On the norm-preservation of squares in real algebra representation

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality ‖a2‖≤‖a2+b2‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$\end{document} for a,b∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b\in {\mathcal {A}}$$\end{document} is sufficient for a commutative real Banach algebra A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} with a unit to be isomorphic to the space CR(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$\end{document} of continuous real-valued functions on a compact Hausdorff space K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}$$\end{document}. Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} onto CR(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$\end{document} is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces CR(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$\end{document} which are (1+ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\epsilon )$$\end{document}-equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} where ‖a2‖≤k‖a2+b2‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert a^2\Vert \le k \Vert a^2+b^2\Vert $$\end{document} for all a,b∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b\in {\mathcal {A}}$$\end{document}, for some k>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>1$$\end{document}, but the inequality fails for k=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document}.


Introduction
The papers [2,3] show how certain very simple inequalities involving either the algebra norm or the spectral radius imply that a real commutative unital Banach algebra is homomorphic, via the Gelfand transform, to C R (K), the algebra of all continuous real-valued functions on a compact Hausdorff space K equipped with the usual norm, In [3,Theorem 1.1] it is shown that if A is a commutative real Banach algebra with unit, then the spectral radius r satisfies the inequality r (a 2 ) ≤ r (a 2 + b 2 ), for all a, b ∈ A, (1.1) if and only if the Gelfand transform What happens if we just know that the spectral radius satisfies instead the (a priori weaker) inequality r (a 2 ) ≤ kr(a 2 + b 2 ), for all a, b ∈ A, (1.2) for some k ≥ 1? The answer is that nothing new happens. Indeed, the fulfilment of condition (1.2) for some k ≥ 1 implies that the spectrum of any a ∈ A is a subset of the real line (see [2,Proposition 3.5]) and hence using [3, Theorem 1.2], we see that inequality (1.2) is satisfied with k = 1.
In regards to isomorphisms we have the following. Proposition 1.1 Let A be a commutative real algebra with unit. Suppose A is isomorphic to C R (K) for some compact Hausdorff space K. Then the spectral radius seminorm r on A is equivalent to the algebra norm (hence in particular r defines a norm on A). Moreover, A equipped with r is isometric to C R (K).
Proof Let · denote the norm on A. Suppose : A → C R (K) is an isomorphism and let k and k 1 be constants so that For a in A we have a n ≤ k 1 (a n ) ∞,K = k 1 (a) n ∞,K .
Taking the nth-root and letting n tend to infinity yields r (a) ≤ (a) ∞,K . Conversely, (a) n ∞,K = (a n ) ∞,K ≤ k a n .
Taking the nth root and letting n tend to infinity yields (a) ∞,K ≤ r (a), so that (a) ∞,K = r (a) as claimed.
In this paper we shall be concerned with normed real algebras satisfying the corresponding inequality (1.2), where the spectral radius is replaced by the algebra norm. Let us assign a tag to such a class of real algebras.

Definition 1.2 Suppose
A is a commutative real Banach algebra with unit and let k ≥ 1. We will say that A satisfies property (A) k , to be denoted A ∈ (A) k , if the following inequality holds In turn, we will say that A satisfies property (B) k , to be denoted A ∈ (B) k , if In [1,Theorem 3.6] the authors also proved that if A ∈ k≥1 (B) k then A is isomorphic to the algebra C R (K) for some compact Hausdorff space K. The next example shows that A can be isomorphic to C R (K) and yet A / ∈ k≥1 (B) k .

