Quasicompact endomorphisms of commutative semiprime Banach algebras
aa r X i v : . [ m a t h . F A ] D ec ****************************************BANACH CENTER PUBLICATIONS, VOLUME **INSTITUTE OF MATHEMATICSPOLISH ACADEMY OF SCIENCESWARSZAWA 200* QUASICOMPACT ENDOMORPHISMS OFCOMMUTATIVE SEMIPRIME BANACH ALGEBRAS
JOEL F. FEINSTEIN
School of Mathematical Sciences, University of NottinghamUniversity Park, Nottingham NG7 2RD, UKE-mail: [email protected]
HERBERT KAMOWITZ
Department of Mathematics, University of Massachusetts at Boston100 Morrissey Boulevard, Boston, MA 02125-3393, USAE-mail: [email protected]
Abstract.
This paper is a continuation of our study of compact, power compact, Riesz, andquasicompact endomorphisms of commutative Banach algebras. Previously it has been shown thatif B is a unital commutative semisimple Banach algebra with connected character space, and T isa unital endomorphism of B , then T is quasicompact if and only if the operators T n converge inoperator norm to a rank-one unital endomorphism of B .In this note the discussion is extended in two ways: we discuss endomorphisms of commutativeBanach algebras which are semiprime and not necessarily semisimple; we also discuss commutativeBanach algebras with character spaces which are not necessarily connected.In previous papers we have given examples of commutative semisimple Banach algebras B andendomorphisms T of B showing that T may be quasicompact but not Riesz, T may be Riesz butnot power compact, and T may be power compact but not compact. In this note we give examplesof commutative, semiprime Banach algebras, some radical and some semisimple, for which everyquasicompact endomorphism is actually compact.
1. Introduction.
Let A be a commutative, complex Banach algebra. We denote by Φ A the character space of A , and, for a ∈ A , we denote by ˆ a the Gelfand transform of a . As in[2], if A has no identity element then we denote by A the usual Banach algebra obtainedby adjoining an identity to A ; otherwise we define A = A . We denote the open unit discby D and the closed unit disc by D . We denote the standard disc algebra (regarded as aBanach space) by A ( D ). Mathematics Subject Classification : Primary 46J05; Secondary 46J45, 46J10.
Key words and phrases : endomorphism, semiprime, semisimple, commutative Banach algebraThe paper is in final form and no version of it will be published elsewhere. [1]
J. F. FEINSTEIN AND H. KAMOWITZ
In previous papers the authors [5, 6, 7] and others [8, 9, 14] have studied endomorphismsof commutative semisimple Banach algebras and have obtained several general theorems,and also a variety of results pertaining to specific classes of algebras. In this note we extendthis discussion to endomorphisms of commutative Banach algebras which are semiprime andnot necessarily semisimple.We recall that a complex algebra B is semiprime if J = { } is the only ideal in B suchthat the product of every pair of elements in J is 0. It is standard that a commutative,complex algebra B is semiprime if and only if B has no non-zero nilpotent elements (see,for example, [4] or [2, pp.77-78]). Certainly semisimple algebras are semiprime.Examples of commutative semiprime Banach algebras which are not semisimple includecertain Banach algebras of formal power series, as discussed in [10]. In particular, A ( D ) and H p ( D ) for p ∈ [1 , ∞ ) are commutative radical semiprime Banach algebras with respect to convolution multiplication defined by( f ∗ g )( z ) = Z γ z f ( z − w ) g ( w ) d w , where the path γ z is a straight line joining 0 to z. Other examples of commutative radicalsemiprime Banach algebras include ℓ p ( ω ) for p ∈ [1 , ∞ ) and radical weights ω . A linear map T from a commutative Banach algebra A to itself is an endomorphism if T preserves multiplication. If the algebra A is unital, then