( A ∗∗ ). If we take Z B ∗∗ ( A ∗∗ ) = Z tB ∗∗ ( A ∗∗ ), then e ′′ ∈ Z tB ∗∗ ( A ∗∗ ). Then themapping b ′′ → π t ∗∗∗ tr ( b ′′ , e ′′ ) is weak ∗ − to − weak ∗ continuous from B ∗∗ into B ∗∗ .Since e α w ∗ → e ′′ , π t ∗∗∗ tr ( b ′′ , e α ) w ∗ → π t ∗∗∗ tr ( b ′′ , e ′′ ). Let b ′ ∈ B ∗ and ( b β ) β ⊆ B such that b β w ∗ → b ′′ . Since Z te ′′ ( B ∗∗ ) = B ∗∗ , we have the following quality h π t ∗∗∗ tr ( b ′′ , e ′′ ) , b ′ i = lim α h π t ∗∗∗ tr ( b ′′ , e α ) , b ′ i = lim α h π t ∗∗∗ r ( e α , b ′′ ) , b ′ i = lim α lim β h π t ∗∗∗ r ( e α , b β ) , b ′ i = lim α lim β h π r ( b β , e α ) , b ′ i = lim α lim β h b ′ , π r ( b β , e α ) i = lim β lim α h b ′ , π r ( b β , e α ) i = lim β h b ′ , b β i = h b ′′ , b ′ i . Thus π t ∗∗∗ tr ( b ′′ , e ′′ ) = π ∗∗∗ r ( b ′′ , e ′′ ) = b ′′ . It follows that B ∗∗ has a right unit. Supposethat b ′′ ∈ B ∗∗ and ( b β ) β ⊆ B such that b β w ∗ → b ′′ . Then for all b ′ ∈ B ∗ we have h b ′′ , b ′ i = h π ∗∗∗ r ( b ′′ , e ′′ ) , b ′ i = h b ′′ , π ∗∗ r ( e ′′ , b ′ ) i = lim β h π ∗∗ r ( e ′′ , b ′ ) , b β i = lim β h e ′′ , π ∗ r ( b ′ , b β ) i = lim β lim α h π ∗ r ( b ′ , b β ) , e α i = lim β lim α h π ∗ r ( b ′ , b β ) , e α i = lim β lim α h b ′ , π r ( b β , e α ) i = lim α lim β h π ∗∗∗ r ( b β , e α ) , b ′ i = lim α lim β h b β , π ∗∗ r ( e α , b ′ ) i = lim α h b ′′ , π ∗∗ r ( e α , b ′ ) i . It follows that weak − lim α π ∗∗ r ( e α , b ′ ) = b ′ . So by Cohen factorization theorem, B ∗ factors on the right that is contradiction. (cid:3) Corollary 2-5.
Let B be a Banach A − bimodule and e ′′ ∈ A ∗∗ be a mixed unit for B ∗∗ . If B ∗ factors on the left, but not on the right, then Z B ∗∗ ( A ∗∗ ) = Z tB ∗∗ ( A ∗∗ ).For a Banach algebra A , if e ′′ is a left mixed unit for A ∗∗ , then it is clear that Z te ′′ ( A ∗∗ ) = A ∗∗ . Then by using preceding theorem, we obtain Proposition 2.10 from[14].We say that a Banach space B is weakly complete, if for every ( b α ) α ⊆ B , we have b α w → b ′′ in B ∗∗ , it follows that b ′′ ∈ B . Theorem 2-6.
Suppose that B is a weakly complete Banach space. Then we havethe following assertions.(1) Let B be a Banach left A − module and B ∗∗ has a left mixed unit e ′′ ∈ A ∗∗ .If AB ∗∗ ⊆ B , then B is reflexive. (2) Let B be a Banach right A − module and B ∗∗ has a right mixed unit e ′′ ∈ A ∗∗ .If Z A ∗∗ ( B ∗∗ ) A ⊆ B , then Z A ∗∗ ( B ∗∗ ) = B . Proof. (1) Assume that b ′′ ∈ B ∗∗ . Since e ′′ is also mixed unit for A ∗∗ , there is a BAI ( e α ) α ⊆ A for A such that e α w ∗ → e ′′ . Then π ∗∗∗ ℓ ( e α , b ′′ ) w ∗ → π ∗∗∗ ℓ ( e ′′ , b ′′ ) = b ′′ in B ∗∗ . Since AB ∗∗ ⊆ B , we have π ∗∗∗ ℓ ( e α , b ′′ ) ∈ B . Consequently π ∗∗∗ ℓ ( e α , b ′′ ) w → π ∗∗∗ ℓ ( e ′′ , b ′′ ) = b ′′ in B . Since B is a weakly complete, b ′′ ∈ B ,and so B is reflexive.(2) Since b ′′ ∈ Z A ∗∗ ( B ∗∗ ), we have π ∗∗∗ r ( b ′′ , e α ) w ∗ → π ∗∗∗ r ( b ′′ , e ′′ ) = b ′′ in B ∗∗ . Since Z A ∗∗ ( B ∗∗ ) A ⊆ B , π ∗∗∗ r ( b ′′ , e α ) ∈ B . Consequently we have π ∗∗∗ r ( b ′′ , e α ) w → π ∗∗∗ r ( b ′′ , e ′′ ) = b ′′ in B . It follows that b ′′ ∈ B , since B is a weakly complete. (cid:3) A functional a ′ in A ∗ is said to be wap (weakly almost periodic) on A if the mapping a → a ′ a from A into A ∗ is weakly compact. The preceding definition is equivalent tothe following condition, see [6, 14, 18].For any two net ( a α ) α and ( b β ) β in { a ∈ A : k a k≤ } , we have lim α lim β h a ′ , a α b β i = lim β lim α h a ′ , a α b β i , whenever both iterated limits exist. The collection of all wap functionals on A isdenoted by wap ( A ). Also we have a ′ ∈ wap ( A ) if and only if h a ′′ b ′′ , a ′ i = h a ′′ ob ′′ , a ′ i for every a ′′ , b ′′ ∈ A ∗∗ . Definition 2-7.
Let B be a Banach left A − module . Then, b ′ ∈ B ∗ is said to be leftweakly almost periodic functional if the set { π ∗ ℓ ( b ′ , a ) : a ∈ A, k a k≤ } is relativelyweakly compact. We denote by wap ℓ ( B ) the closed subspace of B ∗ consisting of allthe left weakly almost periodic functionals in B ∗ .The definition of the right weakly almost periodic functional (= wap r ( B )) is the same.By [18], the definition of wap ℓ ( B ) is equivalent to the following h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = h π t ∗∗∗ tℓ ( a ′′ , b ′′ ) , b ′ i for all a ′′ ∈ A ∗∗ and b ′′ ∈ B ∗∗ . Thus, we can write wap ℓ ( B ) = { b ′ ∈ B ∗ : h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = h π t ∗∗∗ tℓ ( a ′′ , b ′′ ) , b ′ i f or all a ′′ ∈ A ∗∗ , b ′′ ∈ B ∗∗ } . Theorem 2-8.
