Abstract
We study the topological algebra identification of the multiplier algebra of a certain algebra E and that of a closed left ideal in E. The case when one of the algebras is a Segal topological algebra in the other is considered. We also study this problem in the context of locally \(C^{*}\)-algebras.
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1 Introduction
In 1956, S. Helgason dealt with representation theory for commutative Banach algebras with special reference to harmonic analysis [11]. He approached this problem by defining the algebra of multipliers. This algebra consists of certain linear maps, called multipliers which are well behaved. Since after, a considerable number of papers referred to multipliers (see e.g., [11, 15, 17]). Some authors use the term “centralizer” in place of “multiplier” (see for instance [14], among many others). The general theory of multipliers can be found in [18].
In [25], B.J. Tomiuk dealt mainly with the identification up to a topological algebra isomorphism of the left multiplier algebras of two certain Banach algebras. The purpose of this paper is to present non-normed extensions of some results of Tomiuk stated in [ibid.]. We refer to generalizations as much as possible. In [9], a study of the left regular representation of certain topological algebras is given along with the properties of their closed ideals.
All algebras, employed below, are taken over the field \(\mathbb {C}\) of complexes. A topological algebra E is an associative algebra which is a topological vector space and the ring multiplication is separately continuous (see e.g., [20]). We refer to a locally m-convex algebra E, if the topology of E is defined by a family of submultiplicative seminorms [ibid.].
A locally \(C^{*}\)-algebra is a complete involutive locally convex algebra E whose topology is given by a (saturated) family of \(C^{*}\)-seminorms \((q_{i})_{i \in I}\). This means that these seminorms satisfy the \(C^{*}\)-property, namely for each \(i \in I\), \(q_{i}(x)^{2} = q_{i}(x^{*} \, x)\) for every \(x \in E\) (see [7, Chapter II, p. 101] and [13]). It is known that \(C^{*}\)-seminorms are automatically \(*\)-preserving and submultiplicative [ibid.]
Throughout, we employ the symbol (0) to denote either the set containing the zero element or the ideal containing only the zero element. The symbol \(\overline{S}^{E}\) stands for the topological closure of a subset S in a topological algebra E.
2 Preliminaries
In this paper, we consider mostly objects and notions on the left side, but most of all our work could be carried out on the right side and sometimes even on the bilateral setting. If needed, we will make declaratory comments in the appropriate places.
Let E be an algebra. If \(S \subseteq E\), then
denotes the left annihilator of S (with respect to E). A similar definition can be made for \(\mathcal {A}_{r}(S)\), the right annihilator of S. \(\mathcal {A}_{l}(S)\) (resp. \(\mathcal {A}_{r}(S)\)) is a left (right) ideal of E, which in particular, is two-sided if S is a left or right ideal. For a topological algebra (separately continuous multiplication), the previous ideals are closed.
An algebra E is called left (resp. right) preannihilator if \(\mathcal {A}_{l}(E)= (0)\) (resp. \(\mathcal {A}_{r}(E)=(0)\)). If \(\mathcal {A}_{l}(E)=(0)=\mathcal {A}_{r}(E)\), then E is named preannihilator. For more on preannihilator algebras, see [8].
Definition 2.1
An approximate identity in a topological algebra E is a net \(\{ e_{\delta } \}_{\delta \in \Delta }\) in E such that for each \(x \in E\),
An approximate identity is called bounded if it is a bounded subset of E, and it is called uniformly bounded if the set \(\{ (k^{-1}\, e_{\delta })^{n},\delta \in \Delta ,n \in \mathbb {N} \}\) with k a positive constant, is bounded in E.
Any algebra with a unit element or yet a topological algebra with an orthogonal basis, or with an approximate identity, is preannihilator. Thus, a locally \(C^{*}\)-algebra, in particular a \(C^{*}\)-algebra, is preannihilator (see [13, p. 208, Theorem 2.6]). Further, any topologically semisimple topological algebra is semiprime and thus preannihilator (see [8, Lemma 2.3]).
In what follows, L(E) denotes the algebra of all linear operators on an algebra E.