an endomorphism T of A is saidto be unital if T maps the identity to itself. In this case, φ := T ∗ | Φ A is a selfmap of Φ A ; weshall call φ the selfmap of Φ A associated with T . Note that then, for all a ∈ A , we have c T a = ˆ a ◦ φ . In particular, if A is semisimple, then we may recover the endomorphism from the associatedselfmap φ . If A is not semisimple, then φ may give little information about the endomorphism T . Even in the latter case, however, the existence or otherwise of fixed points of φ is relevantto our study of endomorphisms.For commutative semisimple Banach algebras, endomorphisms are automatically contin-uous. However, in the case of commutative semiprime algebras, this need not be the case, atleast if we assume the continuum hypothesis (CH). Indeed, let ω be a radical weight on R + ,and set A = L ( R + , ω ). Assuming CH, it follows from [2, Theorem 5.7.31] and the commentsfollowing that theorem, that there is then a discontinuous, injective, unital endomorphismof the integral domain A . We shall consider only bounded endomorphisms in this note.Let E be an infinite dimensional Banach space, let L ( E ) be the Banach algebra ofbounded linear operators on E , and let K ( E ) be the set of compact linear operators on E .Then K ( E ) is a closed ideal in L ( E ). The quotient algebra L ( E ) / K ( E ) is called the Calkinalgebra . Now let T be a bounded linear operator on E . The essential spectral radius of T , ρ e ( T ), is the spectral radius of T + K ( E ) in the Calkin algebra.We shall discuss operators T such that ρ e ( T ) < . (This holds if and only if there is anatural number n such that the distance from T n to K ( E ) is strictly less than 1.) FollowingHeuser [11] such an operator T is called quasicompact . If ρ e ( T ) = 0 the operator T is a Riesz A real valued function ω on Z + is a weight if ω ( n ) > n ∈ Z + and, for all m and n in Z + , we have ω ( m + n ) ≤ ω ( m ) ω ( n ). The weight is radical if, in addition, lim n →∞ ω ( n ) /n = 0 . NDOMORPHISMS OF SEMIPRIME BANACH ALGEBRAS operator. Quasicompactness is clearly weaker than Riesz, which in turn is weaker than thecondition that an operator be power compact.In [7], the authors investigated quasicompact endomorphisms of commutative semisimpleBanach algebras. One of the main results of that paper was the following. Proposition . Let B be a unital commutative semisimple Banach algebra with connectedcharacter space, let T be a unital endomorphism of B , and let φ be the associated selfmapof Φ B . Then T is quasicompact if and only if the operators T n converge in operator normto a rank-one unital endomorphism of B ; in this case φ has a unique fixed point x ∈ Φ A ,and the rank-one endomorphism above must be the endomorphism b ˆ b ( x )1 . In Section 2, we indicate that Proposition 1.1 is valid for bounded unital endomorphismsof commutative semiprime Banach algebras whose character space is connected. In Section3, we consider the case where the character space need not be connected. Using a fairlystandard technique involving orthogonal idempotents, we will prove the following result,which is a main result of this note.
Theorem . Let B be a unital commutative semiprime Banach algebra, and let T be abounded unital endomorphism of B . Then T is quasicompact if and only if there is a naturalnumber n such that the operators ( T kn ) ∞ k =1 converge in operator norm to a finite-rank unitalendomorphism of B . This result extends earlier results of the authors [5, 6, 7] for commutative semisimpleBanach algebras, and results for uniform algebras of Klein [14] and Gamelin, Galindo andLindstr¨om [8, 9].Section 4 contains some results about commutative radical semiprime Banach algebras,while Section 5 presents two examples of commutative semisimple Banach algebras whereeach quasicompact endomorphism is compact.
