1 Introduction

Let A be a commutative Banach algebra without order with Gelfand spectrum Δ(A) and let \({\mathcal{M}}(A)\) denote the multiplier algebra of A. A bounded continuous function σ on Δ(A) is called a BSE-function if there exists a constant C>0 such that for every finite number of φ 1,…,φ n in Δ(A) and the same number of complex numbers c 1,…,c n , the inequality

$$\Biggl\vert \sum_{j=1}^nc_j \sigma(\varphi_j)\Biggr\vert \leq C. \Biggl\Vert \sum_{j=1}^nc_j \varphi_j\Biggr\Vert _{A^*} $$

holds. The BSE-norm of σ, \(\Vert\sigma\Vert_{\it BSE}\), is defined to be the infimum of all such C. The set of all BSE-functions is denoted by \(C_{\it BSE}(\Delta(A))\). Takahasi and Hatori [15] showed that under the norm \(\Vert. \Vert_{\it BSE}\), \(C_{\it BSE}(\Delta(A))\) is a commutative semisimple Banach algebra. The algebra A is called a BSE-algebra (or said to have the BSE-property) if the BSE-functions on Δ(A) are precisely the Gelfand transforms of the elements of \({\mathcal{M}}(A)\). That is A is a BSE-algebra if and only if

$$C_{\it BSE}\bigl(\Delta(A)\bigr)=\widehat{{\mathcal{M}}(A)}. $$

The abbreviation BSE stands for Bochner-Schoenberg-Eberlin and refers to the famous theorem, proved by Bochner and Schoenberg [4, 13] for the additive group of real numbers and by Eberlein [7] for general locally compact abelian groups G, saying that, in the above terminology, the group algebra L 1(G) is a BSE-algebra (See [12] for a proof).

The notion of BSE-algebra and the algebra of BSE-functions were introduced and studied by Takahasi and Hatori [15, 16], Inoue and Takahasi [8] and later by Kaniuth and Ülger [10].

In [15] the authors raised the question whether the semigroup algebra \(L^{1}({\mathbb{R}}^{+})\) is a BSE algebra.

In the present paper we first give a characterization of the Fourier-Stieltjes algebra B(S) of a foundation ∗-semigroup S and then as a consequence of this result we show that \(L^{1}({\mathbb{R}}^{+})\) is a BSE algebra. However, this is not the case for an arbitrary semigroup algebra. For instance, the semigroup algebra \(l^{1}({\mathbb{N}})\), where \(\mathbb{N}\) is the additive semigroup of positive integers, is not a BSE algebra [15].

2 Preliminaries

In this paper, the term semigroup will describe a set S endowed with an associative and commutative binary operation from S×S into S. If S is also a Hausdorff topological space and the binary operation is continuous for the product topology of S×S, then S is said to be a topological semigroup. If in addition S contains a unit with respect to the operation, we say S has an identity.

A semicharacter γ on a topological semigroup S is a bounded, continuous, complex-valued function on S, not identically zero, such that

$$\gamma(xy)=\gamma(x)\gamma(y)\quad \bigl((x,y)\in S\bigr). $$

The set consisting of all semicharacters of S is denoted by \({\widehat{S}}\). If S has an identity, \({\widehat{S}}\) forms a semigroup under the pointwise multiplication. With the topology of uniform convergence on compact subsets of S, \({\widehat{S}}\) is a topological semigroup. Any semigroup S can be embedded in \({\widehat {\widehat{S}}}\) under the map \(x\rightarrow\tilde{x}\) where \(\tilde {x}(\gamma)=\gamma(x)\) for each γ in \({\widehat{S}}\) and x in S.

An involution on a topological semigroup S is a map ∗ of S into S such that, denoting the images of s,t in S by s ∗,t ∗,

  1. (i)

    s=(s ∗)∗

  2. (ii)

    (st)∗=t ∗ s ∗

  3. (iii)

    ∗ is continuous.

Thus ∗ is an isomorphism and a homeomorphism of S onto S.

The topological semigroup \({\widehat{S}}\) has the involution ∗ defined by \(\gamma^{*}={\bar{\gamma}}\) where \({\bar{\gamma}}\) denotes the semicharacter such that \({\bar{\gamma}}(x)=\overline{\gamma(x)}\) for each x in S.

