A Class of algebras admitting infinitely many norm topologies

It is a well-known fact that a vector space admits either a unique linear norm (up to equivalence) or infinitely many linear norms (up to equivalence). So, it is natural to ask for the case of algebra. It is surprising that the behaviour of submultiplicative norm on algebra is extremely different. From the following references [[DL:97], [DP:22(a)]]. So far there are three types of classification of an algebra norm. These are (i) there is an algebra which has no any algebra norm, e.g. C(R) The set of all continuous functions on R with usual operations and pointwise multiplication, (ii) there is an algebra which has a unique algebra norm, e.g. B(H) (iii) there is an algebra which has infinitely many algebra norms, e.g., the disc algebra A(D). From these classification it naturaly leads the following question: What about an algebra having only finitely many more than one norms !!!. Also this question was asked in author’s doctorate thesis. If such kind of an algebra exist, then the codimension A/A 2 is finite can be conclude from the Corollary 2.5.

Dales and Loy [DL:97] gave a nice example of an algebra with a finite-dimensional radical, and built two different algebra norms on it. Encouraged by them, the current paper generalizes the result further with a proof of the fact that not only two but actually an infinite number of inequivalent algebra norms can be constructed on such an algebra.

Throught A is an infinite dimensional algebra.

Definition 2.1. Let (A, ∥ • ∥) be a normed algebra. An algebra A has the discontinuous square annihilation property (DSAP) if there exists a discontinuous linear functional φ on A such that A 2 ⊆ kerφ.

Proposition 2.2. Let (A, ∥ • ∥) be a normed algebra. Then if A right (left) unital or A has bounded approximate identity. Further, if Proof. In both the cases A = A 2 . Hence, every linear functional having A 2 kerϕ is identically zero. Now, for converse consider c 00 with pointwise linear, scalar, and multiplication operations. Then (c 00 , ∥ • ∥ 1 ) is normed algebra. Now it’s clear that c 2 00 = c 00 but it neither have right (left) identity nor bounded approximate identity. □ Lemma 2.3. Let (A, ∥ • ∥) be a normed algebra. Let φ be a linear functional on A such that the A 2 ⊆ker(φ). For each a ∈ A, define

Proof. The proofs are very easy for both cases. □ Theorem 2.4. Let (A, ∥ • ∥) be a normed algebra. The codimension of A 2 in A is infinite iff A has DAP.

Proof. Let (A, ∥ • ∥) be a normed algebra. Since A 2 has infinite co-dimension in A, there exists a countably infinite linearly independent subset L = {a 1 , a 2 , a 3 , . . .} of A such that A 2 ∩ L = ϕ and ∥a∥ = 1 (a ∈ L).

Let D be a basis of A 2 . Now, consider the (unique) linear map φ : Hence p(•) is inequivalent to ∥ • ∥. Conversely assume that A has DAP, so there is discontinuous linear functional ϕ : A -→ C such that A 2 ⊂ Kerϕ. Now suppose the codimension of A 2 in A is finite, the embeeding map φ :

is continuous as kerϕ = A 2 . Hence ϕ is continuous as ϕ = φ • ψ. □ Corollary 2.5. Let A be a normed algebra, and let the codimension of A 2 in A is infinite. Then A admits infinitely many algebra norm.

Proof. Let (A, ∥ • ∥) be a normed algebra. Since A 2 has infinite co-dimension in A, there exists a countably infinite linearly independent subset L of A such that A 2 ∩ L = ϕ and ∥a∥ = 1 (a ∈ L). For each n ∈ N, choose L n = {a n1 , a n2 , . . .} ⊂ L such that:

(1) Each L n is infinite;

(2) Theorem 2.7. Let A be an algebra having only finitely many norms.

(1) Then A has maximum and minimum norms.

(2) Then A has atmost one complete norm.

(3) The codimension of A 2 in A is finite.

Proof. Example 3.2. Let A = A(D) be a disc algebra. Then it shown that it has infinitely many norms. So, this example says that the converse of the Corollary 2.5 is not true as A(D) is unital algebra.

Example 3.3. Let (A, ∥•∥) be an algebra. Then A×C is an algebra with pointwise operations and multiplication (a, α) × (b, β) = (ab, 0) for ((a, α), (b, β) ∈ A × C). It was shown that A × C has infinitely many norms.

Example 3.4. Let F[x] = {p(x) = a 0 + a 1 x + . . . + a n x n : a 0 , a 1 , . . . + a n ∈ F}.

Consider A = {p(x) ∈ F[x] : p i (0) = 0, (0 ≤ i ≤ n -1)}. Then A is an algebra with the usual linear and scalar multiplication, and pointwise multiplication opearions. Now we can easily conclude that if p(x) ∈ A, then p(x) = x n q(x), for some q(x) ∈ F[x]. So, A 2 = x n A. Hence the co-dimension of A 2 in A is n, as {1, x, x 2 , . . . , x n-1 } is basis of A/A 2 .