aa r X i v : . [ m a t h . F A ] D ec Sahand Communications in Mathematical Analysis (SCMA) Vol. · · ·
No. · · · (2014), · · · - · · · http://scma.maragheh.ac.ir ON CHARACTER SPACE OF THE ALGEBRA OFBSE-FUNCTIONS
MOHAMMAD FOZOUNI ∗ Abstract.
Suppose that A is a semi-simple and commutative Ba-nach algebra. In this paper we try to characterize the characterspace of the Banach algebra C BSE (∆( A )) consisting of all BSE-functions on ∆( A ) where ∆( A ) denotes the character space of A .Indeed, in the case that A = C ( X ) where X is a non-empty lo-cally compact Hausdroff space, we give a complete characterizationof ∆( C BSE (∆( A ))) and in the general case we give a partial answer.Also, using the Fourier algebra, we show that C BSE (∆( A )) isnot a C ∗ -algebra in general. Finally for some subsets E of A ∗ , wedefine the subspace of BSE-like functions on ∆( A ) ∪ E and give anice application of this space related to Goldstine’s theorem. Introduction and Preliminaries
Suppose that A is a semi-simple commutative Banach algebra and∆( A ) is the character space of A , i.e., the space of all non-zero homo-morphisms from A into C .A bounded continuous function σ on ∆( A ) is called a BSE-function ifthere exists a constant C > φ , ..., φ n ∈ ∆( A ) andcomplex numbers c , ..., c n , the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 c i σ ( φ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 c i φ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ∗ holds. For each σ ∈ C BSE (∆( A )) we denote by k σ k BSE the infimum ofsuch C . Let C BSE (∆( A )) be the set of all BSE-functions. We have agood characterization of C BSE (∆( A )) as follows: Mathematics Subject Classification.
Key words and phrases.
Banach algebra, BSE-function, character space, locallycompact group.Received: dd mmmm yyyy, Accepted: dd mmmm yyyy. ∗ Corresponding author.
Theorem 1.1. C BSE (∆( A )) is equal to the set of all σ ∈ C b (∆( A )) forwhich there exists a bounded net { x λ } in A with lim λ φ ( x λ ) = σ ( φ ) forall φ ∈ ∆( A ) Proof. see [11, Theorem 4 (i)]. (cid:3)
Using the above characterization one can see that C BSE (∆( A )) is uni-tal if and only if A has a bounded weak approximate identity in the senseof Lahr and Jones. We recall that a net { x α } in A is called a boundedweak approximate identity (b.w.a.i) for A if { x α } is bounded in A andlim α φ ( x α a ) = φ ( a ) ( φ ∈ ∆( A ) , a ∈ A ) , or equivalently, lim α φ ( x α ) = 1 for each φ ∈ ∆( A ).Also, Theorem 1.1, gives the following definition of k · k BSE : k σ k BSE = inf { β > ∃{ x λ } in A with k x λ k ≤ β, lim λ φ ( x λ ) = σ ( φ ) ( φ ∈ ∆( A )) } . The theory of BSE-algebras for the first time introduced and inves-tigated by Takahasi and Hatori; see [11] and two other notable works[5, 3]. In [3], the authors answered to a question raised in [11]. Ex-amples of BSE-algebras are the group algebra L ( G ) of a locally com-pact abelian group G , the Fourier algebra A ( G ) of a locally compactamenable group G , all commutative C ∗ -algebras, the disk algebra, andthe Hardy algebra on the open unit disk. We recall that a commu-tative and without order Banach algebra A is a type I-BSE algebra if \ M ( A ) = C BSE (∆( A )) = C b (∆( A )), where M ( A ) denotes the multiplieralgebra of A and \ M ( A ) denotes the space of all b T which defined by b T ( ϕ ) ϕ ( x ) = [ T ( x )( ϕ ) for all ϕ ∈ ∆( A ). Note that x ∈ A should satisfies ϕ ( x ) = 0.In this paper, we give a partial characterization of the character spaceof C BSE (∆( A )) where A is a semi-simple commutative Banach algebra.Indeed, we show that if A has a b.w.a.i and C BSE (∆( A )) is an ideal in C b (∆( A )), then ∆( C BSE (∆( A ))) = ∆( A ) w ∗ . Also, we give a negative answer to this question;Whether ( C BSE (∆( A )) , k · k BSE ) is a C ∗ -algebra? At the final sectionof this paper we study the space of BSE-like functions on subsets of A ∗ which containing ∆( A ) and as an application of this space we give a nicerelation with Goldstine’s theorem. HARACTER SPACE OF BSE-FUNCTIONS 3 Character space of C BSE (∆( A ))In view of [11, Lemma 1], C BSE (∆( A )) is a semi-simple commutativeBanach algebra. So, the character space of C BSE (∆( A )) should be non-empty and one may ask: Is there a characterization of ∆( C BSE (∆( A )))for an arbitrary Banach algebra A ?In the sequel of this section we give a partial answer to this question.Let X be a non-empty locally compact Hausdroff space and put C BSE ( X ) := C BSE (∆( C ( X ))) . To proceed further we recall some notions. Let X be a non-emptylocally compact Hausdroff space. A function algebra (FA) on X is asubalgebra A of C b ( X ) that seperates strongly the points of X , that is,for each x, y ∈ X with x = y , there exists f ∈ A with f ( x ) = f ( y ) andfor each x ∈ X , there exists f ∈ A with f ( x ) = 0. A Banach functionalgebra (BFA) on X is a function algebra A on X with a norm k · k suchthat ( A, k · k ) is a Banach algebra.A topological space X is completely regular if every non-empty closedset and every singleton disjoint from it can be separated by continuousfunctions. Theorem 2.1.
