PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 90(104) (2011), 135–143 DOI: 10.2298/PIM1104135S

2-NORMED ALGEBRAS-II

Neeraj Srivastava, S. Bhattacharya, and S. N. Lal

Communicated by Stevan Pilipović

Abstract. In the first part of the paper [5], we gave a new definition of real or complex 2-normed algebras and 2-Banach algebras. Here we give two examples which establish that not all 2-normed algebras are normable and a 2-Banach algebra need not be a 2-Banach space. We conclude by deriving a new and interesting spectral radius formula for 1-Banach algebras from the basic properties of 2-Banach algebras and thus vindicating our definitions of 2-normed and 2-Banach algebras given in [5].

1. Introduction

This paper being the sequel to our earlier paper, for notations and definitions, we refer to the said paper [5].

In the next section we give two examples. The first example establishes that not all 2-normed algebras are normable and the other shows that a 2-Banach algebra need not be a 2-Banach space. In Section 3, some basic properties of a 2-Banach algebra are derived. As it turns out, these properties as well as their proofs go almost parallel to the case of an 1-Banach algebra. In Section 4, we derive, from the results obtained in Section 3, a new and interesting spectral radius formula for an 1-Banach algebra. The results in Sections 2 and 4 vindicate our definitions of a 2-normed and 2-Banach spaces given in [5].

2. Examples

Theorem 2.1. There exist 2-normed algebras (with or without unity) which are not normable.

Proof. Let (E, ∥., .∥) be a 2-normed space which is not normable (for the existence of such a space, see Gähler [1]). We define for x, y ∈ E, xy = 0 and E becomes an algebra. Let a_{1}, a_{2} be any two linearly independent elements of E (dimE ⩾ 2). Then, ∥xy, a_{i}∥ = 0∥x, a_{i}∥ ∥y, a_{i}∥ for i = 1, 2 and for all x, y ∈ E and (E, ∥., .∥) becomes a 2-normed algebra with respect to a_{1}, a_{2} without unity and (E, ∥., .∥) is not normable. Let (E′, ∥., .∥) be the algebra after augmentation of unity. Then as we have observed in [5], (E′, ∥., .∥) is a 2-normed algebra with respect to a_{1}, a_{2} with unity and as (E, ∥., .∥) is not normable, (E′, ∥., .∥) is also not normable and we have the theorem. □

2010 Mathematics Subject Classification: 46A, 46H.

We conclude this section by giving an example which shows that a 2-Banach algebra need not be a 2-Banach space. Let I = [0, 1],

A_{∞} = ℚ ∩ I = \{1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{1}{6}, \frac{5}{6}, \frac{1}{7}, …\} = \{r_{i} : i ∈ ℕ\},

A_{e} = {r_{2i} : i ∈ ℕ}, A_{n} = {r_{1}, …, r_{n}} and define sequence of functions {a_{n}} and {ϕ_{n}} on I by

a_{n}(x) = \{ \begin{matrix}1, & if x = r_{n} \\ 0, & otherwise, \end{matrix} , ϕ_{n}(x) = \{ \begin{matrix}1, & if x ∉ (A_{e} ∪ A_{n}) \\ 0, & otherwise \end{matrix} .

Let ℑ(I) = the set of all bounded 𝕂-valued functions on I having at most countably many points of discontinuity in I. Then the sequences {a_{n}} and {ϕ_{n}} are in ℑ(I). In ℑ(I), let ∥.∥ be the sup 1-norm and ∥., .∥ be the 2-norm defined by, for f, g ∈ ℑ(I), ∥f, g∥ = sup_{x,y∈I} |f(x)g(y) − f(y)g(x)|. The space ℑ(I) is an algebra over 𝕂 with unity with pointwise addition and multiplication. We also have for each n ∈ ℕ and for each f ∈ ℑ(I), fa_{n} = a_{n}f = f(r_{n})a_{n}.

We prove the following lemmas.

Lemma 2.1. The 2-normed space (ℑ(I), ∥., .∥) is a 2-normed algebra with respect to a_{1}, a_{2} (or any pair of distinct elements in {a_{n}}).

