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 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
Previous Page Next Page