Suppose that B is a Banach left A − module . Consider the followingstatements.(1) B ∗ A ⊆ wap ℓ ( B ).(2) AA ∗∗ ⊆ Z B ∗∗ ( A ∗∗ ).(3) AA ∗∗ ⊆ AZ B ∗∗ (( A ∗ A ) ∗ ).Then, we have (1) ⇔ (2) ⇐ (3). Proof. (1) ⇒ (2)Let ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Then for all a ∈ A and a ′′ ∈ A ∗∗ , we have h π ∗∗∗ ℓ ( aa ′′ , b ′′ α ) , b ′ i = h aa ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) i = h a ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) a i = h a ′′ , π ∗∗ ℓ ( b ′′ α , b ′ a ) i = h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ a i → h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ a i = h π ∗∗∗ ℓ ( aa ′′ , b ′′ ) , b ′ i . Hence aa ′′ ∈ Z B ∗∗ ( A ∗∗ ).(2) ⇒ (1)Let a ∈ A and b ′ ∈ B ∗ . Then h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ a i = h aπ ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = h π ∗∗∗ ℓ ( aa ′′ , b ′′ ) , b ′ i = h π t ∗∗∗ tℓ ( aa ′′ , b ′′ ) , b ′ i = h π t ∗∗∗ tℓ ( a ′′ , b ′′ ) , b ′ a i . It follow that b ′ a ∈ wap ℓ ( B ).(3) ⇒ (2)Since AZ B ∗∗ (( A ∗ A ) ∗ ) ⊆ Z B ∗∗ ( A ∗∗ ), proof is hold. (cid:3) In the preceding theorem, if we take B = A , then we obtain Theorem 3.6 from [14]and the same as preceding theorem, we can claim the following assertions:If B is a Banach right A − module , then for the following statements we have(1) ⇔ (2) ⇐ (3).(1) AB ∗ ⊆ wap r ( B ).(2) A ∗∗ A ⊆ Z B ∗∗ ( A ∗∗ ).(3) A ∗∗ A ⊆ Z B ∗∗ (( A ∗ A ) ∗ ) A .The proof of the this assertion is similar to proof of Theorem 2-8. Corollary 2-9.
Suppose that B is a Banach A − bimodule . Then if A is a left [resp.right] ideal in A ∗∗ , then B ∗ A ⊆ wap ℓ ( B ) [resp. AB ∗ ⊆ wap r ( B )]. Example 2-10.
Suppose that 1 ≤ p ≤ ∞ and q is conjugate of p . We know that if G is compact, then L ( G ) is a two-sided ideal in its second dual of it. By precedingTheorem we have L q ( G ) ∗ L ( G ) ⊆ wap ℓ ( L p ( G )) and L ( G ) ∗ L q ( G ) ⊆ wap r ( L p ( G )).Also if G is finite, then L q ( G ) ⊆ wap ℓ ( L p ( G )) ∩ wap r ( L p ( G )). Hence we conclude that Z L ( G ) ∗∗ ( L p ( G ) ∗∗ ) = L p ( G ) and Z L p ( G ) ∗∗ ( L ( G ) ∗∗ ) = L ( G ) . Theorem 2-11.
We have the following assertions.(1) Suppose that B is a Banach left A − module and b ′ ∈ B ∗ . Then b ′ ∈ wap ℓ ( B )if and only if the adjoint of the mapping π ∗ ℓ ( b ′ , ) : A → B ∗ is weak ∗ − to − weak continuous.(2) Suppose that B is a Banach right A − module and b ′ ∈ B ∗ . Then b ′ ∈ wap r ( B )if and only if the adjoint of the mapping π ∗ r ( b ′ , ) : B → A ∗ is weak ∗ − to − weak continuous. Proof. (1) Assume that b ′ ∈ wap ℓ ( B ) and π ∗ ℓ ( b ′ , ) ∗ : B ∗∗ → A ∗ is the adjoint of π ∗ ℓ ( b ′ , ). Then for every b ′′ ∈ B ∗∗ and a ∈ A , we have h π ∗ ℓ ( b ′ , ) ∗ b ′′ , a i = h b ′′ , π ∗ ℓ ( b ′ , a ) i . Suppose ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ and a ′′ ∈ A ∗∗ and ( a β ) β ⊆ A suchthat a β w ∗ → a ′′ . By easy calculation, for all y ′′ ∈ B ∗∗ and y ′ ∈ B ∗ , we have h π ∗ ℓ ( y ′ , ) ∗ , y ′′ i = π ∗∗ ℓ ( y ′′ , y ′ ) . Since b ′ ∈ wap ℓ ( B ), h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i → h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i . Then we have the following statementslim α h a ′′ , π ∗ ℓ ( b ′ , ) ∗ b ′′ α i = lim α h a ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) i = lim α h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = h a ′′ , π ∗ ℓ ( b ′ , ) ∗ b ′′ i . It follow that the adjoint of the mapping π ∗ ℓ ( b ′ , ) : A → B ∗ is weak ∗ − to − weak continuous.Conversely, let the adjoint of the mapping π ∗ ℓ ( b ′ , ) : A → B ∗ is weak ∗ − to − weak continuous. Suppose ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ and b ′ ∈ B ∗ . Thenfor every a ′′ ∈ A ∗∗ , we havelim α h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = lim α h a ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) i = lim α h a ′′ , π ∗ ℓ ( b ′ , ) ∗ b ′′ α i = h a ′′ , π ∗ ℓ ( b ′ , ) ∗ b ′′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i . It follow that b ′ ∈ wap ℓ ( B ).(2) proof is similar to (1). (cid:3) Corollary 2-12.
Let A be a Banach algebra. Assume that a ′ ∈ A ∗ and T a ′ is thelinear operator from A into A ∗ defined by T a ′ a = a ′ a . Then, a ′ ∈ wap ( A ) if and onlyif the adjoint of T a ′ is weak ∗ − to − weak continuous. So A is Arens regular if andonly if the adjoint of the mapping T a ′ a = a ′ a is weak ∗ − to − weak continuous forevery a ′ ∈ A ∗ . Lw ∗ w -property and Rw ∗ w -property In this section, we introduce the new definition as
Lef t − weak ∗ − to − weak propertyand Right − weak ∗ − to − weak property for Banach algebra A and make some relationsbetween these concepts and topological centers of module actions. As some conclu-sion, for locally compact group G , if a ∈ M ( G ) has Lw ∗ w − property [resp. Rw ∗ w − property], then we have L ( G ) ∗∗ ∗ a = L ( G ) ∗∗ [resp. a ∗ L ( G ) ∗∗ = L ( G ) ∗∗ ], and alsowe have Z L ( G ) ∗∗ ( M ( G ) ∗∗ ) = M ( G ) ∗∗ and Z M ( G ) ∗∗ ( L ( G ) ∗∗ ) = L ( G ). For a finitegroup G , we have Z M ( G ) ∗∗ ( L ( G ) ∗∗ ) = L ( G ) ∗∗ and Z L ( G ) ∗∗ ( M ( G ) ∗∗ ) = M ( G ) ∗∗ . Definition 3-1.
