As usual, we will denote by J(A) the Jacobson radical of A, i.e., the intersection
of all maximal left ideals of A (equivalently, the intersection of all maximal right
ideals of A). An algebra is said to be semisimple, if its Jacobson radical consists
only of the zero-element θA of A.
In [8, Proposition 4.24, p. 61], it is proved that, for any algebra A, one has the
(1.1) {y A with ρ(xy) = 0 or ρ(yx) = 0 ∀x A} J(A).
Let U and V be subsets of C. Then, the Hausdorff distance of sets U and V is
defined as d(U, V ) = inf{|u v|: u U, v V }.
2. Results about elements in the center of an algebra
We will start with generalizing Theorem 2.1 from [3, p. 1096].
Proposition 2.1. Let A be a unital semisimple algebra for which
ρ(zx) ρ(z)ρ(x) for every z Z(A) and every x A. Let a A be such that
σ(a) = {α} for some α C. Then a = αeA if and only if a Z(A).
Proof. Let a A be such that σ(a) = {α} for some α C.
Obviously, from a = αeA follows that a Z(A).
Conversely, suppose that a Z(A). Then z := a−αeA Z(A) and σ(z) = {0}.
Hence, ρ(z) = 0 and
ρ(zx) ρ(z)ρ(x) = 0
for all x A. By the inclusion (1), we obtain that z J(A). Since A is semisimple,
then J(A) = {θA}, hence, a = αeA.
In [1, Theorem 5.2.2], it is shown, for a unital (semisimple) Banach algebra A
and a A, that if there exists M 0 such that ρ(a + x) M(1 + ρ(x)) for all
x A, then a Z(A).
We need something similar but in more general setting. More exactly, in some
of the next results, we need algebras, for which the following condition is satisfied.
Condition 2.2. If ρ(a + x) 1 + ρ(x) for all x A with ρ(x) ∞, then
a Z(A).
We would like to mention here that in [4], the authors consider unital semisim-
ple Banach algebras with the condition ρ(ax) ρ(bx) for all x A. They show
that if a A and b Inv(A) are fixed, then from this condition follows that there
exists u Z(A) such that a = ub and ρ(u) 1 (see [4, Theorem 3.1, p. 147]). We
will use something similar but also different, following the ideas of [3], precisely, we
will use the condition ρ(ba) ρ(b) for a fixed a A and all b Inv(A).
The next results generalize Theorem 2.2 from [3, p. 1096].
Proposition 2.3. Let A be a unital algebra and let a A be such that
ρ(ba) ρ(b) for all b Inv(A). Then ρ(a + x) 1 + ρ(x) for all x A with
ρ(x) ∞. Moreover, if A is also an algebra for which Condition 2.2 holds, then
a Z(A).
Proof. Let x A be such that ρ(x) ∞. Take any λ C such that
1 + ρ(x) |λ|. Then c := x λeA is invertible, i.e., there exists b :=
and 1 |λ| −ρ(x) μ| for every μ σ(x). Hence, d({λ},σ(x)) 1.
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