Radicals of Biduals of Beurling Algebras Can Be Different for the Two Arens Products

Let $\operatorname{rad}$ denote the Jacobson radical of a Banach algebra, and let $\Box$ and $\Diamond$ denote the two Arens products on its bidual. We give an example of a Beurling algebra $\mathcal{A}$ for which $\operatorname{rad}(\mathcal{A}^{}, \Box) \neq \operatorname{rad}(\mathcal{A}^{}, \Diamond)$, answering a question of Dales and Lau. The underlying group in our example is the free group on three generators.

💡 Research Summary

The paper addresses a long‑standing question posed by Dales and Lau concerning whether the Jacobson radicals of a Banach algebra’s bidual can differ when the bidual is equipped with the two distinct Arens products (commonly denoted □ and ⋄). The author constructs a concrete counterexample using a weighted ℓ¹‑algebra (a Beurling algebra) on the free group with three generators, F₃.

The construction proceeds in several stages. First, the author recalls the definition of the two Arens products on A**, emphasizing that □ is weak*‑continuous on the right and ⋄ is weak*‑continuous on the left. For commutative Banach algebras the two radicals always coincide, so a non‑abelian group is required.

A non‑standard infinite generating set X for F₃ is introduced. For each natural number j, a finite set X_j consists of elements of the form v b^{2j‑1} a^{2j} b^{2j} where v∈F₃ and |v|>j (|·| denotes the usual word length with respect to the standard generators). The overall generating set is X = (⋃_{j≥1} X_j) ∪ S, where S is the usual symmetric generating set {a,b,c,a^{-1},b^{-1},c^{-1}}. The X‑word length |g|_X is defined as the minimal number of generators from X needed to express g. By construction |g|_X ≥ |g|_S for all g∈F₃.

A weight ω on F₃ is defined by ω(g)=exp(|g|_X). Subadditivity of the X‑length guarantees submultiplicativity of ω, making ℓ¹(F₃,ω) a Banach algebra under convolution. This is the Beurling algebra that will serve as the example.

The core technical work lies in two combinatorial estimates. First, the author proves that for each n∈ℕ, |a^{2ⁿ}b^{2ⁿ}|_X = 2ⁿ‑1+1.
This is established by showing that the element a^{2ⁿ}b^{2ⁿ} can be written as a product of exactly 2ⁿ‑1+1 generators from X, and that no shorter expression exists. The proof occupies Section 5 and relies on careful analysis of reduced words and cancellations.

Second, a more elaborate estimate is proved: for any finite sequences (n₁,…,n_r) and (k₁,…,k_r) with k_i≥3·2^{n_i} (so that each c^{k_i} is long enough), the X‑length of the product a^{2^{n₁}}b^{2^{n₁}} … a^{2^{n_r}}b^{2^{n_r}} c^{k₁} … c^{k_r} equals ∑{i=1}^r (2^{n_i}‑1+1) + ∑{i=1}^r k_i. This is equation (3.2) and is proved in Section 4. The estimate shows that the X‑length grows linearly with the number of a‑b blocks and with the lengths of the c‑blocks.

With these length formulas in hand, the author defines two elements of the bidual (ℓ¹(F₃,ω))**.
Φ₀ is a weak*‑cluster point of the sequence {δ_{a^{2ⁿ}b^{2ⁿ}}/ω(a^{2ⁿ}b^{2ⁿ}) : n∈ℕ}.
Ψ₀ is a weak*‑cluster point of the sequence {δ_{cⁿ}/ω(cⁿ) : n∈ℕ}.

Because |a^{2ⁿ}b^{2ⁿ}|_X grows like 2ⁿ, the normalized point masses become increasingly “small” in the weighted norm, and the weak*‑limit Φ₀ satisfies Φ₀ □ I = 0, where I denotes the identity of the bidual. Consequently Φ₀ belongs to the Jacobson radical with respect to the □‑product.

On the other hand, using the second length estimate, the product Φ₀ ⋄ Ψ₀ has X‑length exactly the sum of the lengths of the a‑b blocks and the c‑blocks, which is large enough to prevent the product from being quasi‑nilpotent. Hence Φ₀ does not lie in the radical for the ⋄‑product.

Thus the two radicals differ: rad((ℓ¹(F₃,ω)),□) ≠ rad((ℓ¹(F₃,ω)),⋄).
This answers Dales‑Lau’s question affirmatively, providing the first example of a group‑based Banach algebra where the two Arens‑product radicals are distinct.

The paper also notes that the weight ω is not symmetric, so ℓ¹(F₃,ω) is not a *‑algebra; whether a symmetric weight can produce the same phenomenon remains open. The result highlights that non‑commutativity of the underlying group is essential (Lemma 3.1 shows equality of radicals for commutative algebras) and that careful manipulation of word lengths with respect to a non‑standard generating set can control the behavior of Arens products in the bidual. The work thus deepens the understanding of the interplay between group structure, weighted convolution algebras, and the dual‑algebraic properties encoded by the Arens products.