Let B be a Banach left A − module . We say that a ∈ A has Lef t − weak ∗ − to − weak property (= Lw ∗ w − property) with respect to B , if forall ( b ′ α ) α ⊆ B ∗ , ab ′ α w ∗ → ab ′ α w →
0. If every a ∈ A has Lw ∗ w − propertywith respect to B , then we say that A has Lw ∗ w − property with respect to B . Thedefinition of the Right − weak ∗ − to − weak property (= Rw ∗ w − property) is thesame.We say that a ∈ A has weak ∗ − to − weak property (= w ∗ w − property) with respectto B if it has Lw ∗ w − property and Rw ∗ w − property with respect to B .If a ∈ A has Lw ∗ w − property with respect to itself, then we say that a ∈ A has Lw ∗ w − property.For preceding definition, we have some examples and remarks as follows.a) If B is a Banach A -bimodule and reflexive, then A has w ∗ w − property with respectto B . Then we have the following statenents for group algebras.i) L ( G ), M ( G ) and A ( G ) have w ∗ w − property when G is finite.ii) Let G be locally compact group. L ( G ) [resp. M ( G )] has w ∗ w − property [resp. Lw ∗ w − property ] with respect to L p ( G ) whenever p > B is a Banach left A − module and e is left unit element of A suchthat eb = b for all b ∈ B . If e has Lw ∗ w − property, then B is reflexive.c) If S is a compact semigroup, then C + ( S ) = { f ∈ C ( S ) : f > } has w ∗ w − property. Theorem 3-2.
Suppose that B is a Banach A − bimodule . Then we have thefollowing assertions.(1) If A ∗∗ = a A ∗∗ [resp. A ∗∗ = A ∗∗ a ] for some a ∈ A and a has Rw ∗ w − property [resp. Lw ∗ w − property], then Z B ∗∗ ( A ∗∗ ) = A ∗∗ .(2) If B ∗∗ = a B ∗∗ [resp. B ∗∗ = B ∗∗ a ] for some a ∈ A and a has Rw ∗ w − property [resp. Lw ∗ w − property] with respect to B , then Z A ∗∗ ( B ∗∗ ) = B ∗∗ . Proof. (1) Suppose that A ∗∗ = a A ∗∗ for some a ∈ A and a has Rw ∗ w − property. Let ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Then for all a ∈ A and b ′ ∈ B ∗ , we have h π ∗∗ ℓ ( b ′′ α , b ′ ) , a i = h b ′′ α , π ∗ ℓ ( b ′ , a ) i → h b ′′ , π ∗ ℓ ( b ′ , a ) i = h π ∗∗ ℓ ( b ′′ , b ′ ) , a i , it follow that π ∗∗ ℓ ( b ′′ α , b ′ ) w ∗ → π ∗∗ ℓ ( b ′′ , b ′ ). Also we can write π ∗∗ ℓ ( b ′′ α , b ′ ) a w ∗ → π ∗∗ ℓ ( b ′′ , b ′ ) a . Since a has Rw ∗ w − property, π ∗∗ ℓ ( b ′′ α , b ′ ) a w → π ∗∗ ℓ ( b ′′ , b ′ ) a .Now let a ′′ ∈ A ∗∗ . Then there is x ′′ ∈ A ∗∗ such that a ′′ = a x ′′ consequentlywe have h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = h a ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) i = h x ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) a i→ h x ′′ , π ∗∗ ℓ ( b ′′ , b ′ ) a i = h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i . We conclude that a ′′ ∈ Z B ∗∗ ( A ∗∗ ). Proof of the next part is the same as thepreceding proof.(2) Let B ∗∗ = a B ∗∗ for some a ∈ A and a has Rw ∗ w − property with respectto B . Assume that ( a ′′ α ) α ⊆ A ∗∗ such that a ′′ α w ∗ → a ′′ . Then for all b ∈ B , wehave h π ∗∗ r ( a ′′ α , b ′ ) , b i = h a ′′ α , π ∗ r ( b ′ , b ) i → h a ′′ , π ∗ r ( b ′ , b ) i = h π ∗∗ r ( a ′′ , b ′ ) , b i . We conclude that π ∗∗ r ( a ′′ α , b ′ ) w ∗ → π ∗∗ r ( a ′′ , b ′ ) then we have π ∗∗ r ( a ′′ α , b ′ ) a w ∗ → π ∗∗ r ( a ′′ , b ′ ) a . Since a has Rw ∗ w − property with respect to B , π ∗∗ r ( a ′′ α , b ′ ) a w → π ∗∗ r ( a ′′ , b ′ ) a .Now let b ′′ ∈ B ∗∗ . Then there is x ′′ ∈ B ∗∗ such that b ′′ = a x ′′ . Hence, wehave h π ∗∗∗ r ( b ′′ , a ′′ α ) , b ′ i = h b ′′ , π ∗∗ r ( a ′′ α , b ′ ) i = h a x ′′ , π ∗∗ r ( a ′′ α , b ′ ) i = h x ′′ , π ∗∗ r ( a ′′ α , b ′ ) a i → h x ′′ , π ∗∗ r ( a ′′ , b ′ ) a i = h b ′′ , π ∗∗ r ( a ′′ , b ′ ) i = h π ∗∗∗ r ( b ′′ , a ′′ ) , b ′ i . It follow that b ′′ ∈ Z A ∗∗ ( B ∗∗ ). The next part is similar to the preceding proof. (cid:3) Example 3-3. (1) By using Theorem 2-3, for locally compact group G , if a ∈ M ( G ) has Lw ∗ w − property [resp. Rw ∗ w − property], then we have L ( G ) ∗∗ ∗ a = L ( G ) ∗∗ [resp. a ∗ L ( G ) ∗∗ = L ( G ) ∗∗ ].(2) If G is finite, then by Theorem 2-3, we have Z M ( G ) ∗∗ ( L ( G ) ∗∗ ) = L ( G ) ∗∗ and Z L ( G ) ∗∗ ( M ( G ) ∗∗ ) = M ( G ) ∗∗ . Theorem 3-4.