Proof. For f ∈ ℑ(I) we have for n ∈ ℕ, ∥f, a_{n}∥ = sup_{x∈I, x≠r_{n}} |f(x)|. Therefore, for each f, g ∈ ℑ(I), and n ∈ ℕ,

∥fg, a_{n}∥ = sup_{x≠r_{n}, x∈I} |f(x)g(x)| ⩽ ( sup_{x≠r_{n}, x∈I} |f(x)|)( sup_{x≠r_{n}, x∈I} |g(x)|) = ∥f, a_{n}∥ ∥g, a_{n}∥

and the lemma is proved. □

Lemma 2.2. The 1-normed space (ℑ(I), ∥.∥) is an 1-Banach space.

Proof. Let {f_{n}} be a Cauchy sequence in (ℑ(I), ∥.∥). Then for each x ∈ I, {f_{n}(x)} converges to some f(x) in 𝕂, and hence {f_{n}} converges to f uniformly in I. To prove the lemma it is required to show that f ∈ ℑ(I). Let F_{n} be the set of all points of discontinuity of f_{n} in I and F be the set of all points of discontinuity of f in I. We prove that F ⊆ ⋃ F_{n} and the lemma will be established. If x_{0} ∈ F, then there exists an ε > 0 such that for all δ > 0, there exists an x_{δ} ∈ I such that

(*)  |x_{0} − x_{δ}| < δ and |f(x_{0}) − f(x_{δ})| ⩾ ε

As {f_{n}} converges to f uniformly in I, there exists an N ∈ ℕ such that

(i)  |f_{n}(x) − f(x)| < ε/3 for all n ⩾ N, for all x ∈ I.

If possible, let x_{0} ∉ ⋃ F_{n}. Then x_{0} ∉ F_{N} and so, as f_{N} is continuous at x_{0}, we have a δ_{0} > 0 such that for |x − x_{0}| < δ_{0}, x ∈ I,

(ii)  |f_{N} (x) − f_{N} (x_{0})| < ε/3.

2-NORMED ALGEBRAS-II 137

Then, for |x − x_{δ_{0}} | < δ_{0}, |f(x_{0}) − f(x_{δ_{0}} )| ⩽ |f(x_{0}) − f_{N} (x_{0})| + |f_{N} (x_{0}) − f_{N} (x_{δ_{0}} )| + |f_{N} (x_{δ_{0}} ) − f(x_{δ_{0}} )| < ε by (i) and (ii) and this contradicts (*). Hence x_{0} ∈ ⋃ F_{n} and the lemma follows. □

Lemma 2.3. A sequence {f_{n}} is Cauchy in (ℑ(I), ∥., .∥) with respect to a_{1}, a_{2} (or any pair of distinct elements in {a_{n}}) (that is, lim_{m,n→∞} ∥f_{m} − f_{n}, a_{i}∥ = 0, for i = 1, 2) if and only if {f_{n}} is Cauchy in (ℑ(I), ∥.∥).

Proof. As for each f ∈ ℑ(I), ∥f, a_{n}∥ = sup_{x∈I, x≠r_{n}} |f(x)|, we have, for i = 1, 2

lim_{m,n→∞} ∥f_{m} − f_{n}, a_{i}∥ = 0 ⇔ lim_{m,n→∞} ( sup_{x∈I, x≠r_{n}} |f_{m}(x) − f_{n}(x)|) = 0

⇔ lim_{m,n→∞} ( sup_{x∈I, x≠r_{i}} |f_{m}(x) − f_{n}(x)|) = 0 ⇔ lim_{m,n→∞} ∥f_{m} − f_{n}∥ = 0

and the lemma is proved. □

Lemma 2.4. A sequence {f_{n}} in ℑ(I) is convergent to an f in (ℑ(I), ∥.∥) iff {f_{n}} is convergent to f in (ℑ(I), ∥., .∥) with respect to a_{1}, a_{2} (or any pair of distinct element in {a_{n}}) (that is, lim_{n→∞} ∥f_{n} − f, a_{i}∥ = 0 for i = 1, 2).