A semicharacter γ is a ∗-semicharacter of S if \(\gamma (x^{*})=\overline{\gamma(x)}\) for every x in S. The subset of \({\widehat{S}}\) consisting of all ∗-semicharacters of S will be denoted by S ∗. If S has an identity, then S ∗ is a subsemigroup of \({\widehat{S}}\). The topological semigroup S with involution ∗ is said to be ∗-reflexive if S≅S ∗ ∗ under the map \(x\rightarrow\tilde{x}\) where \(\tilde{x}(\gamma)=\gamma(x)\) for each γ in S ∗ and x in S.

Let S be a locally compact topological semigroup and M(S) be the space of all bounded complex Borel measures on S. Then with the convolution

$$\mu\star\nu(\psi)=\int\int\psi(xy)d\mu(x)d\nu(y)\quad \bigl(\mu,\nu\in M(S)\bigr),\quad \psi \in C_0(S), $$

M(S) defines a commutative Banach algebra. The space of all measures μ in M(S) for which the mappings x→|μ|⋆δ x from S into M(S) are weakly continuous is denoted by M a (S). Note that M a (S) is an L-ideal of M(S) [3]. A topological semigroup S is called a foundation semigroup if S coincides with the closure of \(\bigcup\{{\rm supp}(\mu):\mu\in M_{a}(S)\}\).

Trivial examples of foundation semigroups are discrete semigroups and topological groups. For more details see [6].

When S is a foundation topological semigroup we have Δ(M a (S)) is topologically isomorphic to \(\widehat{S}\), (see [6] for a proof).

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

A bounded net (e α ) α in A is called a bounded approximate identity for A if it satisfies ∥e α a−a∥→0 for all a∈A.

3 Characterization of B(S)

Let S be a topological ∗-semigroup. A complex valued function \(\varphi:S\rightarrow{\mathbb{C}}\) is called positive definite if for all positive integers n and all \(\lambda_{1},\ldots,\lambda_{n}\in{\mathbb{C}}\) and x 1,…,x n ∈S, we have

$$\sum_{i=1}^n\sum _{j=1}^n\lambda_i\overline{ \lambda_j}\varphi \bigl(x_ix_j^*\bigr) \geq0. $$

Let P(S) denote the set of all continuous bounded positive definite functions on S. We denote the linear span of P(S) by B(S).

As is shown in [11], for a foundation semigroup S with identity, for every φ∈B(S) there is a unique measure μ φ ∈M(S ∗) such that

$$\varphi(x)=\int_{S^*}\chi(x)d\mu_{\varphi}(\chi)\quad (x \in S). $$

Note that under the pointwise multiplication and the norm

$$\Vert\varphi\Vert:=\Vert\mu_{\varphi}\Vert_{M(S^*)}, $$

(B(S),∥.∥) defines a Banach algebra, called Fourier-Stieltjes algebra of S.

Bohr Compactification of a Topological Semigroup

Suppose that S is a topological semigroup. Since \({\widehat{S}}\cup\{0\}\) is closed under pointwise multiplication and complex conjugation (note that here zero is adjoined to \(\widehat{S}\) because if S has no identity, then \(\widehat{S}\) is not necessarily closed under pointwise multiplication, i.e. the multiplication of two nonzero semicharacter may be zero) , the closure A of the linear span of \({\widehat{S}}\) in supremum norm is a C ∗-subalgebra of bounded continuous functions on S. Since 1 S (1 S (x)=1 for all x∈S) is in \(\widehat{S}\), A is a unital Banach algebra under pointwise multiplication. It follows that the spectrum \({\bar{S}}\) of A is a compact Hausdorff space. Furthermore, since points of S determine complex homomorphisms of A, there is a continuous map \(\alpha :S\rightarrow{\bar{S}}\), with dense image, such that \(f\mapsto f\circ \alpha:C({\bar{S}}) \rightarrow A\) is an isometric isomorphism of \(C({\bar{S}})\) onto A.

The map α is injective if and only if \({\widehat{S}}\) separates the points of S. We shall call \({\bar{S}}\) (together with the map α) the Bohr compactification of S. See [17] for more details.

