Identities, Approximate Identities and Topological Divisors of Zero in Banach Algebras
aa r X i v : . [ m a t h . F A ] A ug IDENTITIES, APPROXIMATE IDENTITIES ANDTOPOLOGICAL DIVISORS OF ZERO IN BANACH ALGEBRAS
F. SCHULZ, R. BRITS, AND M. HASSE
Abstract.
In [3] S. J. Bhatt and H. V. Dedania exposed certain classes ofBanach algebras in which every element is a topological divisor of zero. Weidentify a new (large) class of Banach algebras with the aforementioned prop-erty, namely, the class of non-unital Banach algebras which admits either anapproximate identity or approximate units. This also leads to improvementsof results by R. J. Loy and J. Wichmann, respectively. If we observe thatevery single example that appears in [3] belongs to the class identified in thecurrent paper, and, moreover, that many of them are classical examples ofBanach algebras with this property, then it is tempting to conjecture that theclasses exposed in [3] must be contained in the class that we have identifiedhere. However, we show somewhat elusive counterexamples. Furthermore, weinvestigate the role completeness plays in the results and show, by giving asuitable example, that the assumptions are not superfluous. The ideas con-sidered here also yields a pleasing characterization: The socle of a semisimpleBanach algebra is infinite-dimensional if and only if every socle-element is atopological divisor of zero in the socle. Identities, Approximate Identities and TDZ
An element y in a normed algebra ( A, k · k ) is called a topological divisor of zero (or TDZ ) if there exists a sequence ( x n ) ⊆ A such that k x n k = 1 for all n ∈ N andeither yx n → x n y →
0. Furthermore, if yx n → y is called a left TDZ ,and similarly, if x n y → y is a right TDZ . If y is a left and right TDZ (wherethe sequences need not coincide), then y is a two-sided TDZ . The collection of alltopological divisors of zero in a normed algebra A will be denoted by Z ( A ).In [3] S. J. Bhatt and H. V. Dedania established the following result concerningcomplex Banach algebras in which every element is a TDZ: Theorem 1.1. [3, Theorem 1]
Every element of a complex Banach algebra A is aTDZ, if at least one of the following holds: (i) A is infinite dimensional and admits an orthogonal basis. (ii) A is a non-unital uniform Banach algebra in which the Shilov boundary ∂A coincides with the carrier space M ( A ) . (iii) A is a non-unital hermitian Banach ∗ -algebra with continuous involution. We now expose another class of Banach algebras in which every element is a TDZ.Firstly, however, we need to recall the following definitions:
Mathematics Subject Classification.
Key words and phrases. approximate identity, socle, topological divisor of zero.
A net ( e λ ) λ ∈ Λ in a normed space A is called a left approximate identity in A iflim e λ x = x for all x ∈ A . Similarly ( e λ ) λ ∈ Λ is a right or two-sided approximateidentity in A if lim xe λ = x for all x ∈ A or lim e λ x = x and lim xe λ = x for all x ∈ A , respectively. An approximate identity ( e λ ) λ ∈ Λ is said to be bounded if thereexists a positive constant K such that k e λ k ≤ K for each λ ∈ Λ. Theorem 1.2.
Let A be a Banach algebra without a unit (resp. left unit, resp.right unit). Assume that A has a two-sided (resp. left, resp. right) approximateidentity ( e λ ) λ ∈ Λ (which is not necessarily bounded). Then every element of A is atwo-sided (resp. right, resp. left) TDZ.Proof. We shall restrict ourselves to proving the two-sided case, as the other casesare obtained similarly. Firstly note that ( e λ ) λ ∈ Λ does not converge, for then A will have a unit which contradicts our hypothesis on A . Consequently, there existsa fixed ǫ > λ ∈ Λ, there exist λ , λ ∈ Λ such that λ ≥ λ , λ ≥ λ and k e λ − e λ k ≥ ǫ . If this was not thecase then we obtain the contradiction e λ → e λ for some λ ∈ Λ. Let x ∈ A bearbitrary and denote by B ( x, δ ) the open ball centered at x with radius δ > xe λ ) λ ∈ Λ converges to x , for each n ∈ N , we can find a λ n ∈ Λ such that xe λ ∈ B ( x, /n ) whenever λ ≥ λ n . Moreover, by the preceding paragraph, we canfind α n , β n ∈ Λ such that α n ≥ λ n , β n ≥ λ n and k e α n − e β n k ≥ ǫ . For each n ∈ N ,let y n := e α n − e β n . Then xy n → n → ∞ and ( y n ) does not converge to 0.This is sufficient to conclude that x is a left TDZ. Similarly, it can be shown that x is a right TDZ. This completes the proof. (cid:3) Corollary 1.3.