Suppose that B is a Banach A − bimodule and A has a BAI . Thenwe have the following assertions.(1) If B ∗ factors on the left [resp. right] with respect to A and A has Rw ∗ w − property [resp. Lw ∗ w − property], then Z B ∗∗ ( A ∗∗ ) = A ∗∗ .(2) If B ∗ factors on the left [resp. right] with respect to A and A has Rw ∗ w − property [resp. Lw ∗ w − property] with respect B , then Z A ∗∗ ( B ∗∗ ) = B ∗∗ . Proof. (1) Assume that B ∗ factors on the left and A has Rw ∗ w − property. Let( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Since B ∗ A = B ∗ , for all b ′ ∈ B ∗ there are x ∈ A and y ′ ∈ B ∗ such that b ′ = y ′ x . Then for all a ∈ A , we have h π ∗∗ ℓ ( b ′′ α , y ′ ) x, a i = h b ′′ α , π ∗ ℓ ( y ′ , a ) x i = h π ∗∗ ℓ ( b ′′ α , b ′ ) , a i = h b ′′ α , π ∗ ℓ ( b ′ , a ) i → h b ′′ , π ∗ ℓ ( b ′ , a ) i = h π ∗∗ ℓ ( b ′′ , y ′ ) x, a i . Thus, we conclude that π ∗∗ ℓ ( b ′′ α , y ′ ) x w ∗ → π ∗∗ ℓ ( b ′′ , y ′ ) x . Since A has Rw ∗ w − property, π ∗∗ ℓ ( b ′′ α , y ′ ) x w → h π ∗∗ ℓ ( b ′′ , y ′ ) x . Now let a ′′ ∈ A ∗∗ . Then h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = h a ′′ , π ∗∗ ℓ ( b ′′ α , b ′ ) i = h a ′′ , π ∗∗ ℓ ( b ′′ α , y ′ ) x i → h a ′′ , π ∗∗ ℓ ( b ′′ , y ′ ) x i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i . It follow that a ′′ ∈ Z B ∗∗ ( A ∗∗ ).If B ∗ factors on the right with respect to A , and assume that A has Lw ∗ w − property, then proof is the same as preceding proof.(2) Let B ∗ factors on the left with respect to A and A has Rw ∗ w − property withrespect to B . Assume that ( a ′′ α ) α ⊆ A ∗∗ such that a ′′ α w ∗ → a ′′ . Since B ∗ A = B ∗ ,for all b ′ ∈ B ∗ there are x ∈ A and y ′ ∈ B ∗ such that b ′ = y ′ x . Then for all b ∈ B , we have h π ∗∗ r ( a ′′ α , y ′ ) x, b i = h π ∗∗ r ( a ′′ α , b ′ ) , b i = h a ′′ α , π ∗ r ( b ′ , b ) i = h a ′′ , π ∗ r ( b ′ , b ) i = h π ∗∗ r ( a ′′ , y ′ ) x, b i . Consequently π ∗∗ r ( a ′′ α , y ′ ) x w ∗ → π ∗∗ r ( a ′′ , y ′ ) x . Since A has Rw ∗ w − property withrespect to B , π ∗∗ r ( a ′′ α , y ′ ) x w → π ∗∗ r ( a ′′ , y ′ ) x . It follow that for all b ′′ ∈ B ∗∗ , wehave h π ∗∗∗ r ( b ′′ , a ′′ α ) , b ′ i = h b ′′ , π ∗∗ r ( a ′′ α , y ′ ) x i → h b ′′ , π ∗∗ r ( a ′′ , y ′ ) x i = h π ∗∗∗ r ( b ′′ , a ′′ ) , b ′ i . Thus we conclude that b ′′ ∈ Z A ∗∗ ( B ∗∗ ).The proof of the next assertions is the same as the preceding proof. (cid:3) Theorem 3-5.
Suppose that B is a Banach A − bimodule . Then we have the followingassertions.(1) If a ∈ A has Rw ∗ w − property with respect to B , then a A ∗∗ ⊆ Z B ∗∗ ( A ∗∗ )and a B ∗ ⊆ wap ℓ ( B ).(2) If a ∈ A has Lw ∗ w − property with respect to B , then A ∗∗ a ⊆ Z B ∗∗ ( A ∗∗ )and B ∗ a ⊆ wap ℓ ( B ).(3) If a ∈ A has Rw ∗ w − property with respect to B , then a B ∗∗ ⊆ Z A ∗∗ ( B ∗∗ )and B ∗ a ⊆ wap r ( B ).(4) If a ∈ A has Lw ∗ w − property with respect to B , then B ∗∗ a ⊆ Z A ∗∗ ( B ∗∗ )and a B ∗ ⊆ wap r ( B ). Proof. (1) Let ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ . Then for all a ∈ A and b ′ ∈ B ∗ ,we have h π ∗∗ ℓ ( b ′′ α , b ′ ) a , a i = h π ∗∗ ℓ ( b ′′ α , b ′ ) , a a i = h b ′′ α , π ∗ ℓ ( b ′ , a a ) i→ h b ′′ , π ∗ ℓ ( b ′ , a a ) i = h π ∗∗ ℓ ( b ′′ , b ′ ) a , a i . It follow that π ∗∗ ℓ ( b ′′ α , b ′ ) a w ∗ → π ∗∗ ℓ ( b ′′ , b ′ ) a . Since a has Rw ∗ w − propertywith respect to B , π ∗∗ ℓ ( b ′′ α , b ′ ) a w → π ∗∗ ℓ ( b ′′ , b ′ ) a .We conclude that a a ′′ ∈ Z B ∗∗ ( A ∗∗ ) so that a A ∗∗ ∈ Z B ∗∗ ( A ∗∗ ). Since π ∗∗ ℓ ( b ′′ , b ′ ) a = π ∗∗ ℓ ( b ′′ , b ′ a ), a B ∗ ⊆ wap ℓ ( B ).(2) proof is similar to (1). (3) Assume that ( a ′′ α ) α ⊆ A ∗∗ such that a ′′ α w ∗ → a ′′ . Let b ∈ B and b ′ ∈ B ∗ . Thenwe have h π ∗∗ r ( a ′′ α , b ′ ) a , b i = h π ∗∗ r ( a ′′ α , b ′ ) , a b i = h a ′′ α , π ∗ r ( b ′ , a b ) i→ h a ′′ , π ∗ r ( b ′ , a b ) i = h π ∗∗ r ( a ′′ , b ′ ) a , b i . Thus we conclude π ∗∗ r ( a ′′ α , b ′ ) a w ∗ → π ∗∗ r ( a ′′ , b ′ ) a . Since a has Rw ∗ w − prop-erty with respect to B , π ∗∗ r ( a ′′ α , b ′ ) a w → π ∗∗ r ( a ′′ , b ′ ) a . If b ′′ ∈ B ∗∗ , then wehave h π ∗∗∗ r ( a b ′′ , a ′′ α ) , b ′ i = h a b ′′ , π ∗∗∗ r ( a ′′ α , b ′ ) i = h b ′′ , π ∗∗ r ( a ′′ α , b ′ ) a i = h b ′′ , π ∗∗ r ( a ′′ α , b ′ ) a i = h π ∗∗∗ r ( a b ′′ , a ′′ ) , b ′ i . It follow that a b ′′ ∈ Z A ∗∗ ( B ∗∗ ). Consequently we have a B ∗∗ ∈ Z A ∗∗ ( B ∗∗ ).The proof of the next assertion is clear.(4) Proof is similar to (3). (cid:3) Theorem 3-6.