Proof. Follows as in Lemma 2.3. □ Lemma 2.5. The 2-normed space (ℑ(I), ∥., .∥) is a 2-Banach algebra with re-

spect to a_{1}, a_{2} (or any pair of distinct elements in {a_{n}}).

Proof. Follows from Lemmas 2.1, 2.2, 2.3 and 2.4. □ Lemma 2.6. The sequence {ϕ_{n}} is Cauchy in (ℑ(I), ∥., .∥). Proof. Define functions b_{1}, b_{2} on I by

b_{1}(x) = \{ \begin{matrix}1, & if x ∈ A_{e} \\ 0, & otherwise \end{matrix} and b_{2}(x) = \{ \begin{matrix}1, & if x ∈ (A_{e} ∖ \{1/2\}) \\ 0, & otherwise. \end{matrix}

Now for f ∈ ℑ(I), we have,

∥f, b_{1}∥ = max{ sup_{k∈ℕ}|f(r_{2k})|, sup_{k,l∈ℕ} |f(r_{2k}) − f(r_{2l})|},
∥f, b_{2}∥ = max{ sup_{k∈ℕ, k⩾2} |f(r_{2k})|, sup_{k,l∈ℕ, k,l⩾2ℕ} |f(r_{2k}) − f(r_{2l})|}.

Now for each m, n ∈ ℕ and for i = 1, 2 we have ∥ϕ_{m} − ϕ_{n}, b_{i}∥ = 0 and hence {ϕ_{n}} is Cauchy in (ℑ(I), ∥., .∥), and the lemma is proved. □

Lemma 2.7. The sequence {ϕ_{n}} is not Cauchy in (ℑ(I), ∥.∥). Proof. We have, for m, n ∈ ℕ, m < n, A_{m} ⊆ A_{n}, and for x ∈ I,

|ϕ_{m}(x) − ϕ_{n}(x)| = \{ \begin{matrix}1, & if x ∉ (A_{m} ∪ A_{e}), x ∈ (A_{n} ∪ A_{e}) \\ 0, & otherwise. \end{matrix}

As for each pair of m, n ∈ ℕ, m < n and n sufficiently large, there exists an x ∈ I such that x ∉ (A_{m} ∪ A_{e}) but x ∈ (A_{n} ∪ A_{e}), we have ∥ϕ_{m} − ϕ_{n}∥ = 1 and the lemma is proved. □

Lemma 2.8. The 2-normed space (ℑ(I), ∥., .∥) is not a 2-Banach space.

Proof. If (ℑ(I), ∥., .∥) is a 2-Banach space, then by Lemma 2.6 as the sequence {ϕ_{n}} is Cauchy in (ℑ(I), ∥., .∥), there is a ϕ in ℑ(I) so that lim_{n→∞} ∥ϕ_{n} −ϕ, f∥ = 0 for each f ∈ ℑ(I) and hence, in particular, lim_{n→∞} ∥ϕ_{n}−ϕ, a_{i}∥ = 0 for i = 1, 2. But then by Lemma 2.4, {ϕ_{n}} is convergent to ϕ in (ℑ(I), ∥.∥) contradicting Lemma 2.7 and the proof is complete. □

The Lemmas 2.5 and 2.8 imply that the 2-normed space (ℑ(I), ∥., .∥) is a 2-Banach algebra with respect to a_{1}, a_{2} (or any pair of distinct elements in {a_{n}}) though the 2-normed space (ℑ(I), ∥., .∥) is not a Banach space, and we have the following.

Theorem 2.2. A 2-Banach algebra need not be a 2-Banach space.

3. 2-Banach algebras: Some basic properties

Let (E, ∥., .∥) be a 2-Banach algebra with respect to a_{1}, a_{2} over 𝕂 with unity (If E is without unity we augment unity as in [5]) a_{1}, a_{2} ∈ A, where A is an algebra with unity over 𝕂, E is a subalgebra of A and (A, ∥., .∥) is a 2-normed space. As we have seen in [5] (E, ∥., .∥) is a topological vector space, the topology being induced by the 2-norm ∥., .∥ in E. In this section the topological concepts like closed/open sets, continuity etc. in E, are all meant for the topological vector space (E, ∥., .∥). In this context, the following proposition is useful.