Let A be a Banach algebra without a unit (resp. left unit, resp.right unit). Assume that A has a two-sided (resp. left, resp. right) approxi-mate identity ( e λ ) λ ∈ Λ (which is not necessarily bounded). Then there exists a net ( y µ ) µ ∈ Λ in A such that: (i) y µ x → and xy µ → for every x ∈ A (resp. y µ x → , resp. xy µ → ). (ii) y µ .Proof. Suppose that the non-unital Banach algebra A has a two-sided approximateidentity ( e λ ) λ ∈ Λ (the other cases follow similarly). Recall from the proof of Theorem1.2 that there exists an ǫ > λ ∈ Λ, there exist λ , λ ∈ Λ suchthat λ ≥ λ , λ ≥ λ and k e λ − e λ k ≥ ǫ . We now defineΛ := { ( α, γ ) ∈ Λ × Λ : k e α − e γ k ≥ ǫ } and for ( α , γ ) , ( α , γ ) ∈ Λ , ( α , γ ) ≤ ( α , γ ) if and only if α ≤ α and γ ≤ γ in Λ. We claim that (Λ , ≤ ) is a directed set: Reflexivity and transitivity followsreadily from the fact that Λ is directed. So let ( α , γ ) , ( α , γ ) ∈ Λ . Since Λ isdirected, there exists a λ such that λ ≥ α j and λ ≥ γ j for j = 1 ,
2. Now, byhypothesis, there exists β , β ∈ Λ such that β ≥ λ , β ≥ λ and k e β − e β k ≥ ǫ .Hence, ( β , β ) ∈ Λ and ( α j , γ j ) ≤ ( β , β ) for j = 1 ,
2. This proves our claim. Foreach µ := ( α, γ ) ∈ Λ , let y µ := e α − e γ . We now show that ( y µ ) µ ∈ Λ is the desirednet: Certainly (ii) is satisfied since k y µ k ≥ ǫ for all µ ∈ Λ . It therefore remains toverify (i). Let x ∈ A be arbitrary, and recall that ( e λ x ) λ ∈ Λ converges to x . Fix any δ > B (0 , δ ). Since e λ x → x , there exists a λ ∈ Λ suchthat e λ x ∈ B ( x, δ/
2) whenever λ ≥ λ . Moreover, there exists λ , λ ∈ Λ such that λ ≥ λ , λ ≥ λ and k e λ − e λ k ≥ ǫ . Thus, ( λ , λ ) ∈ Λ . Now, if ( α, γ ) ∈ Λ and ( α, γ ) ≥ ( λ , λ ), then α ≥ λ ≥ λ and γ ≥ λ ≥ λ . Consequently, k ( e α − e γ ) x k = k e α x − x + x − e γ x k≤ k e α x − x k + k e γ x − x k < δ δ δ. This shows that ( α, γ ) ≥ ( λ , λ ) implies ( e α − e γ ) x ∈ B (0 , δ ). Hence, we concludethat the net ( y µ x ) µ ∈ Λ converges to 0. Similarly, one can prove that ( xy µ ) µ ∈ Λ converges to 0 since xe λ → x . The result now follows. (cid:3) Remark.
If a net in A satisfies both properties (i) and (ii) in Corollary 1.3, then A is said to consist entirely of joint topological divisors of zero (see [15, p. 88]).Completeness plays a central role in the proof of Theorem 1.2. The next exampleemphasizes this. It exhibits a non-unital normed algebra which has a boundedtwo-sided approximate identity, but which has at least one element which is not aTDZ: Example 1.4.
Let B ( H ) be the Banach algebra of bounded linear operators froma infinite dimensional complex Hilbert space H into itself. Denote by A the C ∗ -subalgebra of B ( H ) generated by the identity operator I H and K ( H ), the ideal ofcompact operators on H . Fix any non-algebraic operator T ∈ K ( H ) (for instance acompact operator with an infinite spectrum). We now define B to be the subalgebraof A generated by the element T + I H . Finally, if we denote by F ( H ) the ideal offinite rank operators in B ( H ), then by [4, Proposition 3] C := F ( H ) + B is a densenon-unital subalgebra of the unital Banach algebra A . So there exits a non-unitalnormed algebra whose completion is unital. Let ( E n ) be the sequence in C suchthat E n → I H as n → ∞ . Then ( E n ) is a bounded two-sided approximate identityin C . However, if every element in C is a TDZ, then by the denseness of C everyelement of A is a TDZ. But this is absurd since I H ∈ A . Hence, C contains at leastone element which is not a TDZ.The closure of a set Y in a topological space X is denoted by cl( Y ). If ( A, k · k )is a normed space then we shall also write cl( A ) for the completion of A under thenorm k · k . It is possible to deduce the following from Theorem 1.2: Corollary 1.5.
