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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:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 104-06

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A Note on Commutative Banach Algebras

Takashi Sano

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J. Ineq. Pure and Appl. Math. 7(2) Art. 68, 2006

http://jipam.vu.edu.au

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.

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Before giving a proof, we recall the definition ofe^AforA∈ A:

e^A :=

∞

X

n=0

1

n!Aⁿ ∈ A.

The assumption that A is a complete, unital normed algebra with a submultiplicative norm guarantees the convergence of this infinite series in A and im- plies

d

dze^zA=Ae^zA (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(e^zABe^−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)|5kϕkke^zABe^−zAk 5κkϕkkBe^−zA·e^zAk

=κkϕkkBk<∞ (z ∈C).

Thus, by the Liouville theorem,f is constant. Hence,

0 = f⁰(z) = ϕ (Ae^zA)Be^−zA+e^zAB(−Ae^−zA) .

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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 ke^zABe^−zAk5κkBk

for all A, B ∈ A and all z ∈ C. In fact, it is essential to the proof of Theorem1that for givenA, B

sup{ke^zABe^−zAk:z ∈C}<∞.

Theorem1and Remark1(2) yield:

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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 M_n(C)) be the set of upper triangular matrices whose diagonal entries are identical; A consists of A := (a_ij) ∈ M_n(C) such that a₁₁=a₂₂=· · ·=a_nn(=:α)anda_ij = 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.

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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).