Remark 1

Let S be topological semigroup with identity. If we consider the Bohr compactification \(\bar{S^{*}}\)of S ∗, then \(\bar {S^{*}}=\Delta(A)\), where A is the ∥.∥∞-closure of \({\rm span}(\widehat{S^{*}})\). Note that if we embed S in (S ∗)∗⊆A, then from the density of S ∗ in \(\bar{S^{*}}\), for every \(\acute{\gamma}\in\bar{S^{*}}\) and x,y∈S we have \(\acute{\gamma}(xy)=\acute{\gamma}(x)\acute{\gamma}(y)\) and \(\overline {\acute{\gamma}(x)}=\acute{\gamma}(x^{*})\).

The following theorem gives a characterization of the Fourier-Stieltjes algebra B(S) of a foundation ∗-semigroup S with identity.

Theorem 1

Let S be a topological foundation ∗-semigroup with identity. Then for a continuous function φ defined on S and β≥0, the following statements are equivalent:

  1. (a)

    φ∈B(S) and ∥φ∥≤β.

  2. (b)

    For every function f on S ∗ of the form

    $$f(\gamma)=\sum_{i=1}^n c_i\widetilde{x_i}(\gamma),\quad \gamma\in S^*, $$

    where c 1,…,c n are complex numbers and x 1,…,x n ∈S, we have

    $$\begin{aligned} \Biggl\vert \sum_{i=1}^n c_i\varphi(x_i)\Biggr\vert \leq\beta \Vert f \Vert_\infty. \end{aligned}$$
    (I)

Proof

Let φ∈B(S) with ∥φ∥≤β. By Bochner Theorem [11], there exists a measure μ∈M(S ∗) such that ∥μ∥=∥φ∥≤β and

$$\varphi(x)=\int_{S^*} \gamma(x)d\mu(\gamma)\quad (x\in S). $$

So that

$$\begin{aligned} \Biggl\vert \sum_{i=1}^n c_i\varphi(x_i)\Biggr\vert =&\Biggl\vert \sum_{i=1}^n c_i \int_{S^*} \gamma (x_i)d\mu(\gamma) \Biggr\vert \\ =&\Biggl\vert \int_{S^*}\sum _{i=1}^n c_i \widetilde{x_i}(\gamma)d\mu(\gamma)\Biggr\vert \\ =&\biggl\vert \int_{S^*}f(\gamma)d\mu(\gamma) \biggr\vert \leq\Vert f\Vert_\infty\Vert\mu\Vert\leq\beta\Vert f \Vert_\infty. \end{aligned}$$

Thus, (a) implies (b).

To prove the reverse implication, we consider the Bohr compactification semigroup \({\bar{S}^{*}}\) of the semigroup S ∗.

We can extend each f on S ∗ of the form

$$f(\gamma)=\sum_{i=1}^n c_i \widetilde{x_i}(\gamma )=\sum _{i=1}^n c_i\gamma(x_i)\quad\bigl(\gamma\in S^*\bigr) $$

to \({\bar{S}^{*}}\) by

$$f({\acute{\gamma}})=\sum_{i=1}^n c_i {\acute{\gamma}}(x_i)\quad\bigl({\acute{\gamma}}\in{ \bar{S}^*}\bigr), $$

and since S ∗ is dense in \({\bar{S}^{*}}\), the norm ∥f∥∞ is not altered by this extension. Let

$${\mathcal{A}}= \Biggl\{ f:{\bar{S}^*}\longrightarrow{\Bbb{C}}\mid\exists x_1,\ldots,x_n\in S,\exists c_1,\ldots,c_n \in{\Bbb{C}} :f({\acute{\gamma}})=\sum_{i=1}^n c_i {\acute{\gamma}}(x_i) \Biggr\} . $$

Then \({\mathcal{A}}\) is a linear subspace in \(C({\bar{S}^{*}})\). Now define a linear functional \({\mathcal{F}}\) on \({\mathcal{A}}\) by

$${\mathcal{F}}f=\sum_{i=1}^n c_i\varphi(x_i), $$

where \(f({\acute{\gamma}})=\sum_{i=1}^{n} c_{i} {\acute{\gamma}}(x_{i})\).