2. Bounded endomorphisms of semiprime Banach algebras with connected char-acter space.
In order to extend the results from [7], we begin by examining the propertiesof semisimplicity which were used in the proof of Lemma 1.1 of [7], and observing that theyare more generally true. Specifically, we note the following. • Let B be a unital commutative Banach algebra. Then Φ B is connected if and only ifthe only idempotent elements in B are 0 and 1. This is an immediate consequence ofthe Shilov Idempotent Theorem [2, Theorem 2.4.33]. • Let B be a commutative unital semiprime Banach algebra, and let T be a unitalendomorphism of B . Then, since B has no non-zero nilpotent elements, the set ofeigenvalues of T is closed under taking powers. • Let A be a finite-dimensional commutative semiprime Banach algebra. Since the radicalof a finite dimensional algebra is nilpotent [2, Theorem 1.5.6(iv)], it follows that A is, in fact, semisimple. Thus A is isomorphic to the finite-dimensional commutativeC*-algebra C m (with coordinate-wise multiplication), where m = dim A .Using these observations it easily follows that the proof of Lemma 1.1 of [7] holds whensemisimple is replaced by semiprime and we have the following lemma. J. F. FEINSTEIN AND H. KAMOWITZ
Lemma . Let B be a unital commutative semiprime Banach algebra with connected char-acter space, and let T be a bounded unital quasicompact endomorphism of B . Then is aneigenvalue of T with multiplicity and eigenspace C · , and σ ( T ) (the spectrum of T ) iscontained in { λ : | λ | < } ∪ { } . Armed with this lemma, the proof of the convergence of the operators T n in Theorem1.2 of [7] is equally valid for semiprime algebras, and we obtain the corresponding result forsemiprime algebras. Theorem . Let B be a unital commutative semiprime Banach algebra with connectedcharacter space, let T be a bounded, unital endomorphism of B , and let φ be the associatedselfmap of Φ B . Then T is quasicompact if and only if the operators T n converge in operatornorm to a rank-one unital endomorphism of B ; in this case φ has a unique fixed point x ∈ Φ A , and the rank-one endomorphism above must be the endomorphism b ˆ b ( x )1 . We immediately obtain the following useful corollary.
Corollary . Let B be a unital commutative semiprime Banach algebra with connectedcharacter space, let T be a bounded unital endomorphism of B , and let φ be the associatedselfmap of Φ B .(i) If φ has no fixed points in Φ B , then T is not quasicompact.(ii) Otherwise, let x ∈ Φ B be a fixed point of φ . Then T is quasicompact if and only ifthe operators T n converge in operator norm to the rank-one unital endomorphism of B defined by b ˆ b ( x )1 . Note that, once we have found a fixed point x of φ , we can apply this corollary withouthaving to check whether this fixed point is unique. However, if we do know that φ has morethan one fixed point, then Theorem 2.2 tells us immediately that T is not quasicompact.Let B be a commutative Banach algebra without identity. Then (by the Shilov Idempo-tent Theorem again) B has no non-zero idempotent elements if and only if Φ B is connected.One trivial special case of this is, of course, when B is radical. Note that it is possible forΦ B to be connected when Φ B is disconnected, and vice-versa. Corollary . Let B be a commutative semiprime Banach algebra which has no non-zeroidempotent elements, and let T be a bounded endomorphism of B . Then T is quasicompactif and only if T n → in operator norm.Proof. Clearly, if T n → T is quasicompact.Conversely, suppose that T is quasicompact. Obviously B has no identity element. Wemay extend T to a bounded unital endomorphism T of the commutative unital semiprimeBanach algebra B , and then T is also quasicompact. By Theorem 2.2, the powers of T converge in operator norm to a rank-one unital endomorphism of B . It follows that T n → NDOMORPHISMS OF SEMIPRIME BANACH ALGEBRAS Corollary . Let R be a commutative radical semiprime Banach algebra, and let T be abounded endomorphism of R . Then T is quasicompact if and only if T n → in the operatornorm.
3. Extension to more general semiprime Banach algebras.
We now wish to gener-alize these results to the setting where the algebra is semiprime and the character spaceneed not be connected. A further examination of the proof of Theorem 1.2 of [7] revealsimmediately that the following more general result holds.