Definition 2.2
An element T in L(E) is called a left (right) multiplier on E if \(T(xy) = T(x)y\) (resp. \(T(xy) = xT(y))\) for all \(x, y \in E\). It is called a two-sided multiplier on E if it is both a left and a right multiplier.
We use the following notation: \(M_{l}(E)\) (resp. \(M_{r}(E)\)) denotes the set of all left (resp. right) multipliers on E and M(E) that of all two-sided multipliers on E. Clearly \(M_{l}(E), M_{r}(E)\) and M(E) are subalgebras of L(E). Now, for \(x \in E\), the operator \(L_{x}\) on E given by \(L_{x}(y) = xy\) (\(y \in E\)) is a left multiplier, due to the associativity of the multiplication on E. Similarly, the right multiplier with respect to \(x \in E\), is defined and it is denoted by \(R_{x}\). For the next, see [10, Proposition 2.2].
Proposition 2.3
Let E be a preannihilator algebra. Then the following hold:
-
(i)
The mapping
$$\begin{aligned} L: E \longrightarrow M_{l}(E) \; \text {given by} \; x \mapsto L_{x} \end{aligned}$$(1)defines an algebra monomorphism which identifies E with a subalgebra of \(M_{l}(E)\).
-
(ii)
E is a left ideal of the algebra \(M_{l}(E)\).
-
(iii)
If E has a left identity, then L is onto.
Similar statements hold, if we consider right or two-sided multipliers. We just note that in the right case, the respective mapping R is an algebra antimonomorphism. If E is a topological algebra, then the linear operators \(L_{x}\) and \(R_{x}\), for all \(x \in E\), are continuous.
3 Segal Algebras
In harmonic analysis, there are examples of Banach algebras being ideals in another Banach algebra. Under certain conditions, this type of topological algebras is known as abstract Segal algebras. Segal algebras were introduced by Reiter in 1968, as subalgebras in the context of group algebras ([22] and [23, p. 173, Definition 6.2.1]). The notion of an abstract Segal algebra is due to J.T. Burnham ([4, p. 551, Definition 1.1]) and it is defined as a Banach algebra \((B, \Vert \cdot \Vert _{B})\) which is continuously embedded as a dense ideal to another Banach algebra \((A, \Vert \cdot \Vert _{A})\) satisfying also a norm-condition with respect to both norms. In particular, a \(C^{*}\)-Segal algebra is a Segal algebra in a \(C^{*}\)-algebra. For the Banach case, see also [18, p. 297, Definition].
In this section, we generalize the notion of an abstract Segal algebra from the context of Banach algebras to that of locally m-convex ones (see Definition 3.1). For the Banach case, see [19, p. 297, Definition]. The same notion, for Banach algebras, is used by B.J. Tomiuk in [25], some results of which we generalize here.
In [1], Mart Abel introduced Segal topological algebras (not necessarily Banach, nor locally m-convex, complete or not, nor even locally m-convex Fréchet algebras). In this framework, he generalizes some results from [2, 16, 21].
Definition 3.1
A left ideal S of a locally m-convex algebra \((E,(p_{\lambda })_{\lambda \in \Lambda })\) is called a left Segal locally m-convex algebra in E, if the following hold true.
-
(i)
S is dense in E,
-
(ii)
S is a locally m-convex algebra with respect to some (saturated) family \((q_{i})_{i \in I}\) of submultiplicative seminorms, and
-
(iii)
for each \(\lambda \in \Lambda \), there exists \(i \in I\) and \(k>0\) such that
$$\begin{aligned} p_{\lambda }(x) \le k \, q_{i}(x) \; \text {for every} \; x \in S. \end{aligned}$$(2)
Remark 3.2
Since the family \((q_{i})_{i \in I}\) is saturated, by condition (iii), we get that the topology defined on S by the previous family is finer than the (subspace) locally m-convex topology defined on S by the family \(\{p_{\lambda }\}_{\lambda \in \Lambda }\). This means that the identical bijection \(x \mapsto x\) from \((S, (q_{i})_{i \in I})\) onto \((S,(p_{\lambda })_{\lambda \in \Lambda })\) is continuous (see e.g., [12, p. 98]).