From the inequality (I), we have

$$\vert{\mathcal{F}}f\vert\leq\beta\Vert f\Vert_\infty. $$

Thus \(\Vert{\mathcal{F}}\Vert\leq\beta\) and \({\mathcal{F}}\) can be extended to a bounded linear functional \({\mathcal{T}}\) on \(C({\bar{S}^{*}})\) of the norm not greeter than β.

By the Riesz representation theorem there is a unique measure \(\mu\in M({\bar{S}^{*}})\) such that ∥μ∥≤β and

$${\mathcal{T}}f=\int_{{\bar{S}^*}}f({\acute{\gamma}})d\mu( \acute{\gamma})\quad \bigl(f\in C\bigl(\bar{S^*}\bigr)\bigr). $$

In particular, \(f(\acute{\gamma})={\acute{\gamma}}(x)\ (x\in S)\). Therefore

$$\varphi(x)=\int_{{\bar{S}^*}}{\acute{\gamma}}(x)d\mu( \acute{\gamma})\quad (x\in S), $$

and ∥φ∥≤∥μ∥≤β.

By Jordan decomposition theorem there are measures \(\mu_{1},\mu_{2},\mu _{3},\mu_{4}\in M({\bar{S}^{*}})^{+}\) such that

$$\mu=\mu_1-\mu_2+i(\mu_3- \mu_4). $$

In order to complete the proof we only need to show that

$$\varphi_i(x)=\int_{{\bar{S}^*}}{\acute{\gamma}}(x)d\mu _i(\acute{\gamma}) $$

is positive definite, for i=1,2,3,4.

For example we show that φ 1 is positive definite and others are similar.

Since, by Remark 1, \(\overline{{\acute{\gamma}}(x)}={\acute{\gamma}}(x^{*})\), for all \(\acute{\gamma}\in{\bar{S}^{*}}\) and x∈S, then we have

$$\begin{aligned} \sum_{i=1}^n\sum _{j=1}^n c_i{\bar{c}}_j \varphi_1\bigl(x_ix_j^*\bigr) =&\sum _{i=1}^n\sum_{j=1}^n c_i{\bar{c}}_j\int_{\bar{S}^*}{\acute{\gamma}}\bigl(x_ix_j^*\bigr)d\mu_1(\acute{\gamma}) \\ =& \int_{\bar{S}^*}\Biggl\vert \sum _{i=1}^n c_i{\acute{\gamma}}(x_i)\Biggr\vert ^2d\mu_1({\acute{\gamma}})\geq0. \end{aligned}$$

So the function φ 1 and similarly φ 2, φ 3 and φ 4 are positive definite and consequently φ∈B(S). □

4 The Banach algebra \(L^{1}({\mathbb{R}}^{+})\)

Let \(\mathbb{R}\) denote the additive group of all real numbers with the usual topology and \({\mathbb{R}}^{+}\) be its subsemigroup of nonnegative real numbers.

For every \(z\in{\mathbb{R}}^{+}\times{\mathbb{R}}\subseteq{\mathbb{C}}\) define

$$\gamma_z(t)=e^{-zt} ,\quad t\in{\mathbb{R}}^+. $$

The mapping z↦γ z makes \({\mathbb{R}}^{+}\times{\mathbb{R}}\) topologically isomorphic with \(\widehat{{\mathbb{R}}^{+}}\) [5].

With involution \(x^{*}=x\ (x\in{\mathbb{R}}^{+})\), \({\mathbb{R}}^{+}\) defines a ∗-semigroup for which \(({\mathbb{R}}^{+})^{*}\cong{\mathbb{R}}^{+}\). In fact the mapping y→γ y , where \(\gamma_{y}(t)=e^{-yt}\ (t,y\in{\mathbb{R}}^{+})\), defines a topological isomorphism between \(({{\mathbb{R}}^{+}})^{*}\) and \({\mathbb{R}}^{+}\). Thus \({\mathbb{R}}^{+}\) is a ∗-reflexive semigroup. See [2] for more details.

