AbstractOne 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 $$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$
‖
a
2
‖
≤
‖
a
2
+
b
2
‖
for $$a,b\in {\mathcal {A}}$$
a
,
b
∈
A
is sufficient for a commutative real Banach algebra $${\mathcal {A}}$$
A
with a unit to be isomorphic to the space $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$
C
R
(
K
)
of continuous real-valued functions on a compact Hausdorff space $${\mathcal {K}}$$
K
. 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 $${\mathcal {A}}$$
A
onto $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$
C
R
(
K
)
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 $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$
C
R
(
K
)
which are $$(1+\epsilon )$$
(
1
+
ϵ
)
-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 $${\mathcal {A}}$$
A
where $$\Vert a^2\Vert \le k \Vert a^2+b^2\Vert $$
‖
a
2
‖
≤
k
‖
a
2
+
b
2
‖
for all $$a,b\in {\mathcal {A}}$$
a
,
b
∈
A
, for some $$k>1$$
k
>
1
, but the inequality fails for $$k=1$$
k
=
1
.
Eigenvalue inequalities for positive block matrices with the inradius of the numerical range