Examples.
-
1.
The case in which \((S,\Vert \cdot \Vert _{S})\), \((E,\Vert \cdot \Vert _{E})\) are Banach algebras and S is a classical Segal algebra in E (see e.g., [19, p. 297, Definition]) is clearly an example for Definition 3.1.
-
2.
Let \((A,\tau _{A})\) be a locally m-convex algebra and I a dense left (right or two-sided) ideal of A, equipped with the subspace topology. Then I is a left (respectively, right or two-sided) Segal locally m-convex algebra in A ([1, Example 2.3]).
For the topological algebra E, let us denote by \(\mathcal {L}(E)\) the algebra of all continuous linear operators on E. Now we need the notion of the left regular representation \(L_{E}\) of E in the locally m-convex case (see also [9]).
Let E be a preannihilator locally m-convex algebra. Denote by \((p_{\alpha })_{\alpha \in \Lambda }\) the set of all continuous seminorms defining the topology on E. We endow \(\mathcal {L}(E)\) with the locally convex topology induced by the family of seminorms \((\tilde{p_{\alpha }})_{\alpha \in \Lambda }\), where, for \(T \in \mathcal {L}(E)\),
Under the assumption that each left multiplier on the topological algebra E is continuous, (this is the case, for instance, for a locally \(C^{*}\)-algebra, see [15, p. 75, Proposition 3.4]), then the algebra \(M_{l}(E)\) of left multipliers with the composition of operators as ring multiplication, is a subalgebra of \(\mathcal {L}(E)\) and becomes a complete locally m-convex algebra with this family of seminorms [ibid, p. 75, Theorem 3.5, (1)]. Moreover, the map L defined in Proposition 2.3 is a seminorm-decreasing homeomorphism.
Consider the set \(\{L_{x}: x \in E\}\) in \(M_{l}(E)\). Put
We call \(L_{E}\) the left regular representation of E. It is a closed left ideal in \(M_{l}(E)\). Being a closed subspace of a complete space, \(L_{E}\) is itself complete.
Since E is a preannihilator algebra, the identification \(x \mapsto L_{x}\) of E as a subalgebra of \(M_{l}(E)\) allows us to identify E with a left ideal of \(L_{E}\). In order to see this, take \(T \in L_{E}\) and \(x \in E\). Then, for each \(y \in E\),
So, \(T \cdot L_{x}= L_{T(x)}\). Since \(T \in L_{E} \subseteq M_{l}(E)\), then \(T(x) \in E\), that is, \(L_{T(x)} \in L_{E}\).
Moreover, for a preannihilator algebra E in which every left multiplier is continuous, we can identify E as a dense left ideal in \(L_{E}\). For, just note that \(E= \{L_{x}:x\in E \}\) in \(M_{l}(E)\) implies \(\overline{E}^{L_{E}} = L_{E} \cap \overline{E}^{M_{l}(E)}= L_{E} \cap \overline{ \{L_{x}:x \in E \}}^{M_{l}(E)} = L_{E}\).
Proposition 3.3
Let \((S,(q_{i})_{i\in I})\) be a preannihilator locally m-convex algebra. Suppose that every left multiplier in S is continuous. Then S is a left Segal algebra in \((L_{S},(\tilde{q_{i}})_{i \in I})\).
Proof
We have already pointed out that, since every left multiplier is continuous, S is a dense left ideal in \(L_{S}\).
Now take \(i \in I\) and \(s \in S\). Then
where \(k_{i}= \sup \{q_{i}(x):x \in S, \, q_{i}(x) \le 1 \}\), which exists because \(q_{i}\) is continuous in S. This completes the proof.
Based on the comment after relation (3), we directly get the next.
Corollary 3.4
Let \((S,(q_{i})_{i\in I})\) be a locally \(C^{*}\)-algebra. Then S is a left Segal algebra in \((L_{S},(\tilde{q_{i}})_{i \in I})\).
4 Isomorphisms
In this section, the fact that E is preannihilator plays a central role, due to Proposition 2.3. The next lemma and statements concern the left regular representation \(L_{S}\) of a locally m-convex algebra S.
