# Bochner-Schoenberg-Eberlein property for abstract Segal algebras.

1. Introduction. Let A be a commutative Banach algebra without order. Denote by [DELTA](A) and M(A) the Gelfand spectrum and the multiplier algebra of A, respectively. A bounded continuous function a on [DELTA](A) is called a BSE-function if there exists a constant C > 0 such that for every finite number of [p.sub.1], ...,[p.sub.n] in [DELTA](A) and the same number of complex numbers [c.sub.1] , ..., [c.sub.n], the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds. The BSE-norm of [sigma], [[parallel[sigma]].sub.BSE], is defined to be the infimum of all such C. The set of all BSE-functions is denoted by Cbse(A(A)). Takahasi and Hatori [19] showed that under the norm \\-\\bse, [C.sub.BSE] ([DELTA](A)) is a commutative semi-simple Banach algebra. The algebra A is called a BSE-algebra (or said to have the BSE-property) if the BSE-functions on [DELTA](A) are precisely the Gelfand transforms of the elements of M(A).That is A is a BSE-algebra if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The abbreviation BSE stands for Bochner-Schoenberg-Eberlein and refers to the famous theorem, proved by Bochner and Schoenberg [2,18] for the additive group of real numbers and in general by Eberlein [6] for locally compact abelian groups G, saying that, in the above terminology, the group algebra [L.sup.1] (G) is a BSE-algebra (See [17] for a proof).

The notion of BSE-algebra and the algebra of BSE-functions were introduced and studied by Takahasi and Hatori [19,20] and later by Kaniuth and Ulger [12]. Also the authors have got some new results on BSE algebras such as BSE property of direct sum of Banach algebras [11].

In 2000, Inoue and Takahasi [8] proved that every Segal algebra S(G) of a locally compact group G is a BSE-algebra if and only if it has a [DELTA]-weak bounded approximate identity

In this paper we generalize this result to abstract Segal algebras. Indeed, we prove that an abstract essential Segal algebra with respect to a BSE-algebra is BSE if and only if it has a [DELTA]-weak bounded approximate identity.

In last section, we study the BSE property for certain abstract Segal algebras which are not discussed before.

2. Preliminaries.

Definition 2.1. Let (A, [parallel] x [[parallel].sub.A]) be a Banach algebra. A Banach algebra (B, [parallel] x [[parallel].sub.B]) is an abstract Segal algebra with respect to A if

(i) B is a dense ideal in A.

(ii) There exists M > 0 such that [parallel]b[[parallel].sub.A] [less than or equal to] M[parallel]b[[parallel].sub.B], for all b [member of] B.

(iii) There exists C > 0 such that [parallel]ab[[parallel].sub.B] [less than or equal to] C[parallel]a[[parallel].sub.A] [parallel]b[[parallel].sub.B], for all a, b [member of] B.

We quote the following result from [3]:

Proposition 2.2. Let (B, [parallel] x [[parallel].sub.B]) be an abstract Segai algebra with respect to the commutative Banach algebra (A, [parallel] x [[parallel].sub.A]).Then [DELTA](A) and [DELTA](B) are homeomorphic.

Definition 2.3. An ideal B in a Banach algebra A is called essential, if

B = {ax : a [member of] A, x [member of] B}.

Dunford [4] proved that any Segal algebra S(G) on a locally compact group G is an essential ideal in [L.sup.1](G).

A linear bounded operator on A is called a multiplier if it satisfies xT(y) = T(xy) for all x, y [member of] A. The set M(A) of all multipliers on A is a unital commutative Banach algebra, called the multiplier algebra of A.

For each T [member of] M(A) there exists a unique continuous function [??] on [DELTA](A) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all a [member of] A and [??] [member of] [DELTA](A). See [15] for a proof.

A bounded net [TEXT NOT REPRODUCIBLE IN ASCII] in A is called a bounded approximate identity for A if it satisfies [parallel][e.sub.[alpha]]a - a[parallel] [right arrow] 0 for all a [member of] A. A bounded net [TEXT NOT REPRODUCIBLE IN ASCII] in A is called a [DELTA]-weak bounded approximate identity for A if it satisfies [TEXT NOT REPRODUCIBLE IN ASCII]. Such approximate identities were studied in [10]. Takahasi and Hatori obtained the following result in [19]:

Proposition 2.4. Let A be a commutative Banach algebra without order. A has a [DELTA]-weak bounded approximate identity if and only if [??] [subset or equal to] [C.sub.BSE] ([DELTA](A)).

