2 MART ABEL

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

inclusion

(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 :=

c−1

∈ Inv(A)

and 1 |λ| −ρ(x) |λ − μ| for every μ ∈ σ(x). Hence, d({λ},σ(x)) 1.