If we define the involution ∗ on \({\mathbb{R}}^{+}\times{\mathbb{R}}\) by

$$*:{\mathbb{R}}^+\times{\mathbb{R}}\rightarrow{\mathbb{R}}^+\times{\mathbb{R}}\quad (x,y)^*=(x,-y), $$

then we have \(({\mathbb{R}}^{+}\times{\mathbb{R}})^{*}=({\mathbb{R}}^{+})^{*}\times {\mathbb{R}}^{*}={\mathbb{R}}^{+}\times{\mathbb{R}}\). In fact if we define

$$z\bullet w=-{\rm Re}(z){\rm Re}(w)+i\,{\rm Im}(z){\rm Im}(w)\quad\bigl(z,w\in { \mathbb{R}}^+\times{\mathbb{R}}\subseteq{\mathbb{C}}\bigr), $$

where \({\rm Re}(z)\) and \({\rm Im}(z)\) are real and imaginary parts of \(z\in{\mathbb{R}}^{+}\times{\mathbb{R}}\subseteq{\mathbb{C}}\), respectively, the mapping z↦γ z , where \(\gamma _{z}(w)=e^{z\bullet w} \ (w\in{\mathbb{R}}^{+}\times{\mathbb{R}})\) defines a topological isomorphism between \({\mathbb{R}}^{+}\times{\mathbb{R}}\) and \(({\mathbb{R}}^{+}\times{\mathbb{R}})^{*}\).

The following which is the main result of this paper gives a positive answer to the question raised in [15].

Theorem 2

\(L^{1}({\mathbb{R}}^{+})\) is a BSE-algebra.

Proof

Since the semigroup \({\mathbb{R}}^{+}\) is a foundation semigroup with identity, it follows that \(L^{1}({\mathbb{R}}^{+})\) has a bounded approximate identity and then by Corollary 5 of [15],

$$\widehat{{\mathcal{M}}}\bigl(L^1\bigl({\mathbb{R}}^+\bigr)\bigr) \subseteq C_{\it BSE} \bigl(\Delta \bigl(L^1\bigl({\mathbb{R}}^+ \bigr)\bigr) \bigr). $$

For each \(z\in{\mathbb{R}}^{+}\times{\mathbb{R}}\) we define

$$\phi_z(f)=\int_{{\mathbb{R}}^+}f(t)e^{-zt}dt\quad \bigl(f\in L^1\bigl({\mathbb{R}}^+\bigr)\bigr). $$

Then the mapping z↦ϕ z is a homeomorphism of \(\widehat {{\mathbb{R}}^{+}}={\mathbb{R}}^{+}\times{\mathbb{R}}\) onto \(\Delta (L^{1}({\mathbb{R}}^{+}))\).

Now let \(\sigma\in C_{\it BSE} (\Delta(L^{1}({\mathbb{R}}^{+})) )=C_{\it BSE}({\widehat{\mathbb{R}}^{+}})=C_{\it BSE}({\mathbb{R}}^{+}\times{\mathbb{R}})\). Then there exists β>0 such that for every finite number of \(z_{1},\ldots,z_{n}\in\widehat{{\mathbb{R}}^{+}}\cong {\mathbb{R}}^{+}\times {\mathbb{R}}\) and \(c_{1},\ldots, c_{n}\in{\mathbb{C}}\),

$$\begin{aligned} \Biggl\vert \sum_{k=1}^n c_k\sigma(z_k)\Biggr\vert \leq&\beta\Biggl\Vert \sum_{k=1}^n c_k \phi_{z_k}\Biggr\Vert _{L^{\infty}({\mathbb{R}}^+)} \\ =&\beta\sup \Biggl\{ \Biggl\vert \sum_{k=1}^nc_k \phi_{z_k}(f)\Biggr\vert :f\in L^1\bigl({\mathbb{R}}^+ \bigr), \Vert f\Vert_1\leq1 \Biggr\} \\ =&\beta\sup \Biggl\{ \Biggl\vert \sum_{k=1}^nc_k \int_{{\mathbb{R}}^+}f(t)e^{-z_kt}dt\Biggr\vert :f\in L^1\bigl({\mathbb{R}}^+\bigr), \Vert f\Vert _1\leq1 \Biggr\} \\ \leq&\beta\sup \Biggl\{ \int_{{\mathbb{R}}^+}\bigl\vert f(t)\bigr\vert \Biggl\vert \sum_{i=1}^nc_ke^{-z_kt} \Biggr\vert dt:f\in L^1\bigl({\mathbb{R}}^+\bigr), \Vert f \Vert_1\leq1 \Biggr\} \\ \leq&\beta\sup \Biggl\{ \Biggl\vert \sum_{k=1}^nc_k e^{-z_kt}\Biggr\vert :t\in{\mathbb{R}}^+ \Biggr\} \\ \leq&\beta\sup \Biggl\{ \Biggl\vert \sum_{k=1}^n c_ke^{z_k\bullet w}\Biggr\vert :w\in{\mathbb{R}}^+\times{\mathbb{R}} \Biggr\} \\ =&\beta\Biggl\Vert \sum_{k=1}^nc_k \widetilde{z_k}\Biggr\Vert _{\infty}, \end{aligned}$$

