ON SPECTRUM AND SPECTRAL RADIUS 3

In every algebra we have

σ(x−1)

=

1

μ

: μ ∈ σ(x) .

Since x−1 − λ−1eA = (−λ−1x−1)(x − λeA) = (x − λeA)(−λ−1x−1), then

λ−1 ∈ σ(x−1) iff λ ∈ σ(x).

It is also clear that σ(λeA + x) = λ + σ(x) for every λ ∈ C and x ∈ A. Hence,

ρ(b) =

ρ(c−1)

= sup

1

v

: v ∈ σ(c) =

1

inf |v| : v ∈ σ(c)

.

Since σ(c) = σ(x) − λ, then v = μ − λ, where μ ∈ σ(x). Therefore,

ρ(b) =

1

inf |μ − λ| : μ ∈ σ(x)

=

1

d({λ},σ(x))

.

Notice, that

(a + x) − λeA = c(ba + eA).

Since

ρ(ba) ρ(b) =

1

d({λ},σ(x))

1,

then −1 ∈ σ(ba) and (ba + eA) is invertible. Thus, (a + x) − λeA is invertible for

every λ ∈ C with 1 + ρ(x) |λ|.

Suppose that μ ∈ σ(a + x). Then |μ| must be less or equal to 1 + ρ(x), because

for every λ ∈ C with 1 + ρ(x) |λ| we showed that λ ∈ σ(a + x). Hence, we have

ρ(a + x) 1+ ρ(x) for every x ∈ A with ρ(x) ∞. Now, the Condition 2.2 implies

that a ∈ Z(A).

From Propositions 2.1 and 2.3 we get the next corollary, which generalizes

Corollary 2.3 from [3, p. 1097].

Corollary 2.4. Let A be a unital semisimple algebra for which Condition 2.2

holds and let a ∈ A be such an element that ρ(ba) ρ(b) for all b ∈ Inv(A). If

σ(a) = {α} for some α ∈ C, then a = αeA with |α| 1.

Proof. From Proposition 2.3 we get that a ∈ Z(A). Now, Proposition 2.1

gives us a = αeA. Since ba = αb and ρ(αb) =|α|ρ(b) for every α ∈ C and b ∈ A,

then we get |α|ρ(b) = ρ(ba) ρ(b) for all b ∈ Inv(A). Hence, we get that either

ρ(b) = 0 for all b ∈ Inv(A) or |α| 1. Since eA ∈ Inv(A) and ρ(eA) = 1 = 0, then

|α| 1.

3. Results about the case when two elements of an algebra coincide

Next, we generalize Theorem 2.2 (i) [2, pp. 157–158]. We would like to point

out that a similar result was also obtained in [4] for unital commutative semisimple

Banach algebras (Theorem 2.5, p. 146) or unital

C∗-algebras

(Theorem 2.6, p. 146).

Proposition 3.1. Let A be a unital semisimple algebra and a, b ∈ A such that

ρ(a) ∞ or ρ(b) ∞. Then a = b if and only if σ(ax) = σ(bx) for all x ∈ A with

ρ(x) ∞.

Proof. It is clear that from a = b follows that σ(ax) = σ(bx) for all x ∈ A

with ρ(x) ∞.

Suppose that a, b ∈ A are such that σ(ax) = σ(bx) for all x ∈ A with ρ(x) ∞.

Since ρ(eA) = 1 ∞, then we obtain that σ(a) = σ(b) and ρ(a) = ρ(b), which