Proposition 3.1. Let (E, ∥., .∥) be a 2-normed linear space over 𝕂, X be a nonempty subset of E. Then X is open if and only if for each a_{0} ∈ X, there exists ε_{a_{0}} > 0 and b ∈ E such that for each c ∈ E with ρ_{b}(c) = ∥b, c∥ < ε_{a_{0}} implies a_{0} + c ∈ X.

Proof. For a proof see [4]. □

Before we proceed further, let us agree with the following notations. For a 2-Banach algebra (E, ∥., .∥) with unity e with respect to a_{1}, a_{2} over 𝕂, G(E) denotes the group of all invertible elements of E. For a ∈ E, σ(a), ωa and r(a) denote the spectrum, resolvent and spectral radius of a respectively.

Theorem 3.1. Let (E, ∥., .∥) be a 2-Banach algebra with unity e over 𝕂 with respect to a_{1}, a_{2}.

(i) If a ∈ E is such that ∥a, a_{i}∥ < 1 for i = 1, 2, then e − a ∈ G(E) and if ϕ be a nontrivial 𝕂-homomorphism on E, |ϕ(a)| < 1.

(ii) The group G(E) is open in (E, ∥., .∥), and the mapping f : G(E) → G(E) defined by f(a) = a^{−1} , a ∈ G(E) is a homeomorphism on G(E).

Proof. (i) For each a ∈ E, associate a sequence {s_{n}(a)} in E defined by s_{n}(a) = e + a + a^{2} + ⋯+ a^{n}. Now if ∥a, a_{i}∥ < 1, i = 1, 2 we have for n ∈ ℕ, n ⩾ 2, i = 1, 2:

∥a^{n}, a_{i}∥ ⩽ ∥a^{n−1} , a_{i}∥ ∥a, a_{i}∥ ⩽ ∥a^{n−2} , a_{i}∥ ∥a, a_{i}∥^{2} ⩽ ⋯⩽ ∥a, a_{i}∥^{n} → 0 as n → ∞.

So,

∥s_{n+p} (a) − s_{n}(a), a_{i}∥ ⩽ ∥a^{n+1} , a_{i}∥ + ⋯+ ∥a^{n+p} , a_{i}∥

⩽ ∥a, a_{i}∥^{n+1} + ⋯+ ∥a, a_{i}∥^{n+p}

⩽ \frac{∥a, a_{i}∥^{n+1}}{1 − ∥a, a_{i}∥} for i = 1, 2; n, p ∈ ℕ, n ⩾ 2, → 0 as n, p → ∞.

But (E, ∥., .∥) being 2-Banach algebra with respect to a_{1}, a_{2} there exists an s(a) in E such that lim_{n→∞} ∥s_{n}(a) − s(a), a_{i}∥ = 0.

Now, s_{n}(a)(e − a) = e − a^{n+1} = (e − a)s_{n}(a) for all n ∈ ℕ and therefore for i = 1, 2,

∥s(a)(e − a) − e, a_{i}∥ = ∥s(a)(e − a) − s_{n}(a)(e − a) − a^{n+1} , a_{i}∥
⩽ ∥(s(a) − s_{n}(a))(e − a), a_{i}∥ + ∥a^{n+1} , a_{i}∥
⩽ ∥s(a) − s_{n}(a), a_{i}∥ ∥e − a, a_{i}∥ + ∥a^{n+1} , a_{i}∥

becomes zero as n → ∞. Hence ∥s(a)(e − a) − e, a_{i}∥ = 0 for i = 1, 2 and so s(a)(e − a) = e. Similarly, (e − a)s(a) = e and therefore, s(a) = (e − a)^{−1} and e − a ∈ G(E). (We call the series s(a) = e + a + a^{2} + ⋯, the associate series of a).