where \(\widetilde{z_{k}}(\gamma_{w})=\gamma_{w}(z_{k})=e^{z_{k}\bullet w}\), for all \(w\in{\mathbb{R}}^{+}\times{\mathbb{R}}\). That is for every function f of the form

$$f(\gamma_w)=\sum_{k=1}^n c_i\widetilde{z_k}(\gamma_w), $$

\(\gamma_{w}\in({\mathbb{R}}^{+}\times{\mathbb{R}})^{*}(={\mathbb{R}}^{+}\times {\mathbb{R}})\), we have

$$\Biggl\vert \sum_{k=1}^n c_k\sigma(z_k)\Biggr\vert \leq\beta\Vert f\Vert _{\infty}. $$

Then by Theorem 1, \(\sigma\in B({\mathbb{R}}^{+}\times{\mathbb{R}})\).

\({\mathbb{R}}^{+}\times{\mathbb{R}}\) is a foundation semigroup with identity. So the Bochner Theorem [11] implies that there exists a unique nonnegative measure \(\mu\in M(({\mathbb{R}}^{+}\times{\mathbb{R}})^{*})= M({\mathbb{R}}^{+}\times{\mathbb{R}})\) such that

$$\sigma(z)=\int_{{\mathbb{R}}^+\times{\mathbb{R}}}e^{z\bullet w}d\mu(w), $$

for all \(z\in{\mathbb{R}}^{+}\times{\mathbb{R}}\).

The function z↦σ(z) is analytic on the domain \(D=(0,+\infty)\times{\mathbb{R}}\) and its restriction to the set (0,+∞) coincides with the function

$$f(x)=\int_{\mathbb{R}^+}e^{-xt}d\nu(t)\quad\bigl(x \in(0,+\infty)\bigr) $$

where \(\nu\in M({\mathbb{R}}^{+})\) is the restriction of measure μ to \({\mathbb{R}}^{+}\). So, σ is an analytic extension of f to the domain D.

Now recall from page 38 of [17] that the Gelfand transform (Fourier-Stieltjes transform) of ν is the Laplace transform

$$\widehat{\nu}(z)=\int_{\mathbb{R}^+}e^{-zt}d\nu(t)\quad \bigl(z\in {\mathbb{R}}^+\times{\mathbb{R}}\bigr), $$

Then \(\widehat{\nu}\) is also an analytic extension of f. Since (0,+∞) has an accumulation point in D, so by Corollary 6.10 of [1], \(\sigma=\widehat{\nu}\) on D. The functions σ and \(\widehat{\nu}\) are continuous on \({\mathbb{R}}^{+}\times{\mathbb{R}}\), then \(\sigma=\widehat{\nu}\) on \(\overline{D}={\mathbb{R}}^{+}\times {\mathbb{R}}\). It follows that \(\sigma\in\widehat{M}({\mathbb{R}}^{+})\). Hence, from [14], \(\sigma\in\widehat{{\mathcal{M}}}(L^{1}({\mathbb{R}}^{+}))=\widehat{M}({\mathbb{R}}^{+})\).

Consequently

$$C_{\it BSE}\bigl(\widehat{{\mathbb{R}}^+}\bigr)\subseteq\widehat{{\mathcal{M}}}\bigl(L^1\bigl({\mathbb{R}}^+\bigr)\bigr), $$

and the proof is complete. □

Corollary 1

Any finite ℓ 1-direct sum \(\bigoplus_{1}^{n} L^{1}({\mathbb{R}}^{+})\) is a BSE algebra.

Proof

Since \(L^{1}({\mathbb{R}}^{+})\) is a semisimple Banach algebra, it follows from Theorem 2 and Theorem 2.4 of [9]. □