Lemma . Let B be a unital commutative semiprime Banach algebra, and let T be abounded unital quasicompact endomorphism of B . Suppose that σ ( T ) ⊆ { λ ∈ C : | λ | < } ∪ { } and that the eigenvalue of T has multiplicity . Then the operators T n converge in operatornorm to a rank-one unital endomorphism S of B . The method we use to obtain results when the character space is disconnected is basedon a standard technique involving orthogonal idempotents.
Theorem . Let B be a unital commutative semiprime Banach algebra, and let T be abounded unital quasicompact endomorphism of B . Then there exists an n ∈ N such that σ ( T n ) ⊆ { λ ∈ C : | λ | < } ∪ { } . (1) For such n , the unital quasicompact endomorphism T n of B has the following properties.(i) The eigenspace of T n corresponding to eigenvalue is a finite-dimensional, unitalsubalgebra of B isomorphic to C m for some m ∈ N , and hence spanned by m orthogonalidempotents, say e , e , . . . , e m .(ii) Set B i = e i B (1 ≤ i ≤ m ) . Then (under an equivalent norm) each B i is a commutative,unital semiprime Banach algebra, with identity e i , and B = m M i =1 B i . (iii) For ≤ i ≤ m , T n | B i is a unital quasicompact endomorphism of B i , and T n | B i satisfies the conditions of Lemma 3.1. The operators ( T kn | B i ) ∞ k =1 converge in operatornorm to a rank- unital endomorphism of B i , say S i .(iv) The operators ( T kn ) ∞ k =1 converge in operator norm to the rank- m endomorphism S of B given by S ( b ) = m X i =1 S i ( be i ) ( b ∈ B ) . Proof.
As in the proof of Lemma 1.1 of [7], the existence of an n satisfying (1) is an easyconsequence of the following pair of facts: the set of eigenvalues of T is closed under takingpowers and the spectrum of T has no limit point on the unit circle.Now suppose that we have fixed such an n satisfying (1). Then (i) follows immediatelyfrom the fact that ker( I − T n ) is a finite-dimensional, commutative semiprime algebra. Now(ii) is a standard construction. For (iii), it is clear that T n | B i is a unital endomorphismof B i , and the multiplicity of the eigenvalue 1 of this endomorphism is 1 by construction. J. F. FEINSTEIN AND H. KAMOWITZ
The quasicompactness of T n | B i is standard. Then, since every eigenvalue of T n | B i is alsoin σ ( T n ), it follows that σ ( T n | B i ) ⊆ { λ ∈ C : | λ | < } ∪ { } . Thus T n | B i satisfies theconditions of Lemma 3.1. The rest of (iii) now follows by applying Lemma 3.1 to T n | B i .Finally, (iv) follows immediately from (i), (ii) and (iii).Theorem 1.2 is now an immediate corollary, since one implication is part of the resultabove, while the converse is trivial.