To prove the second part of (i), let, if possible, ϕ be a nontrivial 𝕂-homo-morphism on E, |ϕ(a)| ⩾ 1. Let λ ∈ 𝕂 be such that ϕ(a) = λ. Then |λ| ⩾ 1 and ϕ(λ^{−1} a) = 1 and so ϕ(e − λ^{−1} a) = 0 as ϕ being nontrivial, ϕ(e) = 1. Let b = e − λ^{−1} a. Then ϕ(b) = 0. But as |λ| ⩾ 1 and ∥a, a_{i}∥ < 1 for i = 1, 2, we have ∥λ^{−1} a, a_{i}∥ < 1 for i = 1, 2 and hence b = e − λ^{−1} a ∈ G(E). But then ϕ(b) ≠ 0 and we have a contradiction and (i) is completely proved.

(ii) To see that G(E) is open, let a ∈ G(E). Note that, as a_{1}, a_{2} are linearly independent, for a ∈ E, ∥a, a_{i}∥ = 0 for i = 1, 2 if and only if a = 0, see [5]. Take ε_{a} = \frac{1}{2}(max_{i=1,2} \{∥a^{−1} , a_{i}∥\})^{−1} . Then for b ∈ E with ∥b, a_{i}∥ < ε_{a}, i = 1, 2, we have ∥ − a^{−1} b, a_{i}∥ ⩽ ∥a^{−1} , a_{i}∥ ∥b, a_{i}∥ < \frac{1}{2}, which implies, e + a^{−1} b ∈ G(E) by (i), and as a + b = a(e + a^{−1} b), we have a + b ∈ G(E) which using Proposition 3.1 proves that G(E) is open.

To prove that f is a homeomorphism, let b ∈ E, a ∈ G(E) and ∥b−a, a_{i}∥ < ε_{a}, for i = 1, 2; then, as a + (b − a) = b and G(E) is open, b ∈ G(E). Write c = a^{−1} (b − a). Then for i = 1, 2, ∥c, a_{i}∥ ⩽ ∥a^{−1} , a_{i}∥ ∥b − a, a_{i}∥ ⩽ \frac{1}{2} and hence by (i), e + c ∈ G(E). We also have, for each n ∈ ℕ, s_{n}(−c)(e + c) = e − (−1)^{n+1} c^{n+1} and s(−c) = (e + c)^{−1} ∈ G(E) and for i = 1, 2,

∥s_{n}(−c) − e, a_{i}∥ ⩽ ∥c, a_{i}∥ + ∥c, a_{i}∥^{2} + ⋯+ ∥c, a_{i}∥^{n} = \frac{∥c, a_{i}∥(1 − ∥c, a_{i}∥^{n})}{1 − ∥c, a_{i}∥}
⩽ 2∥c, a_{i}∥(1 − ∥c, a_{i}∥^{n}) (as ∥c, a_{i}∥ ⩽ 1/2)

and so ∥s(−c)−e, a_{i}∥ ⩽ 2∥c, a_{i}∥ for i = 1, 2. Since b^{−1} −a^{−1} = [(e + c)^{−1} − e]a^{−1} = [s(−c) − e]a^{−1} , we have for i = 1, 2:

∥f(b) − f(a), a_{i}∥ = ∥b^{−1} − a^{−1} , a_{i}∥ = ∥(s(−c) − e)a^{−1} , a_{i}∥
⩽ ∥s(−c) − e, a_{i}∥ ∥a^{−1} , a_{i}∥ ⩽ 2∥a^{−1} , a_{i}∥ ∥b − a, a_{i}∥ ∥a^{−1} , a_{i}∥
= 2∥a^{−1} , a_{i}∥^{2} ∥b − a, a_{i}∥.

