volume 7, issue 2, article 68, 2006.
Received 09 March, 2006;
accepted 06 April, 2006.
Communicated by:C.-K. Li
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Journal of Inequalities in Pure and Applied Mathematics
A NOTE ON COMMUTATIVE BANACH ALGEBRAS
TAKASHI SANO
Department of Mathematical Sciences Faculty of Science
Yamagata University Yamagata 990-8560, Japan.
EMail:sano@sci.kj.yamagata-u.ac.jp
c
2000Victoria University ISSN (electronic): 1443-5756 104-06
A Note on Commutative Banach Algebras
Takashi Sano
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Abstract
LetAbe a unital Banach algebra overCwith normk · k. In this note, several characterizations of commutativity ofAare given. For instance, it is shown that Ais commutative if
kABk=kBAk
for allA, B∈ A, or if the spectral radius onAis a norm.
2000 Mathematics Subject Classification:46J99, 47A30.
Key words: Commutative Banach algebra; Norm; Similarity transformation; Spectral radius.
The author is grateful to the referee for careful reading of the manuscript and for helpful comments.
LetAbe a unital Banach algebra overCwith normk · k. In this note, several characterizations of the commutativity ofAare studied.
The following theorem is a simple characterization of commutativity in terms of norm inequalities, whose proof depends on complex analysis as the well- known one for the Fuglede-Putnum theorem, for instance, see [2, p. 278].
Theorem 1. LetAbe a unital Banach algebra overCwith normk · k0.If there is a normk · konAand positive constantsγ, κsuch that
kAk5γkAk0, kABk5κkBAk
for allA, B ∈ A,thenAis commutative, that is,AB =BAfor allA, B ∈ A.
A Note on Commutative Banach Algebras
Takashi Sano
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Before giving a proof, we recall the definition ofeAforA∈ A:
eA :=
∞
X
n=0
1
n!An ∈ A.
The assumption that A is a complete, unital normed algebra with a submulti- plicative norm guarantees the convergence of this infinite series in A and im- plies
d
dzezA=AezA (z∈C).
Proof. Let A, B ∈ A. Let us consider the normed space (A,k · k). For each bounded linear functionalϕon this normed space, we define a complex-valued functionf onCby
f(z) :=ϕ(ezABe−zA) (z ∈C).
Then the first assumption ofk · kguarantees thatf is an entire analytic function.
f is also bounded: in fact, by the second assumption
|f(z)|5kϕkkezABe−zAk 5κkϕkkBe−zA·ezAk
=κkϕkkBk<∞ (z ∈C).
Thus, by the Liouville theorem,f is constant. Hence,
0 = f0(z) = ϕ (AezA)Be−zA+ezAB(−Ae−zA) .
A Note on Commutative Banach Algebras
Takashi Sano
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Puttingz = 0yields
ϕ(AB−BA) = 0
for each bounded linear functional ϕ on A. By the Hahn-Banach theorem, AB =BAand the proof is completed.
Remark 1.
1. By considering completion, we find it sufficient to assume in Theorem 1 that A is a unital normed algebra over C with submultiplicative norm k · k0.
2. The assumption that
kABk5κkBAk for allA, B ∈ Acan be replaced with a weaker one
kSAS−1k5κkAk
for allA∈ Aand all invertibleS ∈ A, or even further kezABe−zAk5κkBk
for all A, B ∈ A and all z ∈ C. In fact, it is essential to the proof of Theorem1that for givenA, B
sup{kezABe−zAk:z ∈C}<∞.
Theorem1and Remark1(2) yield:
A Note on Commutative Banach Algebras
Takashi Sano
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Corollary 2. LetAbe a unital Banach algebra overCwith normk · k.Suppose that there is a positive constantγsuch that
kABk5γkBAk
for allA, B ∈ A.ThenAis commutative. In particular, ifkABk =kBAkfor allA, B ∈ A,thenAis commutative.
Corollary 3 ([1, Exercise IV 4.1]). On the set of all complexn-square matrices forn=2no norm is invariant under all similarity transformations.
See [1, p.102] for similarity transformations.
Corollary 4. Let Abe a unital Banach algebra over Cwith normk · k.If the spectral radius is a norm onA,thenAis commutative.
This follows from Theorem1and the properties of the spectral radiusr(A) thatr(AB) = r(BA)andr(A)5kAkforA, B ∈ A.
Remark 2. There is a unital Banach algebra whose spectral radius is not a norm but a semi-norm. This semi-norm condition is not sufficient for commuta- tivity.
In fact, let A (j Mn(C)) be the set of upper triangular matrices whose diagonal entries are identical; A consists of A := (aij) ∈ Mn(C) such that a11=a22=· · ·=ann(=:α)andaij = 0 (i > j). For thisA,r(A) =|α|and the spectral radius on A is a semi-norm. Therefore, the unital Banach algebra Ais a non-commutative example.
A Note on Commutative Banach Algebras
Takashi Sano
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References
[1] R. BHATIA, Matrix Analysis, Springer-Verlag (1996).
[2] J.B. CONWAY, A Course in Functional Analysis, 2nd Ed., Springer-Verlag, (1990).