4. Radical Banach algebras of power series.
In this section we look briefly at radicalBanach algebras of power series.We recall the following terminology and notation from [10]. The algebra of complexformal power series in one variable is denoted by C [[ z ]]. The coordinate projections on C [[ z ]]are ( π n ) ∞ n =0 . Let B be a subalgebra of C [[ z ]] with z ∈ B and such that B ⊆ ker π (i.e.,all elements of B have constant coefficient 0). Then B is a generalized Banach algebra ofpower series if it is a Banach algebra under some norm for which all of the functionals π n | B are continuous. In this case, for each n ∈ N , we denote by k π n k the operator norm of thecontinuous linear functional π n | B . If B is a generalized Banach algebra of power series suchthat the polynomials are dense in B , then B is a Banach algebra of power series . Since C [[ z ]]is an integral domain, these algebras of power series are certainly semiprime.The reader should note that there are variations in the terminology and notation usedin the literature. In [2, Section 4.6], for example, the algebras are allowed to be unital, andgeneralized Banach algebras of power series are called simply Banach algebras of power series(with no requirement that the polynomials be dense).Let B be a generalized Banach algebra of power series. For each non-negative integer j , S − j ( B ) is the set of those formal power series f with zero constant term for which f z j belongs to B . In fact S − j ( B ) is a Banach space when we define the norm of f ∈ S − j ( B ) tobe the norm of f z j in B .Let B be a Banach algebra of power series. Then every non-zero endomorphism of B hasthe form f → f ◦ g (formal composition of power series) for some g ∈ B [10, p.7]. For those g ∈ B which give rise to an endomorphism of B in this way, we denote the correspondingendomorphism by T g . In this case, we have g = T g z. A result of Loy [2, Theorem 5.2.20] shows that endomorphisms of Banach algebras ofpower series are automatically continuous. (See also [3] for some striking recent developmentsconcerning Fr´echet algebras of power series.)It was previously shown that for a wide class of radical Banach algebras of power series,every endomorphism is either an automorphism or compact [10, Theorem (2.6)]. In such casesevery quasicompact endomorphism is (trivially) compact. In particular, for many radicalweights ω , the Banach algebras ℓ p ( ω ) are examples of radical semiprime commutative Banachalgebras for which every quasicompact endomorphism is compact. Lemma . Let B be a radical Banach algebra of power series, and let T be a quasicompactendomorphism of B . Set g = T z (so that T = T g ). Then | π ( g ) | < . In fact, surprising recent results from [3] show that the continuity of the functionals π n | B inthis setting is automatic, while the corresponding statement for formal power series in two variablesis false. NDOMORPHISMS OF SEMIPRIME BANACH ALGEBRAS Proof.
Since the endomorphism T = T g is quasicompact, by Corollary 2.5, T ng → T ng z → . But π ( T ng z ) = π ( g ) n , and so we must have | π ( g ) | < . The following proposition is [10, Theorem 5.7].
Proposition . Suppose that B and S − ( B ) are both radical generalized Banach algebrasof power series, and that R := lim sup( k π n kk z n k ) /n is finite. Let g ∈ B with | Rπ ( g ) | < .Then T g is a compact endomorphism of B . Combining the previous two results, we have the following.
Corollary . Suppose that S − ( B ) and B are radical Banach algebras of power se-ries, and that lim sup( k π n kk z n k ) /n = 1 . Then every quasicompact endomorphism of B iscompact. Let A be A ( D ) or H p ( D ) for some p ∈ [1 , ∞ ). Using the definition of convolution multi-plication on A from Section 1, it was shown in [10, Section 13], that ( A, ∗ ) is a commutativeradical semiprime Banach algebra which can be identified with a radical Banach algebraof power series B satisfying the hypotheses of Corollary 4.3 (see, in particular, [10, Theo-rem (13.10)]). Thus, for B , and hence also for ( A, ∗ ), every quasicompact endomorphism iscompact.
5. Two semisimple examples.
We have just seen several examples of commutative radi-cal semiprime Banach algebras where every quasicompact endomorphism is compact. In thissection we give two examples of commutative semisimple Banach algebras where this holds.This is in contrast to the commutative semisimple Banach algebra C [0 ,
1] where there exista quasicompact endomorphism which is not Riesz, a Riesz endomorphism which is not powercompact and a power compact endomorphism which is not compact.