3. Main result.

Theorem 3.1. Let (A, [parallel] x [[parallel].sub.A]) be a BSE-algebra and (B, [parallel] x [[parallel].sub.B]) an essential abstract Segal algebra with respect to A. Then B is a BSE-algebra if and only if it has a [DELTA]-weak bounded approximate identity.

Proof. Suppose that B is a BSE-algebra. Then by Proposition 2.4 it has a [DELTA]-weak bounded approximate identity. Conversely, suppose that B has a [DELTA]-weak bounded approximate identity, then by the same proposition,

[??] [subset or equal to] [C.sub.BSE] ([DELTA](B)) .

So it remains to show that Cbse(A(B)) C M(B). Suppose that a 2 [C.sub.BSE]([DELTA](B)). Then there exists a positive number C such that for any finite number of [[??].sub.1],... ,[[??].sub.n] [member of] [DELTA](B) and [c.sub.1], ...,[c.sub.n] [member of] C,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now for every [f] [member of] [A.sup.*] and x [member of] A, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By definition of abstract Segal algebra, there exists M > 0 such that [[parallel]x[parallel].sub.A] [less than or equal to] M[[parallel]x[parallel].sub.B] (x [member of] B). It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Especially, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Proposition 2.2, [DELTA](B) is homeomorphic to [DELTA](A) and we may consider [[??].sub.1] ... [[??].sub.n] [member of] [DELTA](A) .It means that [sigma] [member of] [C.sub.BSE] ([DELTA](A)). Since A is a BSE-algebra, a [member of] M(A). Therefore there exists T [member of] M(A such that [sigma] = [??] .We have to show that [T|.sub.B] [member of] M(B). Since T [member of] M(A), it is obvious that T(xy) = T(x)y(x,y [member of] B). So it is enough to show that TB [subset or equal to] B. Indeed, if it is shown that TB [subset or equal to] B, then T is continuous in the [parallel] x [[parallel].sub.B]-topology by the closed graph theorem because B has no nonzero annihilators. Let x 2 B. Since B is an essential ideal of A, there exist a [member of] A and y [member of] B such that x = ay and hence

T(x) = T(ay) = T(a) y [member of] B.

Thus [sigma] [member of] [??]. Hence B is a BSE-algebra. []

Remark 3.2. When B is an abstract Segal algebra with respect to A, as it is shown in the proof of Theorem 3.1, we have

[C.sub.BSE]([DELTA](B)) [subset or equal to] [C.sub.BSE]([DELTA](A)).

4. BSE property of certain abstract Segal algebras. In this section we study the BSE property of some abstract Segal algebras which are not discussed in [8].

4.1. Segal algebras of compact abelian groups. A dense ideal S(G) of the convolution group algebra [L.sup.1](G) of a locally compact group G is said to be a Segal algebra if it satisfies the following conditions:

(a) S(G) is a Banach space under some norm [parallel] x [[parallel].sub.S] and [[parallel][f][parallel].sub.S] [greater than or equal to] [[parallel][f][parallel].sub.1]

(b) S(G) is left translation invariant, i.e. [[parallel][L.sub.x][f][parallel].sub.s] = [[parallel]f[parallel].sub.s] for all x [member of] G and f [member of] S(G), and the map x [??] [L.sub.x]f from G into S(G) is continuous.

Every Segal algebra is an abstract segal algebra with respect to [L.sup.1](G) (see [9], Proposition 1).

Proposition 4.1. Let G be an abelian compact group. Then a Segal algebra S(G is a BSE algebra if and only if S(G) = [L.sup.1] (G) .

Proof. The "if" part is clear, since [L.sup.1](G) is a BSE algebra. Conversely, suppose that S(G) is a BSE algebra. Since G is an abelian compact group, then S(G is an ideal in its second dual [16]. By semi-simplicity of S(G and by Theorem 3.1 of [12], it has a bounded approximate identity which by Theorem 1.2 of [3] implies that S(G) = [L.sup.1](G). [ ]

For a locally compact group G, let A(G) be the Fourier algebra defined in [5] and let

LA(G) = A(G) [intersection] [L.sup.1](G)

with norm

[parallel]f[parallel] = [[parallel]f[parallel].sub.A(G)] + [[parallel]f[parallel].sub.1]

(LA(G), [parallel].[parallel]) with convolution product is a Segal algebra, called Lebesgue-Fourier algebra. Note that LA(G) with point-wise multiplication is an abstract Segal algebra of A(G .

The concept of Lebesgue-Fourier algebra was introduced and extensively studied by Ghahramani and Lau [7].

Corollary 4.2. Let G be an abelian compact group. Then the Banach algebra LA(G) is a BSE algebra if and only if G is finite.