Let B be a Banach A − bimodule . Then we have the followingassertions.(1) Suppose lim α lim β h b ′ β , b α i = lim β lim α h b ′ β , b α i , for every ( b α ) α ⊆ B and ( b ′ β ) β ⊆ B ∗ . Then A has Lw ∗ w − property and Rw ∗ w − property with respect to B .(2) If for some a ∈ A ,lim α lim β h ab ′ β , b α i = lim β lim α h ab ′ β , b α i , for every ( b α ) α ⊆ B and ( b ′ β ) β ⊆ B ∗ , then a has Rw ∗ w − property withrespect to B . Also if for some a ∈ A ,lim α lim β h b ′ β a, b α i = lim β lim α h b ′ β a, b α i , for every ( b α ) α ⊆ B and ( b ′ β ) β ⊆ B ∗ , then a has Lw ∗ w − property withrespect to B . Proof. (1) Assume that a ∈ A such that ab ′ β w ∗ → b ′ β ) β ⊆ B ∗ . Let b ′′ ∈ B ∗∗ and ( b α ) α ⊆ B such that b α w ∗ → b ′′ . Thenlim β h b ′′ , ab ′ β i = lim β lim α h b α , ab ′ β i = lim β lim α h ab ′ β , b α i = lim α lim β h ab ′ β , b α i = 0 . We conclude that ab ′ β w →
0, so A has Lw ∗ w − property. It also easy that A has Rw ∗ w − property. (2) Proof is easy and is the same as (1). (cid:3) Definition 3-7.
Let B be a Banach left A − module . We say that B ∗ strong factorson the left [resp. right] if for all ( b ′ α ) α ⊆ B ∗ there are ( a α ) α ⊆ A and b ′ ∈ B ∗ suchthat b ′ α = b ′ a α [resp. b ′ α = a α b ′ ] where ( a α ) α has limit the weak ∗ topology in A ∗∗ .If B ∗ strong factors on the left and right, then we say that B ∗ strong factors on theboth side.It is clear that if B ∗ strong factors on the left [resp. right], then B ∗ factors on theleft [resp. right]. Theorem 3-8.
Suppose that B is a Banach A − bimodule . Assume that AB ∗ ⊆ wap ℓ B . If B ∗ strong factors on the left [resp. right], then A has Lw ∗ w − property[resp. Rw ∗ w − property ] with respect to B . Proof.
Let ( b ′ α ) α ⊆ B ∗ such that ab ′ α w ∗ →
0. Since B ∗ strong factors on the left, thereare ( a α ) α ⊆ A and b ′ ∈ B ∗ such that b ′ α = b ′ a α . Let b ′′ ∈ B ∗∗ and ( b β ) β ⊆ B suchthat b β w ∗ → b ′′ . Then we havelim α h b ′′ , ab ′ α i = lim α lim β h b β , ab ′ α i = lim α lim β h ab ′ α , b β i = lim α lim β h ab ′ a α , b β i = lim α lim β h ab ′ , a α b β i = lim β lim α h ab ′ , a α b β i = lim β lim α h ab ′ α , b β i = 0It follow that ab ′ α w → (cid:3) Problems . (1) Suppose that B is a Banach A − bimodule . If B is left or right factors withrespect to A , dose A has Lw ∗ w − property or Rw ∗ w − property, respectively?(2) Suppose that B is a Banach A − bimodule . Let A has Lw ∗ w − property withrespect to B . Dose Z B ∗∗ ( A ∗∗ ) = A ∗∗ ?
4. Factorization properties and topological centers ofleft module actions
Let B be a left Banach A − module and e be a left unit element of A . We saythat e is a left unit (resp. weakly left unit) A − module for B , if π ℓ ( e, b ) = b (resp. h b ′ , π ℓ ( e, b ) i = h b ′ , b i for all b ′ ∈ B ∗ ) where b ∈ B . The definition of right unit (resp.weakly right unit) A − module is similar.We say that a Banach A − bimodule B is a unital A − module , if B has left and rightunit A − module that are equal then we say that B is unital A − module . Let B be a left Banach A − module and ( e α ) α ⊆ A be a LAI [resp. weakly leftapproximate identity(=WLAI)] for A . We say that ( e α ) α is left approximate identity(= LAI )[ resp. weakly left approximate identity (=
W LAI )] for B , if for all b ∈ B , π ℓ ( e α , b ) → b ( resp. π ℓ ( e α , b ) w → b ). The definition of the right approximate identity(= RAI )[ resp. weakly right approximate identity (=
W RAI )] is similar.( e α ) α ⊆ A is called a approximate identity (= AI )[ resp. weakly approximate identity( W AI )] for B , if B has the same left and right approximate identity [ resp. weaklyleft and right approximate identity ].Let ( e α ) α ⊆ A be weak ∗ left approximate identity for A ∗∗ . We say that ( e α ) α is weak ∗ left approximate identity as A ∗∗ − module (= W ∗ LAI as A ∗∗ − module ) for B ∗∗ , if for all b ′′ ∈ B ∗∗ , we have π ∗∗∗ ℓ ( e α , b ′′ ) w ∗ → b ′′ . The definition of the weak ∗ rightapproximate identity (= W ∗ RAI ) is similar.( e α ) α ⊆ A is called a weak ∗ approximate identity (= W ∗ AI ) for B ∗∗ , if B ∗∗ has thesame weak ∗ left and right approximate identity.Let B be a Banach A − bimodule . We say that B is a left [resp. right] factors withrespect to A , if BA = B [resp. AB = B ]. Theorem 4-1.