Hence, ∥f(b) − f(a), a_{i}∥ ⩽ 2∥a^{−1} , a_{i}∥^{2}∥b − a, a_{i}∥ for i = 1, 2; whenever b ∈ E with ∥b − a, a_{i}∥ < ε_{a}. But this proves that f is continuous on G(E). As f is one to one on G(E), f^{−1} = f. The mapping f is a homeomorphism on G(E) and (ii) is proved. This completes the proof of the theorem. □

Theorem 3.2. Let (E, ∥., .∥) be a 2-Banach algebra with unity e over 𝕂 with respect to a_{1}, a_{2}, and a ∈ E. Then,

(i) σ(a) is closed in 𝕂, (ii) r(a) ⩽ max_{i=1,2} {∥a, a_{i}∥}, (iii) σ(a) is compact in 𝕂, (iv) σ(a) is nonempty if 𝕂 = ℂ, and

(v) r(a) = lim_{n→∞} [max_{i=1,2} {∥a^{n}, a_{i}∥}]^{1/n} .

Proof. (i) For a ∈ E define f_{a} : 𝕂 → E by f_{a}(λ) = λe−a for λ ∈ 𝕂. Then f_{a} is continuous on 𝕂, and so f_{a}^{−1} (G(E)) is open in 𝕂 as G(E) is open in (E, ∥., .∥) by Theorem 3.1. We claim that Ω_{a} = f_{a}^{−1} (G(E)). To prove the claim, let λ ∈ Ω_{a}; then λe − a ∈ G(E) and so f_{a}(λ) ∈ G(E) which implies λ ∈ f_{a}^{−1} (G(E)). Conversely, let λ ∈ f_{a}^{−1} (G(E)). Then f_{a}(λ) = λe − a ∈ G(E) and so λ ∉ σ(a) and the claim is proved. This proves that σ(a) is closed.

(ii) Write k = max_{i=1,2} ∥a, a_{i}∥ and let, if possible, r(a) > k. Then there exists λ ∈ σ(a) such that |λ| > k, and therefore ∥λ^{−1} a, a_{i}∥ < 1 for i = 1, 2. But this implies by Theorem 3.1, e − λ^{−1} a ∈ G(E) and hence λ ∉ σ(a), and (ii) is proved.

(iii) Combining (i) and (ii), we get (iii).

(iv) For a ∈ E, define a mapping R_{a} : Ω_{a} → G(E) by R_{a}(λ) = (λe − a)^{−1} , λ ∈ Ω_{a}. Let λ ∈ Ω_{a}, δ = [max_{i=1,2} ∥R_{a}(λ), a_{i}∥]^{−1} . Let μ ∈ Ω_{a} be such that

(3.1)  |λ − μ| < \frac{1}{2} δ,

and b = (μ − λ)R_{a}(λ). Then, as ∥b, a_{i}∥ < 1 for i = 1, 2, e − b ∈ G(E), by Theorem 3.1. For i = 1, 2 we have

∥s_{n}(b) − e − b, a_{i}∥ ⩽ ∥b, a_{i}∥^{2} + ⋯+ ∥b, a_{i}∥^{n} = \frac{∥b, a_{i}∥^{2}(1 − ∥b, a_{i}∥^{n})}{1 − ∥b, a_{i}∥}

(see the proof of Theorem 3.1) and so, for i = 1, 2,

(3.2) ∥(e − b)^{−1} − e − b, a_{i}∥ = ∥s(b) − e − b, a_{i}∥ ⩽ \frac{∥b, a_{i}∥^{2}}{1 − ∥b, a_{i}∥} ,  i = 1, 2.

Now,

R_{a}(μ) − R_{a}(λ) + (μ − λ)(R_{a}(λ))^{2}
= [(μe − a)^{−1} (λe − a) − e + (μ − λ)R_{a}(λ)]R_{a}(λ)
= [[(λe − a)^{−1} {(μ − λ)e + (λe − a)}]^{−1} − e + (μ − λ)R_{a}(λ)]R_{a}(λ)
= [{e + (μ − λ)R_{a}(λ)}^{−1} − e + (μ − λ)R_{a}(λ)]R_{a}(λ).