Let X be a non-empty locally compact Hausdroff space.Then C BSE ( X ) is a unital BFA and its character space is homeomorphicto X w ∗ , that is, ∆( C BSE ( X )) = { φ x : x ∈ X } w ∗ . Proof.
By [11, Lemma 1], C BSE ( X ) is a subalgebra of C b ( X ) and k · k BSE is a complete algebra norm. Since C ( X ) has a bounded approxi-mate identity, C BSE ( X ) is unital. So, for each x ∈ X , there exists f ∈ C BSE ( X ) with f ( x ) = 0. On the other hand, using the Urysohnlemma, for each x, y ∈ X with x = y one can see that there exists f ∈ C BSE ( X ) such that f ( x ) = f ( y ).Finally, since X is a locally compact Hausdroff space, it is completelyregular by [1, Corollary 2.74]. On the other hand, by [11, Theorem 3],we know that C ( X ) is a type I-BSE algebra. Therefore, C BSE ( X ) = C b (∆( C ( X ))) = C b ( X ). Also, for every f ∈ C BSE ( X ), by the remarkafter Theorem 4 of [11], we have k f k X ≤ k f k BSE . Also, by the Openmapping theorem there exists a positive constant M such that k f k BSE ≤ M k f k X . So, C BSE ( X ) and C b ( X ) are topologically isomorphic, andso ∆( C BSE ( X )) and ∆( C b ( X )) are homeomorphic. Now, by using [4,Theorem 2.4.12], we have∆( C BSE ( X )) = ∆( C b ( X )) = X w ∗ = { φ x : x ∈ X } w ∗ . (cid:3) M. FOZOUNI
Remark . In general for a commutative Banach algebra A , we havethe following conditions concerning the character space of C BSE (∆( A )):(i) If C BSE (∆( A )) = C b (∆( A )), then∆( C BSE (∆( A ))) = ∆( A ) w ∗ . Examples of Banach algebras A satisfying C BSE (∆( A )) = C b (∆( A ))are finite dimensional Banach algebras and commutative C ∗ -algebras; see the remark on page 609 of [12]. Also, see [11,Lemma 2] for a characterization of Banach algebras A for whichsatisfying C BSE (∆( A )) = C b (∆( A )).(ii) If A has a b.w.a.i, then C BSE (∆( A )) is unital and so ∆( C BSE (∆( A )))is compact and hence it is w ∗ -closed. On the other hand, weknow that ∆( A ) ⊆ ∆( C BSE (∆( A ))) , in the sense that for each ϕ ∈ ∆( A ), f ϕ : C BSE (∆( A )) −→ C defined by f ϕ ( σ ) = σ ( ϕ ) is an element of ∆( C BSE (∆( A ))). Notethat f ϕ = 0, since in this case C BSE (∆( A )) is unital and f ϕ (1) =1. So∆( A ) w ∗ ⊆ ∆( C BSE (∆( A ))) w ∗ = ∆( C BSE (∆( A ))) . (iii) On can see that if ( B, k · k B ) is a Banach algebra which contain-ing the Banach algebra ( C, k·k C ) as a two-sided ideal, then every ϕ ∈ ∆( C ) extends to one e ϕ ∈ ∆( B ). Now, let C = C BSE (∆( A ))and B = C b (∆( A )). If C is an ideal in B , then∆( C BSE (∆( A ))) = ∆( C ) ⊆ ∆( B ) = ∆( A ) w ∗ . (iv) Suppose that B is a commutative semi-simple Banach algebrasuch that ∆( B ) is compact. Then B is unital; see [4, Theorem3.5.5]. Now, If A has no b.w.a.i, then∆( C BSE (∆( A ))) = ∆( A ) w ∗ . Because if ∆( C BSE (∆( A ))) = ∆( A ) w ∗ , by using the above as-sertion, C BSE (∆( A )) is unital, since ∆( A ) w ∗ = ∆( C b (∆( A ))) iscompact and C BSE (∆( A )) is a semi-simple commutative Banachalgebra. Therefore, A has a b.w.a.i which is impossible.(v) If A has a b.w.a.i and C BSE (∆( A )) is an ideal of C b (∆( A )), thenusing parts (ii) and (iii), we have∆( C BSE (∆( A ))) = ∆( A ) w ∗ . HARACTER SPACE OF BSE-FUNCTIONS 5