Proof. By Proposition 2.3 of [7], LA(G) = [L.sup.1](G) if and only if G is discrete. Then by Proposition 4.1, LA(G) is BSE if and only if G is discrete and by compactness of G, if and only if it is finite. ?

Remark 4.3. Let G be a discrete group and suppose that LA(G) is equipped with the point-wise product. Then LA(G) is a BSE algebra if and only if G is finite. In fact, when G is discrete, LA(G) = [l.sup.1](G) with point-wise multiplication and this algebra is BSE if and only if G is finite [20].

4.2. [W.sup.p]-algebras. Consider the additive group of vectors in [R.sup.n] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For t [member of] [R.sup.n], define [Q.sub.t] : = {t + x : x [member of] Q} and [f.sub.t] denotes the translated function [f.sub.t](x) =f(x - t). For an arbitrary set A, [[chi].sub.a] will denote the characteristic function of A. For simplicity we write %t instead of [[chi].sub.Qt].

For 1 < p < [infinite], let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Proposition 3.1 of [14], [W.sup.p] is a Segal algebra with respect to [L.sup.1] ([R.sup.n]) by the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 4.4. The Segal algebra [W.sup.p] is not BSE.

Proof. By Corollary 3.8 of [14], there is a multiplier in M([W.sup.p]) which is not a measure. It means that there exists T 2 M([W.sup.p]) and T [not member of] M([R.sup.n]). Since M([R.sup.n]) is a semi-simple Banach algebra, T [member of] M([W.sup.p]) and T [not member of] [??]. [L.sup.1]([R.sup.n]) is a BSE algebra, then [??] [not member of] [C.sub.BSE] ([DELTA]([L.sup.1]([R.sup.n])) and consequently by Remark 3.2, T 2 [C.sub.BSE] ([DELTA]([W.sup.p])) .It follows that [??] [not equal to] [C.sub.BSE] ([DELTA]([W.sup.p])) and [W.sup.p] is not a BSE algebra. [ ]

Corollary 4.5. [W.sup.p] has no [DELTA]-weak bounded approximate identity.

Proof. By Proposition 4.4 and Theorem 3.1, the result is obvious. [ ]

4.3. [C.sup.*]-Segal algebra [C.sup.w.sub.0] (X). Let X be a locally compact Hausdorff space, and let w : X [right arrow] R be an upper semi-continuous function such that w(t) [greater than or equal to] 1 for every t [member of] X. Define [C.sup.w.sub.0](X) : = {f [member of] C(X) : fw vanishes at infinity on X}, where C(X) denotes the set of all continuous complex-valued functions on X.

Equipped with point-wise operations and the weighted supremum norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[C.sup.w.sub.0](X) is a self-adjoint [C.sup.*]-Segal algebra (abstract segal algebra with respect to [C.sub.0](X) [13].

By Proposition 2.2, [DELTA]([C.sup.w.sub.0] (X) = [DELTA]([C.sub.0](X) = X. In fact, the function x [??] [[phi].sub.x], where [[phi].sub.x](f) = f(x) (x [member of] X, f [member of] [C.sup.w.sub.0] (X), is a homeomorphism from X onto [DELTA]([C.sup.w.sub.0](X).

Proposition 4.6. ([C.sup.w.sub.0](X) is a BSE algebra if and only if w is bounded.

Proof. When w is bounded, by [1], ([C.sup.w.sub.0](X) = [C.sub.0](X) is a [C.sup.*]-algebra and so by Theorem 3 of [19] is a BSE algebra. Now suppose that ([C.sup.w.sub.0](X) is a BSE algebra. Then by Proposition 2.4, it h as a [DELTA]-weak approximate identity. It means that there exists a bounded net [{[f.sub.[alpha]}.sub.[alpha]] [member of] [C.sup.w.sub.0] (X) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all t [member of] X and there exists [beta] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus w(t) [less than or equal to][beta] (t [member of] X) which means that w is bounded. []

As it is shown in Corollary 3.6 of [1], ([C.sup.w.sub.0](X) has a bounded approximate identity if and only if w is bounded. By Theorem 3.1 and Proposition 4.6, we conclude the following result:

Corollary 4.7. ([C.sup.w.sub.0](X) has a [DELTA]-weak bounded approximate identity ifand only ifw is bounded.

doi: 10.3792/pjaa.89.107

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Zeinab KAMALI and Mahmood LASHKARIZADEH BAMI

Department of Mathematics, Faculty of Science, University of Isfahan, Isfahan, Iran

(Communicated by Masaki Kashiwara, M.J.A., Oct. 15, 2013)

2010 Mathematics Subject Classification. Primary 46Jxx; Secondary 22D15.