Let B be a Banach left A − module and ( e α ) α ⊆ A be a LBAI for B . Then the following assertions hold.(1) For each b ′ ∈ B ∗ , we have π ∗ ℓ ( b ′ , e α ) w ∗ → b ′ .(2) B ∗ factors on the left with respect to A if and only if B ∗∗ has a W ∗ LBAI ( e α ) α ⊆ A .(3) B ∗∗ has a W ∗ LBAI ( e α ) α ⊆ A if and only if B ∗∗ has a left unit element e ′′ ∈ A ∗∗ such that e α w ∗ → e ′′ . Proof. (1) Let b ∈ B and b ′ ∈ B ∗ . Since | h b ′ , π ℓ ( e α , b ) |≤k b ′ kk π ℓ ( e α , b ) k , wehave the following equalitylim α h π ∗ ℓ ( b ′ , e α ) , b i = lim α h b ′ , π ℓ ( e α , b ) i = 0 . It follows that π ∗ ℓ ( b ′ , e α ) w ∗ → B ∗ factors on the left with respect to A . Then for every b ′ ∈ B ∗ , thereare x ′ ∈ B ∗ and a ∈ A such that b ′ = x ′ a . Then for every b ′′ ∈ B ∗∗ , we have h π ∗∗∗ ℓ ( e α , b ′′ ) , b ′ i = h e α , π ∗∗ ℓ ( b ′′ , b ′ ) i = h π ∗∗ ℓ ( b ′′ , b ′ ) , e α i = h b ′′ , π ∗ ℓ ( b ′ , e α ) i = h b ′′ , π ∗ ℓ ( x ′ a, e α ) i = h b ′′ , π ∗ ℓ ( x ′ , ae α ) i = h π ∗∗ ℓ ( b ′′ , x ′ ) , ae α i → h π ∗∗ ℓ ( b ′′ , x ′ ) , a i = h b ′′ , b ′ i . It follows that π ∗∗∗ ℓ ( e α , b ′′ ) w ∗ → b ′′ , and so B ∗∗ has W ∗ LBAI .Conversely, let b ′ ∈ B ∗ . Then for every b ′′ ∈ B ∗∗ , we have h b ′′ , π ∗ ℓ ( b ′ , e α ) i = h π ∗∗∗ ℓ ( e α , b ′′ ) , b ′ i → h b ′′ , b ′ i . It follows that π ∗ ℓ ( b ′ , e α ) w → b ′ , and so by Cohen factorization theorem, we are done.(3) Assume that B ∗∗ has a W ∗ LBAI ( e α ) α ⊆ A . Without loss generality, let e ′′ ∈ A ∗∗ be a left unit for A ∗∗ with respect to the first Arens product suchthat e α w ∗ → e ′′ . Then for each b ′ ∈ B ∗ , we have h π ∗∗∗ ℓ ( e ′′ , b ′′ ) , b ′ i = h e ′′ , π ∗∗ ℓ ( b ′′ , b ′ ) i = lim α h e α , π ∗∗ ℓ ( b ′′ , b ′ ) i = lim α h π ∗∗ ℓ ( b ′′ , b ′ ) , e α i = lim α h b ′′ , π ∗ ℓ ( b ′ , e α ) i = lim α h π ∗∗∗∗ ℓ ( b ′ , e α ) , b ′′ i = lim α h b ′ , π ∗∗∗ ℓ ( e α , b ′′ ) i = lim α h π ∗∗∗ ℓ ( e α , b ′′ ) , b ′ i = h b ′′ , b ′ i . Thus e ′′ ∈ A ∗∗ is a left unit for B ∗∗ .Conversely, let e ′′ ∈ A ∗∗ be a left unit for B ∗∗ and assume that e α w ∗ → e ′′ in A ∗∗ . Then for every b ′′ ∈ B ∗∗ and b ′ ∈ B ∗ , we have h π ∗∗∗ ℓ ( e α , b ′′ ) , b ′ i = h e α , π ∗∗ ℓ ( b ′′ , b ′ ) i→ h e ′′ , π ∗∗ ℓ ( b ′′ , b ′ ) i = h π ∗∗∗ ℓ ( e ′′ , b ′′ ) , b ′ i = h b ′′ , b ′ i . It follows that π ∗∗∗ ℓ ( e α , b ′′ ) w ∗ → b ′′ . (cid:3) Corollary 4-2.
Let B be a Banach left A − module and A has a BLAI . If B ∗∗ hasa W ∗ LBAI , then { a ′′ ∈ A ∗∗ : Aa ′′ ⊆ A } ⊆ Z B ∗∗ ( A ∗∗ ) . Proof.
By using the preceding theorem, since B ∗∗ has W ∗ LBAI , B ∗ factors on theleft with respect to A . Suppose that b ′ ∈ B ∗ . Then there are x ′ ∈ B ∗ and a ∈ A such that b ′ = x ′ a . Assume that a ′′ ∈ A ∗∗ such that Aa ′′ ⊆ A . Let b ′′ ∈ B ∗∗ and( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ in B ∗∗ . Then we have the following equalitylim α h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = lim α h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , x ′ a i = lim α h aπ ∗∗∗ ℓ ( a ′′ , b ′′ α ) , x ′ i = h π ∗∗∗ ℓ ( aa ′′ , b ′′ ) , x ′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i . It follows that a ′′ ∈ Z B ∗∗ ( A ∗∗ ) . (cid:3) In the preceding corollary, if we take B = A , then we have the following conclusion { a ′′ ∈ A ∗∗ : Aa ′′ ⊆ A } ⊆ Z ( A ∗∗ ) . Theorem 4-3.
Let B be a Banach left A − module and suppose that b ′ ∈ wap ℓ ( B ).Let a ′′ ∈ A ∗∗ and ( a α ) α ⊆ A such that a α w ∗ → a ′′ in A ∗∗ . Then we have π ∗ ℓ ( b ′ , a α ) w → π ∗∗∗∗ ℓ ( b ′ , a ′′ ) . Proof.
Assume that b ′′ ∈ B ∗∗ . Then we have the following equality h π ∗∗∗∗ ℓ ( b ′ , a ′′ ) , b ′′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = lim α h π ∗∗∗ ℓ ( a α , b ′′ ) , b ′ i = lim α h b ′′ , π ∗ ℓ ( b ′ , a α ) i . Now suppose that ( b ′′ β ) β ⊆ B ∗∗ such that b ′′ β w ∗ → b ′′ . Since b ′ ∈ wap ℓ ( B ), we have h π ∗∗∗∗ ℓ ( b ′ , a ′′ ) , b ′′ β i = h π ∗∗∗ ℓ ( a ′′ , b ′′ β ) , b ′ i → h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i = h π ∗∗∗∗ ℓ ( b ′ , a ′′ ) , b ′′ i . Thus π ∗∗∗∗ ℓ ( b ′ , a ′′ ) ∈ ( B ∗∗ , weak ∗ ) ∗ = B ∗ . So we conclude that π ∗ ℓ ( b ′ , a α ) w → π ∗∗∗∗ ℓ ( b ′ , a ′′ ) in B ∗∗ . (cid:3) In the preceding corollary, if we take B = A , then we obtain the following result.Suppose that a ′ ∈ wap ( A ) and a ′′ ∈ A ∗∗ such that a α w ∗ → a ′′ where( a α ) α ⊆ A . Then we have a ′ a α w → a ′ a ′′ . Theorem 4-4.
Let B be a Banach left A − module and it has a BLAI ( e α ) α ⊆ A .Suppose that b ′ ∈ wap ℓ ( B ). Then we have π ∗ ℓ ( b ′ , e α ) w → b ′ . Proof.
Let b ′′ ∈ B ∗∗ and ( b β ) β ⊆ B such that b β w ∗ → b ′′ in B ∗∗ . Then for every b ′ ∈ wap ℓ ( B ), we have the following equalitylim α h b ′′ , π ∗ ℓ ( b ′ , e α ) i = lim α h π ∗∗∗∗ ℓ ( b ′ , e α ) , b ′′ i = lim α h b ′ , π ∗∗∗ ℓ ( e α , b ′′ ) i = lim α h π ∗∗∗ ℓ ( e α , b ′′ ) , b ′ i = lim α lim β h π ℓ ( e α , b β ) , b ′ i = lim β lim α h π ℓ ( e α , b β ) , b ′ i = lim β h b β , b ′ i = h b ′′ , b ′ i . It follows that π ∗ ℓ ( b ′ , e α ) w → b ′ . (cid:3) Corollary 4-5.
Let B be a Banach left A − module and it has a BLAI ( e α ) α ⊆ A .Suppose that wap ℓ ( B ) = B ∗ . Then B ∗ factors on the left with respect to A . Corollary 4-6.