3. ( C BSE (∆( A )) , k · k BSE ) is not a C ∗ -algebra The theory of C ∗ -algebras is very fruitful and applied. As an advan-tage of this theory, especially in Harmonic Analysis, one can see the C ∗ -algebra approach for defining a locally compact quantum group; see[6]. So, verifying the Banach algebras from a C ∗ -algebraic point of viewis very helpful. In this section, using a result due to Kaniuth and ¨Ulgerin [5], we show that ( C BSE (∆( A )) , k · k BSE ) is not a C ∗ -algebra in gen-eral. On the other hand, there is a question which left open that, underwhat conditions on A , ( C BSE (∆( A )) , k · k BSE ) is a C ∗ -algebra?In the sequel for each locally compact group G , let A ( G ) denote theFourier algebra and B ( G ) denote the Fourier-Stieltjes algebra introducedby Eymard; see [9, § b G denote the dual group of G and M ( G ) denote the Measure algebra; see [2, § A ( G ) and B ( G ) as follows:Let G be a locally compact group. Suppose that A ( G ) denotes thesubspace of C ( G ) consisting of functions of the form u = P ∞ i =1 f i ∗ e g i where f i , g i ∈ L ( G ), P ∞ i =1 || f i || || g i || < ∞ and e f ( x ) = f ( x − ) for all x ∈ G . The space A ( G ) with the pointwise operation and the followingnorm is a Banach algebra, || u || A ( G ) = inf { ∞ X i =1 || f i || || g i || : u = ∞ X i =1 f i ∗ e g i } , which we call it the Fourier algebra. It is obvious that for each u ∈ A ( G ), || u || ≤ || u || A ( G ) where || u || is the norm of u in C ( G ).Now let Σ denote the equivalence class of all irreducible represen-tations of G . Then B ( G ) consisting of all functions φ of the form φ ( x ) = < π ( x ) ξ, η > where π ∈ Σ and ξ, η are elements of H π , theHilbert space associated to the representation π . It is well-known that A ( G ) is a closed ideal of B ( G ).Also, recall that an involutive Banach algebra A is called a C ∗ -algebraif its norm satisfies k aa ∗ k = k a k for each a ∈ A . We refer the reader to[8] to see a complete description of C ∗ -algebras.In the following remark we give the main result of this section. Remark . In general ( C BSE (∆( A )) , k · k BSE ) is not a C ∗ -algebra, thatis, there is not any involution ”*” on C BSE (∆( A )) such that k σ ∗ σ k BSE = k σ k ∀ σ ∈ C BSE (∆( A )) . Because we know that every commutative C ∗ -algebra is a BSE-algebra.For a non-compact locally compact Abelian group G take A = A ( G ).By [5, Theorem 5.1], we know that C BSE (∆( A )) = B ( G ) and for each u ∈ B ( G ), k u k B ( G ) = k u k BSE . But B ( G ) = M ( b G ) and it is shown M. FOZOUNI in [11] that M ( b G ) and hence B ( G ) is not a BSE-algebra. Therefore,( C BSE (∆( A )) , k · k BSE ) is not a C ∗ -algebra.As the second example, let G be a locally compact Abelian group.It is well-known that C BSE (∆( L ( G ))) is isometrically isomorphic to M ( G ), where L ( G ) denotes the group algebra; see the last remark onpage 151 of [11]. On the other hand, by the Gelfand-Nimark theoremwe know that every commutative C ∗ -algebra should be symmetric. Butin general M ( G ) is not symmetric, i.e., the formula c µ ∗ ( ξ ) = b µ ( ξ ) doesnot hold for every ξ ∈ ∆( M ( G )). For example if G is non-discrete thenby [10, Theorem 5.3.4], M ( G ) is not symmetric and hence fails to be a C ∗ -algebra.It is a good question to characterize Banach algebras A for which( C BSE (∆( A )) , k · k BSE ) is a C ∗ -algebra.4. BSE-functions on subsets of A ∗ Suppose that A is a Banach algebra and E ⊆ A ∗ \ ∆( A ). A complex-valued bounded continuous function σ on ∆( A ) ∪ E is called a BSE-likefunction if there exists an M > f , f , f , . . . f n ∈ ∆( A ) ∪ E and complex numbers c , c , c , . . . , c n ,(4.