Lemma 4.1
Let \((S,(q_{i})_{i\in I})\) be a locally m-convex algebra which is a dense left ideal of a preannihilator complete locally m-convex algebra \((E,(p_{\lambda })_{\lambda \in \Lambda })\). Suppose that for each \(\lambda \in \Lambda \), there exist \(i \in I\) and \(k>0\) such that
for every \(s \in S\). Then, for each \(i \in I\), there exist \(\lambda \in \Lambda \) and \(C>0\) such that
for every \(s \in S\) and \(x \in E\).
Proof
Take \(s \in S\) and define the map \(R_{s}:E \rightarrow S\) as \(R_{s}(x)=x s\). We claim that \(R_{s}\) is continuous. Indeed, for \(\lambda \in \Lambda \), take \(i \in I\) that satisfies (4). For \(\{x_{n}\} \subseteq E\), \(t \in S\) and \(\{p_{\lambda }(x_{n})\} \rightarrow 0\), \(q_{i}(R_{s}(x_{n})-t) \rightarrow 0\), we get \(q_{i}(x_{n}s-t) \rightarrow 0\). Since \(q_{i}\) dominates \(p_{\lambda }\), we also get \(p_{\lambda }(x_{n}s-t) \rightarrow 0\). But \(p_{\lambda }(x_{n}s) \rightarrow 0\) too, therefore \(t=0\).
Take \(i \in I\). Now, since \(R_{s}\) is continuous, for each \(s \in S\), there exists \(M_{\lambda ,s}>0\) such that
for \(x \in E\).
Take \(x \in E\) and note that the linear map \(L_{x}:S \rightarrow S\), \(s \mapsto xs\) is also continuous.
Put \(\vert L_{x} \vert _{i}= \sup \{ q_{i}(L_{x}(s)):q_{i}(s) \le 1 \}\) and \(\mathcal {E_{\lambda }}=\{L_{x}:x \in E, \; p_{\lambda }(x)\le 1\}\). Then
for each \(s \in S\) and \(L_{x} \in \mathcal {E_{\lambda }}\). Since E is complete, we can apply the Uniform Boundedness Theorem (see for instance [24, p. 43, Theorem 2.5]), and we get that there exists \(C>0\) such that, for \(\lambda \in \Lambda \) and \(L_{x} \in \mathcal {E_{\lambda }}\),
From this, we get that, if \(s \in S\), \(q_{i}(s) \ne 0\) and \(x \in E\) with \(p_{\lambda }(x) \le 1\),
and therefore
which also holds if \(q_{i}(s) = 0\), due to the continuity of \(L_{x}\). Now, take an arbitrary \(x \in E\). We can conclude that, if \(p_{\lambda }(x) \ne 0\), then
which implies
and this inequality also holds if \(p_{\lambda }(x) = 0\), due to relation (6).
Remark 4.2
Note that if \((S,(q_{i})_{i\in I})\) is a locally m-convex left Segal algebra in a locally m-convex algebra \((E,(p_{\lambda })_{\lambda \in \Lambda })\), then condition (4) is automatically fulfilled, by the very definition.
Theorem 4.3
Let \((S,(q_{i})_{i\in I})\) be a preannihilator complete locally convex algebra which is a left Segal algebra in a preannihilator complete locally m-convex algebra \((E,(p_{\lambda })_{\lambda \in \Lambda })\). Then there exists a topological algebra isomorphism of E onto \(L_{S}\) which maps x into \(L_{x}\), for all \(x \in S\).
Proof
Since S is preannihilator, the mapping \(\Phi : S \rightarrow L_{S}\), \(s \mapsto L_{s}\) is an algebraic monomorphism. Now, due to Lemma 4.1 and Remark 4.2, for each \(i \in I\), there exist \(\lambda \in \Lambda \) and \(C>0\) such that
for every \(s \in S\) and \(x \in E\). In particular, this also holds if \(x \in S\).
This shows that
for \(x \in S\).
