E extracta mathematicae Vol. 31, Núm. 2, 119 – 121 (2016)

Spectral Rank of Maximal Finite-Rank Elements in
Banach Jordan Algebras

Abdelaziz Maouche

Department of Mathematics and Statistics Faculty of Science,
Sultan Qaboos University, Oman, maouche@squ.edu.om

Presented by Jesús M.F. Castillo Received November 15, 2015

Abstract: We give a new proof to a spectral characterisation of the spectral rank established
by Aupetit by replacing his deep analytic arguments by the new characterisation of the
connected component of the group of invertible elements obtained by O. Loos.
Key words: Banach Jordan algebra, spectral rank, maximal finite-rank element.
AMS Subject Class. (2010): 17A15, 46H70.

1. Preliminaries

Let A be a semisimple complex unital Banach Jordan algebra and Ω(A) its
set of invertible elements. For x ∈ A we denote Sp(x) = {λ : λ1 − x /∈ Ω(A)}
and ρA(x) = sup{|λ| : λ ∈ Sp(x)} the spectrum and spectral radius of x.

For each nonnegative integer m, let

Fm = {a ∈ A : ♯(Sp Ux a \ {0}) ≤ m for all x ∈ A},

where the symbol ♯K denotes the number of distinct elements in a set K ∈ C.
Following [1], we define the rank of an element a of A as the smallest integer
m such that a ∈ Fm, if it exists; otherwise the rank is infinite. In other words,

rank(a) = {sup ♯(Sp Ux a \ {0}), x ∈ A}.

If a ∈ A is a finite-rank element, then

E (a) = {x ∈ A : ♯(Sp Ux a \ {0}) = rank(a)}

is a dense open subset of A [2, Theorem 2.1].
It is shown in [1] that the socle, denoted Soc A, of a semisimple Banach

Jordan algebra A coincides with the collection ∪∞m=0Fm of finite-rank ele-
ments.

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120 a. maouche

We first recall a very important theorem obtained by O. Loos [2], saying
that the connected component of Ω(A) is arcwise connected as in the case of
Banach algebras.

Theorem 1. (O. Loos [2]) Let A be a real or complex Banach Jordan
algebra with unit element 1. Then

Ω1 = {U(exp x1) · · · U(exp xn)(1) : xi ∈ A, n ≥ 1}

is the connected component of 1 of the set Ω of invertible elements of A.

With the help of this Theorem 1 we are able now to eliminate the deep
and difficult analytic arguments used by Aupetit to prove the next theorem.

2. The rank in Banach Jordan algebras

Theorem 2. ([1], Theorem 3.1) Let A be a Banach Jordan algebra with
identity. Suppose that a ∈ A and that m ≥ 0 is an integer. The following
properties are equivalent:

(1) ♯(Sp(Ux a) \ {0}) ≤ m for every x ∈ A,
(2) {t ∈ C : 0 ∈ Sp(y + ta)) ≤ m} for every y invertible in A,
(3)

∩
t∈F Sp(y + ta) ⊂ Sp y for every y ∈ A and every subset F of C having

m + 1 non-zero elements.

Proof. (1) ⇒ (2) First suppose that 0 /∈ σ(y). By the Holomorphic Func-
tional Calculus Theorem applied to y and a branch of

√
z, there exists an

invertible x such that y = x2. Since

y + ta = x2 + ta = Ux(1 + tUx−1 a)

we get y + ta is non-invertible if and only if − 1
t

∈ Sp(Ux−1 a). By Hypothesis
(1) this set Sp(U−1x a) contains at most m non-zero points. Thus (2) is proved
in this situation. Now {y ∈ A : 0 /∈ σ(y)} is an open subset of Ω, by upper
semicontinuity of the spectrum. Let y ∈ Ω1, then by O. Loos’s Theorem 1:

y = Uexp(x1) · · · Uexp(xn)1

so
y + ta = Uexp(x1) · · · Uexp(xn)

[
1 + t · Uexp(−xn) · · · Uexp(−x1)a

]
.



spectral rank 121

Then
0 ∈ Sp(y + ta) ⇐⇒ −

1
t

∈ Sp(Uexp(−xn) · · · Uexp(−x1)a).

The set Sp(Uexp(−xn) · · · Uexp(−x1)a) contains at most m points by (1) because
exp(−xi) is invertible in A.

(2) ⇒ (3) If λ ∈ ε(y) then λ − y ∈ Ω1, so ♯{t : λ ∈ Sp(y + ta)} ≤ m, hence
λ /∈ ∩t∈F Sp(y + ta).

(3) ⇒ (1) Same as in [1].

References

[1] A. Bernard, Spectral characterization of the socle in Jordan-Banach algebras,
Math. Proc. Cambridge Philos. Soc. 117 (3) (1995), 479 – 489.

[2] O. Loos, On the set of invertible elements in Banach Jordan algebras, Results
Math. 29 (1) (1996), 111 – 114.