Let A be an Arens regular Banach algebra with LBAI . Then A ∗ factors on the left. Example 4-7. i) Let G be finite group. Then we have the following equality M ( G ) ∗ L ( G ) = M ( G ) ∗ and L ∞ ( G ) L ( G ) = L ∞ ( G ) . ii) Consider the Banach algebra ( ℓ , . ) that is Arens regular Banach algebra with unitelement. Then we have ℓ ∞ .ℓ = ℓ ∞ .iii) Let K ( E ) be a compact operators from E into E . Let E = ℓ p with p ∈ (1 , ∞ ).Then by using [6, Theorem 2.6.23] and [19 , Example 2.3.18] , K ( ℓ p ) is Arens regularand amenable, respectively. Since K ( ℓ p ) is amenable, by using [19 , Proposition 2.2.1], K ( ℓ p ) has a BAI. Therefore by preceding corollary, K ( ℓ p ) ∗ factors on the left. Theorem 4-8.
Let B be a Banach left A − module and B ∗ factors on the left withrespect to A . If AA ∗∗ ⊆ Z B ∗∗ ( A ∗∗ ), then Z B ∗∗ ( A ∗∗ ) = A ∗∗ . Proof.
Let b ′′ ∈ B ∗∗ and ( b ′′ α ) α ⊆ B ∗∗ such that b ′′ α w ∗ → b ′′ in B ∗∗ . Suppose that a ′′ ∈ A ∗∗ . Since B ∗ factors on the left with respect to A , for every b ′ ∈ B ∗ , there are x ′ ∈ B ∗ and a ∈ A such that b ′ = x ′ a . Since aa ′′ ∈ Z B ∗∗ ( A ∗∗ ), we have h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , b ′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ α ) , x ′ a i = h aπ ∗∗∗ ℓ ( a ′′ , b ′′ α ) , x ′ i = h π ∗∗∗ ℓ ( aa ′′ , b ′′ α ) , x ′ i→ h π ∗∗∗ ℓ ( aa ′′ , b ′′ ) , x ′ i = h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , b ′ i . It follows that π ∗∗∗ ℓ ( a ′′ , b ′′ α ) w ∗ → h π ∗∗∗ ℓ ( a ′′ , b ′′ ) , and so a ′′ ∈ Z B ∗∗ ( A ∗∗ ). (cid:3) Corollary 4-9.
Let A be a Banach algebra and A ∗ factors on the left. If AA ∗∗ ⊆ Z ( A ∗∗ ), then A is Arens regular. Theorem 4-10.
Let B be a Banach left A − module and B ∗∗ has a LBAI withrespect to A ∗∗ . Then B ∗∗ has a left unit with respect to A ∗∗ . Proof.
Assume that ( e ′′ α ) α ⊆ A ∗∗ is a LBAI for B ∗∗ . By passing to a subnet, we maysuppose that there is e ′′ ∈ A ∗∗ such that e ′′ α w ∗ → e ′′ in A ∗∗ . Then for every b ′′ ∈ B ∗∗ and b ′ ∈ B ∗ , we have h π ∗∗∗ ℓ ( e ′′ , b ′′ ) , b ′ i = h e ′′ , π ∗∗ ℓ ( b ′′ , b ′ ) i = lim α h e ′′ α , π ∗∗ ℓ ( b ′′ , b ′ ) i = lim α h π ∗∗∗ ℓ ( e ′′ α , b ′′ ) , b ′ i = h b ′′ , b ′ i . It follows that π ∗∗∗ ℓ ( e ′′ , b ′′ ) = b ′′ . (cid:3) Corollary 4-11.
Let A be a Banach algebra and A ∗∗ has a LBAI . Then A ∗∗ has aleft unit with respect to the first Arens product. Theorem 4-12.
Let B be a Banach left A − module and it has a LBAI with respectto A . Then we have the following assertions.(1) B ∗ factors on the left with respect to A if and only if for each b ′ ∈ B ∗ , wehave π ∗ ℓ ( b ′ , e α ) w → b ′ in B ∗ .(2) B factors on the left with respect to A if and only if for each b ∈ B , we have π ∗ ℓ ( b, e α ) w → b in B . Proof. (1) Assume that B ∗ factors on the left with respect to A . Then for every b ′ ∈ B ∗ , there are x ′ ∈ B ∗ and a ∈ A such that b ′ = x ′ a . Then for every b ′′ ∈ B ∗∗ , we have h b ′′ , π ∗ ℓ ( b ′ , e α ) i = h b ′′ , π ∗ ℓ ( x ′ a, e α ) i = h b ′′ , π ∗ ℓ ( x ′ , ae α ) i = h π ∗∗ ℓ ( b ′′ , x ′ ) , ae α ) i → h π ∗∗ ℓ ( b ′′ , x ′ ) , a i = h b ′′ , b ′ i . It follows that π ∗ ℓ ( b ′ , e α ) w → b ′ in B ∗ . The converse by Cohen factorizationtheorem hold.(2) It is similar to the preceding proof. (cid:3) In the preceding theorem, if we take B = A , we obtain Lemma 2.1 from [14]. Theorem 4-13.
Let B be a Banach left A − module and A has a LBAI ( e α ) α ⊆ A such that e α w ∗ → e ′′ in A ∗∗ where e ′′ is a left unit for A ∗∗ . Suppose that Z te ′′ ( B ∗∗ ) = B ∗∗ . Then, B factors on the right with respect to A if and only if e ′′ is a left unit for B ∗∗ . Proof.
Assume that B factors on the right with respect to A . Then for every b ∈ B ,there are x ∈ B and a ∈ A such that b = ax . Then for every b ′ ∈ B ∗ , we have h π ∗ ℓ ( b ′ , e α ) , b i = h b ′ , π ℓ ( e α , b ) i = h π ∗∗∗ ℓ ( e α , b ) , b ′ i = h π ∗∗∗ ℓ ( e α , ax ) , b ′ i = h π ∗∗∗ ℓ ( e α a, x ) , b ′ i = h e α a, π ∗∗ ℓ ( x, b ′ ) i = h π ∗∗ ℓ ( x, b ′ ) , e α a i→ h π ∗∗ ℓ ( x, b ′ ) , a i = h b ′ , b i . It follows that π ∗ ℓ ( b ′ , e α ) w ∗ → b ′ in B ∗ . Let b ′′ ∈ B ∗∗ and ( b β ) β ⊆ B such that b β w ∗ → b ′′ in B ∗∗ . Since Z te ′′ ( B ∗∗ ) = B ∗∗ , for every b ′ ∈ B ∗ , we have the following equality h π ∗∗∗ ℓ ( e ′′ , b ′′ ) , b ′ i = lim α lim β h b ′ , π ℓ ( e α , b β ) i = lim β lim α h b ′ , π ℓ ( e α , b β ) i = lim β h b ′ , b β i = h b ′′ , b ′ i . It follows that π ∗∗∗ ℓ ( e ′′ , b ′′ ) = b ′′ , and so e ′′ is a left unit for B ∗∗ .Conversely, let e ′′ be a left unit for B ∗∗ and suppose that b ∈ B . Thren for every b ′ ∈ B ∗ , we have h b ′ , π ( e α , b ) i = h π ∗∗∗ ( e α , b ) , b ′ i = h e α , π ∗∗ ( b, b ′ ) i = h π ∗∗ ( b, b ′ ) , e α i = h e ′′ , π ∗∗ ( b, b ′ ) i = h π ∗∗∗ ( e ′′ , b ) , b ′ i = h b ′ , b i . Then we have π ∗ ℓ ( b ′ , e α ) w → b ′ in B ∗ , and so by Cohen factorization theorem we aredone. (cid:3) Corollary 4-14.