Then, due to relation (10), the map \(\Phi \) is also continuous when S is endowed with the subspace topology of E. Since S is dense in E, \(\Phi \) extends to a continuous map \(\Psi :E \rightarrow L_{S}\), \(x \mapsto L_{x}\).
Let K be the kernel of \(\Psi \). Then K is a closed ideal in E and \(K \cap S=\{0\}\). But \(KS \subseteq K \cap S \) and so \(KS=\{0\}\); therefore \(KE=\{0\}\). Since E is preannihilator, this implies that \(K=\{0\}\) and so, \(\Psi \) is also a monomorphism.
Next, we claim that \(Im(\Psi )\) is closed. So, take \((T_{\delta })_{\delta \in \Delta } \subseteq Im(\Psi )\). Then \(T_{\delta }=L_{s_{\delta }}\) for some \(s_{\delta } \in S\). Take \(i \in I\). Applying relation (10), there exist \(i \in I\) and \(C>0\) such that
so \((T_{\delta })_{\delta \in \textbf{N}}\) is a Cauchy net in \(L_{S}\), which is complete. Then there exists \(U \in L_{S}\) such that \(\Psi (L_{T_{\delta }}) \longrightarrow U\). By the continuity of \(\Psi \), \(U=L_{\lim (s_{\delta })}\), so \(U \in Im(\Psi )\).
Finally, since \(Im(\Psi )\) is closed and dense in \(L_{S}\), the map \(\Psi \) is onto, and therefore bicontinuous.
Proposition 4.4
Let \((S,(q_{i})_{i \in I})\) be a preannihilator locally m-convex algebra. Then every continuous left multiplier U on S has a unique extension to a continuous left multiplier T on \(L_{S}\) and, for each \(i \in I\), there exist a constant \(k_{i}\) such thatFootnote 1
Proof
For each \(s \in S\) and \(i \in I\), as in the proof of Proposition 3.3, there exists a constant \(k_{i}\) such that \(\tilde{q_{i}}(L_{s}) \le k_{i} \cdot q_{i}(s)\). Therefore,
Then U is continuous considered as a map from S to \(L_{S}\). Since S is dense in \(L_{S}\), there is a unique continuous map \(T:L_{S} \rightarrow L_{S}\) that extends U and satisfies (11). Clearly \(T \in M_{l}(L_{S})\) and \(T\vert _{S}=U\).
Remark 4.5
Note that it is also true that if \(T \in M_{l}(L_{S})\) is continuous, then \(U=T \vert _{S}\) belongs to \(M_{l}(S)\) and is continuous. This fact, in addition to Proposition 4.4, implies that, for a preannihilator locally m-convex algebra S, every left multiplier in S is continuous if and only if every left multiplier in \(L_{S}\) is continuous.
An approximate identity \(\{e_{\delta }\}_{\delta }\) in a topological algebra E is said to be quasi-bounded if the set \(\{L_{e_{\delta }}\}_{\delta }\) is bounded in \(L_{E}\). It is easy to see that, if \(\{e_{\delta }\}_{\delta }\) is a quasi-bounded left (right) approximate identity in E, then \(\{L_{e_{\delta }}\}_{\delta }\) is a bounded left (right) approximate identity in \(L_{E}\).
Moreover, if \(\{ e_{\delta } \}_{\delta }\) is a bounded approximate identity in a locally m-convex algebra E, and if q is a continuous seminorm in E, then \(\tilde{q}(L_{e_{\delta }}) \le q(e_{\delta })\) and therefore the set \(\{ L_{e_{\delta }}\}_{\delta }\) is also bounded (in \(L_{E}\)), that is, \(\{ e_{\delta }\}_{\delta }\) is a (left) quasi-bounded approximate identity in E. In particular, every locally \(C^{*}\)-algebra has a quasi-bounded approximate identity.
Definition 4.6
A topological algebra A is called left regular factorable, if
Among left regular factorable topological algebras we can find locally bounded algebras (see [26]), locally convex F-algebras (see [5]) and fundamental F-algebras (see [3]), when there is a uniformly bounded approximate identity in each of them. For more information about this topic, the reader is also referred to [6, 8].