Therefore, for i = 1, 2,

∥R_{a}(μ) − R_{a}(λ) + (μ − λ)(R_{a}(λ))^{2}, a_{i}∥
⩽ ∥{e + (μ − λ)R_{a}(λ)}^{−1} − e + (μ − λ)R_{a}(λ), a_{i}∥ ∥R_{a}(λ), a_{i}∥
⩽ \frac{|(λ − μ)| ∥R_{a}(λ), a_{i}∥^{2}}{1 − |(μ − λ)| ∥R_{a}(λ), a_{i}∥} ∥R_{a}(λ), a_{i}∥  (by (3.2))
⩽ 2|μ − λ|^{2}∥R_{a}(λ), a_{i}∥^{3} ⩽ \frac{1}{2} δ∥R_{a}(λ), a_{i}∥  (by (3.1))

as ∥b, a_{i}∥ = |μ − λ| ∥R_{a}(λ), a_{i}∥ ⩽ \frac{1}{2}.

Therefore, for i = 1, 2, for μ ∈ Ω_{a}, μ ≠ λ and |μ − λ| < \frac{1}{2}δ,

∥ \frac{R_{a}(μ) − R_{a}(λ)}{μ − λ} + (R_{a}(λ))^{2}, a_{i} ∥ ⩽ 2|μ − λ| ∥R_{a}(λ), a_{i}∥^{3}.

So, lim_{μ→λ} \frac{R_{a} (μ)−R_{a} (λ)}{μ−λ} exists in the topological linear space (E, ∥., .∥) and equals to −(R_{a}(λ))^{2} for λ ∈ Ω_{a} and we conclude that R_{a} is analytic in Ω_{a}.

Now, if possible, let σ(a) be empty. Then Ω_{a} = 𝕂 = ℂ and R_{a} is an entire function. Let λ ∈ ℂ be such that k < |λ|, that is, ∥λ^{−1} a, a_{i}∥ < 1 for i = 1, 2; k be as in (ii). Then by Theorem 3.1, e−λ^{−1} a ∈ G(E) and as s(λ^{−1} a) = (e−λ^{−1} a)^{−1} = e + (λ^{−1} a) + (λ^{−1} a)^{2} + ⋯, we have

(3.3)  R_{a}(λ) = λ^{−1} (e − λ^{−1} a)^{−1} = λ^{−1} e + λ^{−2} a + λ^{−3} a^{2} + ⋯

Let Γ_{r} be the circle on the complex plane with center at origin and radius r, where k < r. Then the series on the right-hand side of (3.3) converges uniformly on Γ_{r} and so term by term integration over Γ_{r} is allowed to the right hand-side of the series in (3.3), and we conclude that for n = 0, 1, 2, … and for r > k,

(3.4)  a^{n} = \frac{1}{2πi} ∫_{Γ_{r}} λ^{n}R_{a}(λ) dλ

and in particular

(3.5)  e = \frac{1}{2πi} ∫_{Γ_{r}} R_{a}(λ) dλ.

But as R_{a} is entire, by Cauchy theorem, the integral on the right-hand side of (3.5) is zero, which is a contradiction and the proof of (iv) is complete.

(v) Let a ∈ E. Then for r > r(a) also (3.4) holds. The continuity of R_{a} in Γ_{r} implies that for r > r(a), B(r) = max_{j=1,2, θ∈[0,2π]} ∥R_{a}(re^{iθ}), a_{j} ∥ is finite. Hence by (3.4) we have max_{j=1,2} {∥a^{n}, a_{j} ∥} ⩽ r^{n+1} B(r) for all n ∈ ℕ, which then implies that

(3.6)  lim_{n→∞} sup[ max_{j=1,2}{∥a^{n}, a_{j} ∥}]^{1/n} ⩽ r(a)

Again, for λ ∈ σ(a), as we have, for all n ∈ ℕ,

λ^{n}e − a^{n} = (λe − a)(λ^{n−1} e + λ^{n−2} a + ⋯+ λa^{n−2} + a^{n−1} ),

we see that λ^{n}e − a^{n} ∉ G(E) and hence λ^{n} ∈ σ(a^{n}) for all n ∈ ℕ. Then by (ii) we have for all n ∈ ℕ, |λ^{n}| ⩽ r(a^{n}) ⩽ [max_{j=1,2} {∥a^{n}, a_{j} ∥}]. Hence, for all λ ∈ σ(a), and for all n ∈ ℕ, |λ| ⩽ [max_{j=1,2} {∥a^{n}, a_{j} ∥}]^{1/n} implying,

(3.7)  r(a) ⩽ lim_{n→∞} inf [ max_{j=1,2}{∥a^{n}, a_{j} ∥}]^{1/n} .