Let B be a Banach left A − module and A has a LBAI ( e α ) α ⊆ A such that e α w ∗ → e ′′ in A ∗∗ where e ′′ is a left unit for A ∗∗ . Suppose that Z te ′′ ( B ∗∗ ) = B ∗∗ . Then π ∗ ℓ ( b ′ , e α ) w → b ′ in B ∗ if and only if e ′′ is a left unit for B ∗∗ . Problem.
Let B be a Banach A − bimodule . Theni) If B factors, when B ∗ factors?ii) If B is separable, dose B necessarily factor on the one side?
5. Involution ∗− algebra and Arens regularity of module actionsDefinition 5-1. Let B be a Banach left A − module and let ( a α ) α ⊆ A has weak ∗ limit in A ∗∗ . We say that a α ) α is left regular with respect to B , if for every ( b β ) β ⊆ B , w ∗ − lim α w ∗ − lim β a α b β = w ∗ − lim β w ∗ − lim α a α b β , where ( b β ) β has weak ∗ limit in B ∗∗ .The definition of the right regular is similar. For a Banach A − bimodule B , if( a α ) α ⊆ A is left and right regular with respect to B , we say that ( a α ) α is regularwith respect to B . If ( a α ) α is left or right regular with respect to A , we write ( a α ) α is left or right regular, respectively. Example 5-2. i) Let B be a right Banach A − module and a ′′ ∈ A ∗∗ . Suppose that Z ra ′′ ( B ∗∗ ) = B ∗∗ and w ∗ − lim α = a ′′ where ( a α ) α ⊆ A . Then ( a α ) α is right regular with respect to B . ii) Let A be a Banach algebra and ( a α ) α ⊆ A weak ∗ convergence to some point of Z ( A ∗∗ ). Then it is clear that ( a α ) α is regular. Theorem 5-3. i) Let B be a Banach left (resp. right) A − module and suppose that( e α ) α ⊆ A is a LBAI for B . If ( e α ) α ⊆ A is left (resp. right) regular with respect to B , then B ∗ factors on the left (resp. right).ii) Let B be a Banach left (resp. right) A − module and suppose that B ∗∗ has a left(resp. right) unit as A ∗∗ -module. If B ∗ A (resp. AB ∗ ) is closed subspace of B ∗ , then B ∗ factors on the left (resp. right). Proof.
Let e ′′ ∈ A ∗∗ and e α w ∗ → e ′′ . Assume that ( b β ) β ⊆ B such that b β w ∗ → b ′′ in B ∗∗ .Then e ′′ b ′′ = w ∗ − lim α e α b ′′ = w ∗ − lim α w ∗ − lim β e α b β = w ∗ − lim β w ∗ − lim α e α b β = w ∗ − lim β b β = b ′′ . Thus for every b ′ ∈ B ∗ , we have h b ′′ , b ′ i = h e ′′ b ′′ , b ′ i = lim α h e α b ′′ , b ′ i = lim α h b ′′ , b ′ e α i . It follows that b ′ e α w ∗ → b ′ , and so by Cohen factorization theorem, we are done.ii) Assume that B ∗ A = B ∗ . Let e ′′ ∈ A ∗∗ be a left unit element for B ∗∗ and supposethat there is a net ( e α ) α ⊆ A such that e α w ∗ → e ′′ . By Hahn Banach theorem take0 = b ′′ ∈ B ∗∗ such that h b ′′ , B ∗ A i = 0. Then for every b ′ ∈ B ∗ , we have h b ′′ , b ′ i = h e ′′ b ′′ , b ′ i = lim α h e α b ′′ , b ′ i = lim α h b ′′ , b ′ e α i . That is contradiction. Thus B ∗ A = B ∗ .Proof of the next part is similar. (cid:3) Definition 5-4.
Let B be a Banach A − bimodule and suppose that A is a Banach ∗ − involution algebra. We say that B is a Banach ∗ − involution algebra as A − module , if the mapping b → b ∗ from B into B satisfies in the following conditions( ab ) ∗ = b ∗ a ∗ , ( ba ) ∗ = a ∗ b ∗ , ( λb ) ∗ = ¯ λb ∗ , ( b ∗ ) ∗ = b , k b ∗ k , for all a ∈ A , b ∈ B and λ ∈ C . Theorem 5-5.
Let B be a Banach A − bimodule and suppose that ( e α ) α ⊆ A isa LBAI for B . Let ( e α ) α ⊆ A be a left regular with respect to B . If B ∗∗ is aBanach ∗ − involution algebra as A ∗∗ -module, then B ∗∗ is unital as A ∗∗ -module and B ∗ factors on the both side. Proof.
By using Theorem 5-3, B ∗ factors on the left and without loss generally, thereis a left unit for B ∗∗ as e ′′ ∈ A ∗∗ such that e α w ∗ → e ′′ . Let b ′′ ∈ B ∗∗ . Then b ′′ ( e ′′ ) ∗ = ( e ′′ ( b ′′ ) ∗ ) ∗ = (( b ′′ ) ∗ ) ∗ = b ′′ . Since e ′′ = ( e ′′ ) ∗ , B ∗∗ is unital. Thus it is similar to Theorem 5-3, B ∗ factors on theright. (cid:3) Corollary 5-6.
Let B be a Banach A − bimodule and suppose that ( e α ) α ⊆ A is a BAI for B . Then if B ∗∗ is a Banach ∗ − involution algebra as A ∗∗ -module, then B ∗∗ is unital as A ∗∗ -module and B ∗ factors on the both side. Example 5-7.
Let G be a locally compact group. Then, on L ( G ) there is a naturalinvolution ∗ defined by f ∗ ( x ) = ∆( x − ) f ( x − ), where ∆ is modular function and x ∈ G . By preceding theorem if L ( G ) ∗∗ is a Banach ∗ − involution algebra, then L ( G ) ∗∗ is unital with respect to first Arens product, and so LU C ( G ) = L ∞ ( G ). Itfollows that G is discrete. Problem .
Let G be a locally compact group. We know that M ( G ) by µ ∗ ( f ) = R f ( x − ) dµ for all f ∈ C ( G ) is ∗ -involution algebra. When there is extension of itinto M ( G ) ∗∗ where M ( G ) ∗∗ equipped first Arens product. References
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Department of Mathematics, University of Mohghegh Ardabili, Ardabil, Iran