Example . A theorem of Beurling and Helson [13, Theorem 4.5 and exercise 4.12] tellsus that every non-zero endomorphism of the group algebra L ( R ) is an automorphism.Thus, for this commutative semisimple Banach algebra, there are no non-zero quasicompactendomorphisms at all.For the next example, the proof is based on our results concerning the powers of quasi-compact endomorphisms. Example . Let A be the Banach algebra Ea [ − ,
1] described in [1] and [12], and definedas follows. Let M ( C ) denote the set of finite regular Borel measures on C and M ω ( C ) theset of measures µ ∈ M ( C ) for which R C e | Reλ | d | µ | ( λ ) < ∞ . For each µ ∈ M ω ( C ), we maydefine a continuous function f µ : [ − , → C by f µ ( x ) = R C e xλ d µ ( λ ) ( x ∈ [ − , Ea [ − ,
1] = { f µ : µ ∈ M ω ( C ) } . With norm defined by k f k A = inf (cid:26)Z C e | Reλ | d | µ | ( λ ) : µ ∈ M ω ( C ) with f µ = f (cid:27) ,A = Ea [ − ,
1] is a regular commutative semisimple Banach algebra [1, 15]. Further Φ A is[ − , extremal algebra for [ − , J. F. FEINSTEIN AND H. KAMOWITZ it possesses relative to the study of numerical ranges of elements in complex unital Banachalgebras. The Banach algebra A is generated by the Hermitian element u , where u ( x ) = x for x ∈ [ − , k e itu k A = 1 for all real t [1, 15].Let T be an endomorphism of A , and let φ be the associated selfmap of [ − , φ must have the form x αx + β , where α and β are real numbers with | α | + | β | ≤
1. If β = 0 and | α | = 1, then T is an automorphism, while if α = 0, then theendomorphism T has rank one, and so T is compact. Also it was shown in [12] that therank-one endomorphisms are the only nonzero compact endomorphisms of A . We claim thatevery quasicompact endomorphism of A is compact.To see this, let T be a quasicompact endomorphism of A , with associated selfmap φ . Forsome α and β as above, we have φ ( x ) = αx + β . Since T is not an automorphism, we have α = 1. Set x = β − α , so that x is the fixed point of φ . Let S be the rank-one endomorphism f f ( x )1. Since T is quasicompact, Corollary 2.3 implies that k T n − S k → . We shallshow that α = 0, and hence that x = β and T = S .Suppose, towards a contradiction, that α = 0. For each n ∈ N , define f n ∈ A by f n ( x ) = (cid:18) i ( α − n ( x − x )))2 (cid:19) n . As mentioned above, for each real number t , the function x e itx has norm 1 in A . Thuswe have 1 = k f n k ∞ ≤ k f n k A ≤ , and so k f n k A = 1 . It is routine to show that for each positive integer n , T n f ( x ) = f ( α n ( x − x ) + x ) andso T n f n ( x ) = (cid:18) e i ( x − x ) (cid:19) n . We also note that Sf n ( x ) = f n ( x ) = 1. Now k f n k A k T n − S k ≥ k ( T n − S ) f n k A ≥ | ( T n − S ) f n ( x ) | for all x in [ − , x = x . Then k T n − S k = k f n k A k T n − S k ≥ k ( T n − S ) f n k ≥ | ( T n − S ) f n ( x ) | . This implies that k T n − S k ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) e i ( x − x ) (cid:19) n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ / n . Therefore k T n − S k does not converge to 0, and so T is not quasicompactaccording to Corollary 2.3 (or Proposition 1.1). This contradiction shows that α = 0, andso T = S .Therefore for this algebra, every quasicompact endomorphism is compact. Acknowledgements
We would like to thank the referee for useful comments. We wouldalso like to thank Garth Dales and Sandy Grabiner for some very helpful discussions. Ourdiscussions with Sandy Grabiner took place at the 19 th International Conference on BanachAlgebras held at B¸edlewo, July 14–24, 2009. The support for the meeting by the Polish
NDOMORPHISMS OF SEMIPRIME BANACH ALGEBRAS Academy of Sciences, the European Science Foundation under the ESF-EMS-ERCOM part-nership, and the Faculty of Mathematics and Computer Science of the Adam MickiewiczUniversity at Pozna´n is gratefully acknowledged. In addition to the support for both au-thors from the Conference’s funds, the first author’s attendance at this meeting was furthersupported by a grant from the Royal Society.
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