Example 1.3 Consider the algebra of matrices
endowed with the norm on each a ∈ A regarded as an operator on (R 2 , A natural question arises: Do we have a similar situation as with the spectral radius? i.e., does it hold that k≥1 (A) k = (A) 1 or k≥1 (B) k = (B) 1 ? The answer to this question for property (A) k is clearly negative. Recall that if A ∈ (A) 1 , i.e., (1.5) then A is isometrically isomorphic to the algebra C R (K) for some compact Hausdorff space K (see [4,5]). Hence it suffices to equip C R (K) with some equivalent algebra norm. On the other hand, the condition that A ∈ (B) 1 , i.e., only guarantees that A is isomorphic to the algebra C R (K) for some compact Hausdorff space K, although in general it needs not be isometric. In the example where A = C R (K) equipped with the algebra norm f = f + ∞,K + f − ∞,K the condition (1.6) is satisfied but A is not isometric to any C R (K). However, we have the following extra information.
which preserves the norm of squares, i.e., The question arises whether an R-algebra isomorphism of A onto C R (K) is always normpreserving on squares.
As the alert reader might have guessed, if a commutative real Banach algebra with unit A is isomorphic to a space C R (K) for some compact Hausdorff space K then K must agree with the set R A of all real homomorphisms of the algebra, and the isomorphism must be the Gelfand transform (see [2, Remark 2.8]). So the above question will be answered negatively, by constructing in Theorem 2.1 some algebra norm in C R (K) equivalent to · ∞,K which does not preserve the norm of squares.
Of course the above result also exhibits an example of a real normed algebra A ∈ k>1 (B) k \ (B) 1 . Now, Theorem 2.5 will allow us to produce a number of such examples simply by considering two-dimensional normed algebras A such that there exists v ∈ A with v 2 = v and v > 1. We will prove this in the following section.
For notation and background we refer the reader to the recent article [3], which this note aims to complement.

Main theorems
Theorem 2.1 Let K be a compact Hausdorff space with more than two points. For each > 0 we can construct a norm · on C R (K) with the following properties: Moreover, the constants 1 + 2 in (2.1) and 1 + in (2.2) are sharp.
Let us observe also that if f , g ∈ C R (K), and k 1 , Hence, The constant 1 + 2 in (i) is sharp since we can pick points k 1 = k 2 in K and a function To see that the constant 1 + in the last inequality is sharp, choose f ∈ C R (K) with 0 ≤ f ≤ 1 taking the values 0 and 1, and We now give some results concerning the class (B) k for k ≥ 1.

Remark 2.3 Notice that if
On the other hand, since a = α+β 2 e + α−β 2 u and u ≥ 1 we obtain 2 )w and we can write Using also (2.5) we obtain that u = 1 if and only if a = max{|α|, |β|}.
Let us now present some examples of norms on two-dimensional commutative real algebras with a unit satisfying the properties in Corollary 2.6. The construction is inspired by examples of two-dimensional Hilbert space operator algebras from [6].

Example 2.8 Let A be the set of matrices
with the usual matrix multiplication. For each λ > 0 define It is easy to check that · λ is a norm.

Theorem 2.9 Let
We need to show that ab λ ≤ a λ b λ . The case x 2 y 2 = 0 follows trivially since either a = x 1 e or b = y 1 e where e = 1 0 0 1 and a = |x 1 | or b = |y 1 |. We may assume that x 2 = 0 and y 2 = 0, so that it suffices to check the above inequality for where x, y ∈ R \ {0}. Thus we need to show that or, equivalently, We may assume that x y > 0. Observe that 4x y which gives (2.6) for all λ ≥ √ 2.
Assume now that A λ is a normed algebra. In particular for a t = t 1 1 t we have a 2 t λ ≤ a t 2 λ for all t > 0. Since Therefore, Taking limits as t → ∞ gives λ 2 ≥ 2.

Example 2.10 Let
where, as usual, (x 1 , Using Examples 2.7 and 2.8 we can enunciate the following result about the normed algebra A = (R 2 , · λ, p ).

Proposition 2.11
Let A be R 2 equipped with the norm · λ, p defined in (2.7).
When |x| ≤ 1 and |y| ≤ 1 the right hand-side is equal to (1 + λ) 2 and the left hand-side is at most 2 + 2λ, so the inequality holds.
In the case when |x| ≤ 1 ≤ |y| the inequality becomes which follows again under the assumption of λ ≥ 1.
To finish the proof we just need to apply Theorem 2.5 and Corollary 2.6 with v = e+u 2 .