Theorem 4.7
Let \((S,(q_{i})_{i \in I})\) be a preannihilator left regular factorable complete locally m-convex algebra with a quasi-bounded approximate identity. Suppose that every multiplier in S is continuous. Then \(M_{l}(S)=M_{l}(L_{S})\) within a topological algebra isomorphism.
Proof
Take \(T \in M_{l}(L_{S})\). Since S is left regular factorable, \(S=L_{S} \cdot S\). If \(T'=T \vert _{S}\), then \(T'(S)=T(S)=T(L_{S} \cdot S)=T(L_{S}) \cdot S \subseteq L_{S} \cdot S=S\) and hence \(T'\) maps S into S, that is, \(T' \in M_{l}(S)\).
Let \(\{e_{\delta }\}_{\delta }\) be a quasi-bounded left approximate identity in S. Due to Remark 4.5, \(T'\) is continuous and therefore the set \(\{ T'(e_{\delta }) \}_{\delta }\) is bounded in \(L_{S}\). So, for each \(i \in I\), there is a constant \(D_{i}>0\) such that, \(\tilde{q_{i}} (T'(e_{\delta })) \le D_{i}\) for every \(\delta \).
Due to Proposition 3.3 and Remark 4.2, we can apply Lemma 4.1. Therefore, for each \(i \in I\) there is \(j \in I\) and \(C>0\) such that \(q_{i}(b \, s) \le C \, \tilde{q_{i}}(b) \, q_{j}(s)\) for every \(b \in L_{S}\) and \(s \in S\). So, we have that for each \(s\in S\), given \(i \in I\), there exist \(j \in I\) and \(C>0\) such that
Whence the map \(T \mapsto T'\) is continuous.
Now, by Proposition 4.4, every \(U \in M_{l}(S)\) has a unique extension T to \(L_{S}\), \(T \in M_{l}(L_{S})\) and \(\tilde{q_{i}}(T) \le k_{i} \cdot \tilde{q_{i}}(U)\) (for each \(i \in I\) and a constant \(k_{i}\) that depends on i). Hence, the mapping \(T \rightarrow T'\) is an algebraic and topological isomorphism from \(M_{l}(L_{S})\) onto \(M_{l}(S)\).
Corollary 4.8
Let \((S,(q_{i})_{i \in I})\) be a left regular factorable locally \(C^{*}\)-algebra. Then \(M_{l}(S)=M_{l}(L_{S})\) within a topological algebra isomorphism.
Proof
We have already remarked that left multipliers in locally \(C^{*}\)-algebras are continuous. Also, every locally \(C^{*}\)-algebra is preannihilator and has a left quasi-bounded approximate identity. Then apply Theorem 4.7.
Corollary 4.9
Let \((S,(q_{i})_{i \in I})\) be a preannihilator left regular factorable Segal locally m-convex algebra in a preannihilator complete locally m-convex algebra E. Suppose that every multiplier in S is continuous and that E has a bounded left approximate identity contained in S. Then \(M_{l}(S)=M_{l}(E)\) within a topological algebra isomorphism.
Proof
First, due to Theorem 4.3, E and \(L_{S}\) are isomorphic as topological algebras. If \(\{ e_{\delta } \}_{\delta }\) is a bounded left approximate identity in E that lies in S, then it is also a left approximate identity in S. Therefore \(\{ e_{\delta } \}_{\delta }\) is a quasi-bounded left approximate identity in S. The conclusion follows from Theorem 4.7.
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Notes
We are using the same notation \(\tilde{q_{i}}\) for the uniform seminorm in \(\mathcal {L}(S)\) and in \(L_{S}\).
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Haralampidou, M., Palacios, L. & Signoret, C. Isomorphisms Between the Multiplier Algebras of Certain Topological Algebras. Mediterr. J. Math. 21, 101 (2024). https://doi.org/10.1007/s00009-024-02647-8
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DOI: https://doi.org/10.1007/s00009-024-02647-8
Keywords
- Left (right) multiplier
- multiplier algebra
- Segal topological algebra
- preannihilator algebra
- left regular representation
- locally \(C^{*}\)-algebra