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 c i σ ( f i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 c i f i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ∗ . We show the set of all the BSE-like functions on ∆( A ) ∪ E by C BSE (∆( A ) , E )and let k σ k BSE be the infimum of all M satisfying relation 4.1. Obvi-ously, C BSE (∆( A ) , E ) is a linear subspace of C b (∆( A ) ∪ E ) and we have { σ | ∆( A ) : σ ∈ C BSE (∆( A ) , E ) } ⊆ C BSE (∆( A )) . Clearly, ι A ( A ) ⊆ C BSE (∆( A ) , E ) where ι A : A −→ A ∗∗ is the naturalembedding. For a ∈ A , we let b a = ι A ( a ) and b A = ι A ( A ).To proceed further, we recall the Helly theorem. Theorem 4.1. ( Helly) Let ( X, k · k ) be a normed linear space over C and let M >
0. Suppose that x ∗ , . . . , x ∗ n are in X ∗ and c ∗ , . . . , c ∗ n are in C . Then the following are equivalent:(i) for all ǫ >
0, there exists x ǫ ∈ X such that k x ǫ k ≤ M + ǫ and x ∗ k ( x ǫ ) = c k for k = 1 , . . . , n .(ii) for all a , . . . , a n ∈ C , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 a i c i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 a i x ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ∗ . Proof.
See [7, Theorem 4.10.1]. (cid:3)
HARACTER SPACE OF BSE-FUNCTIONS 7
As an application of Helly’s theorem, we give the following character-ization which is similar to [11, Theorem 4 (i)].
Theorem 4.2. C BSE (∆( A ) , E ) is equal to the set of all σ ∈ C b (∆( A ) ∪ E ) for which there exists a bounded net { x α } in A with lim α f ( x α ) = σ ( f ) for all f ∈ ∆( A ) ∪ E .Proof. Suppose that σ ∈ C b (∆( A ) ∪ E ) is such that there exists β < ∞ and a net { x α } ⊆ X with k x α k < β for all α and lim α f ( x α ) = σ ( f )for all f ∈ ∆( A ) ∪ E . Let f , . . . , f n be in ∆( A ) ∪ E and c , . . . , c n becomplex numbers. Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 c i σ ( f i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 c i f i ( x α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 c i ( f i ( x α ) − σ ( f i )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 c i f i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + n X i =1 | c i || f i ( x α ) − σ ( f i ) | Taking the limit with respect to α , we conclude that σ ∈ C BSE (∆( A ) , E ).Conversely, let σ ∈ C BSE (∆( A ) , E ). Suppose that Λ is the net con-sisting of all finite subsets of ∆( A ) ∪ E . By Helly’s theorem, for each ǫ > λ ∈ Λ, there exists x ( λ,ǫ ) ∈ A with k x ( λ,ǫ ) k ≤ k σ k BSE + ǫ and f ( x ( λ,ǫ ) ) = σ ( f ) for all f ∈ λ . Clearly, { ( λ, ǫ ) : λ ∈ Λ , ǫ > } is adirected set with ( λ , ǫ ) (cid:22) ( λ , ǫ ) iff λ ⊆ λ and ǫ ≤ ǫ . Therefore,we have lim ( λ,ǫ ) f ( x ( λ,ǫ ) ) = σ ( f ) ( f ∈ ∆( A ) ∪ E ) . (cid:3) Remark . As an application of Theorem 4.2, if E = A ∗ \ ∆( A ), thenone can see that b A w ∗ = A ∗∗ , i.e., we conclude Goldstine’s theorem. Thatis, b A with the w ∗ -topology of A ∗∗ is dense in A ∗∗ . Remark . We say that A has a b.w.a.i respect to E if there exists abounded net { x α } in A withlim α f ( x α ) = 1 ( f ∈ ∆( A ) ∪ E ) . Using Theorem 4.2,one can check that 1 ∈ C BSE (∆( A ) , E ) if and only if A has a b.w.a.i respect to E .We conclude this section with the following question. Question Is C BSE (∆( A ) , E ) a commutative and semi-simple Banachalgebra? If it is what is its character space? Acknowledgment.
The author wish to thank the referee for his \ hersuggestions. The author partially supported by a grant from GonbadKavous University. M. FOZOUNI
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