Now (3.6) and (3.7) implies that lim_{n→∞} [max_{j=1,2} {∥a^{n}, a_{j} ∥}]^{1/n} exists and equals to r(a). This establishes (v) and the proof of the theorem is complete. □

4. Spectral radius formula for 1-Banach algebras

The following theorem contains a new spectral radius formula for 1-Banach algebras.

Theorem 4.1. Let (E, ∥.∥) be an 1-Banach algebra with unity e over ℂ, dimE ⩾ 2 such that a nontrivial ℂ-homomorphism on E exists. Then there exists an 1-Banach algebra (B, ∥.∥_{1}) of which (E, ∥.∥) is a closed subalgebra, the 1-norm ∥.∥_{1} on B when restricted on E becomes the 1-norm ∥.∥ on E and a_{1}, a_{2} ∈ B such that for all a ∈ E, (4.1)

r(a) = lim_{n→∞} ∥a^{n}∥^{1/n} = lim_{n→∞} [ max_{i=1,2} \{ sup_{\begin{matrix}ϕ,Ψ∈B^{∗} \\ ∥ϕ∥=∥Ψ∥=1 \end{matrix}} |ϕ(a^{n})Ψ(a_{i}) − Ψ(a^{n})ϕ(a_{i})|\} ]^{1/n}

Proof. Follows from Lemma 5.4, Theorem 5.1 of [5] and Theorem 3.2. □

We conclude this section by stating the following 2-norm version of the Gelfand– Mazur theorem [2, 3].

Theorem 4.2. There does not exist a 2-Banach division algebra over ℂ.

Proof. If possible, let (E, ∥., .∥) be a 2-Banach division algebra on ℂ with respect to a_{1}, a_{2}. Then dimE ⩾ 2. For each 0 ≠ a ∈ E, we claim that σ(a) is a singleton. To prove this claim, we observe that σ(a) is nonempty by Theorem 3.2 and if λ_{1}, λ_{2} ∈ σ(a), λ_{1} ≠ λ_{2}, then as λ_{1}e − a and λ_{2}e − a both are noninvertible, we have λ_{1}e − a = λ_{2}e − a = 0 as E is a division algebra. So, λ_{1} = λ_{2}, and our claim is proved. Now for each a in E, let σ(a) = {λ(a)} and by definition of σ(a), a = λ(a)e, that is, E is generated by e and hence dimE = 1 and the theorem follows. □

References

1. S. Gähler, Lineare 2-normierte Räume, Math. Nachr. 28 (1965), 1–43.

2. I. M. Gel’fand, Normierte Ringe, Mat. Sb., Nov. Ser. 9(51) (1941), 3–21.

3. I. M. Gel’fand and M. A. Neumar, On the imbedding of normed rings into the ring of operators in Hilbert space, Mat. Sb., Nov. Ser. 12(54) (1943), 197–213.

4. S. N. Lal, S. Bhattacharya, and E. Keshava Reddi, 2-norm on an algebra, Prog. Math., Varanasi 35 (2001), 27–32.

5. Neeraj Srivastava, S. Bhattacharya, and S. N. Lal, 2-normed algebras–I, Publ. Inst. Math., Nouv. Sér. 88(102) (2010), 111–121.

National Academy for Scientific Development (Received 20 05 2010) Education and Research “Shashi Villa” Mahamanapuri Colony B.H.U. Post Office Varanasi 221 005 India

neeraj_bhuin@yahoo.co.in

Department of Mathematics Faculty of Science Banaras Hindu University Varanasi 221 005 India

sbhatta@bhu.ac.in

National Academy for Scientific Development, Education and Research “Shashi Villa” Mahamanapuri Colony B.H.U. Post Office Varanasi 221 005 India

snlmath@yahoo.com