Let A be a normed algebra and assume that cl( A ) does not havea unit (resp. left unit, resp. right unit). If A has a two-sided (resp. left, resp.right) approximate identity ( e λ ) λ ∈ Λ (which is not necessarily bounded), then everyelement of A is a two-sided (resp. right, resp. left) TDZ (in A ).Proof. If ( e λ ) λ ∈ Λ converges in cl( A ), then we obtain a contradiction with the hy-pothesis on cl( A ). The argument in the proof of Theorem 1.2 now readily establishesthe result. (cid:3) A similar result holds true for algebras with approximate units. A normed algebra A is said to have left approximate units if for every x ∈ A and every ǫ > u ∈ A (depending on x and ǫ ) such that k x − ux k < ǫ . Similarly, in theobvious natural way, we can define the notions of right and two-sided approximateunits in A . Furthermore we note that A is said to have, for instance, bounded left F. SCHULZ, R. BRITS, AND M. HASSE approximate units if there exists a positive constant K such that for every x ∈ A and every ǫ > u ∈ A (depending on x and ǫ ) such that k u k ≤ K and k x − ux k < ǫ. Finally we mention that A has, for instance, pointwise-bounded left approximateunits if for every x ∈ A and every ǫ > u ∈ A (depending on x and ǫ ) and a positive constant K ( x ) such that k u k ≤ K ( x ) and k x − ux k < ǫ. Theorem 1.6.
Let A be a normed algebra such that cl( A ) is not unital. If A hasleft (or right or two-sided) approximate units then every element of cl( A ) is a TDZ.In particular, every element of a non-unital Banach algebra with left approximateunits is a TDZ.Proof. Suppose first a ∈ A is not a TDZ in cl( A ). Since A has left approximateunits there exists a sequence ( u n ) in A such that lim u n a = a . If ( u n ) is not Cauchythen we can find ǫ >
0, and two subsequences of ( u n ), say ( u n k ) and ( u m k ), suchthat k u n k − u m k k ≥ ǫ for all k ∈ N . But thenlim k ( u n k − u m k ) a k u n k − u m k k = 0shows that a is a TDZ which contradicts the hypothesis. Thus ( u n ) must be Cauchy,with limit say u ∈ cl( A ). From this it follows that ( u − u ) a = 0 and hence, if u isnot an idempotent, a is a divisor of zero in cl( A ) and thus a TDZ which contradictsthe assumption on a . So u is an idempotent satisfying ua = a . Necessarily u commutes with a because otherwise, if au − a = 0, we get that ( au − a ) a = 0 sothat a is a divisor of zero in cl( A ) which again contradicts the assumption on a .Thus ua = au = a . But, by assumption, there must exist some x ∈ cl( A ) such thateither ux − x = 0 or xu − x = 0. If the first instance occurs we have a ( ux − x ) = 0;and if the second case holds ( xu − x ) a = 0. Again this gives a contradiction. Socl( A ) contains a dense set of topological divisors of zero, and it follows that eachelement of cl( A ) is a TDZ. (cid:3) Remark.
Example 1.4 also shows that the assumption that cl( A ) is not unital inTheorem 1.6 is not superfluous.In [10, Proposition 3] R. J. Loy proves that if a Banach algebra A does not consistentirely of right (left) topological divisors of zero and has a left (right) approximateidentity, then it has a bounded left (right) approximate identity. In light of Theorem1.2 much more is true: Corollary 1.7.
Let A be a Banach algebra which does not consist entirely of right(resp. left) topological divisors of zero. If A has a left (resp. right) approximateidentity, then A has a left (resp. right) unit. In particular, if moreover A iscommutative, then A is unital. Loy remarks further, in his paper, that the converse of [10, Proposition 3] is nottrue. In particular he gives an example of a Banach algebra and then observes thefollowing: “..., so that all the elements are topological divisors of zero, but has a(countable) bounded approximate identity”. We now know that every element is aTDZ (in his example) because it has an approximate identity. In a similar vein J.
DENTITIES, APPROXIMATE IDENTITIES AND TDZ 5
Wichmann states and proves in [14, Theorem 2] that a commutative normed algebra A which does not consist entirely of topological divisors of zero has pointwise-bounded approximate units if and only if A has a bounded approximate identity.So suppose A has at least one element, say a , which is not a TDZ in A . Then a is not a TDZ in the completion of A either. Now if A has a bounded approximateidentity ( e λ ) in A , then it is easy to see that ( e λ ) is also a bounded approximateidentity for cl( A ). So again, by Theorem 1.2, every element of cl( A ) must be aTDZ. But this contradicts the fact that a is not a TDZ in cl( A ). Hence, under theconditions above, it must in fact be the case that cl( A ) is unital. This then is animprovement of Wichmann’s result: Corollary 1.8.
Let A be a commutative normed algebra which does not consistentirely of topological divisors of zero. If A has approximate units or an approximateidentity, then cl( A ) is unital. It is easy to see that the class of Banach algebras identified in Theorem 1.2 containsthose Banach algebras mentioned in (i) of Theorem 1.1. Indeed, let ( e n ) be theorthogonal basis of A . For each positive integer n , set k n := P ni =1 e i . Then, sinceeach x ∈ A can be expressed as x = P ∞ m =1 α m e m , where the α m ’s are scalars, andsince e m e n = δ mn e n , δ mn being the Kronecker delta, it readily follows thatlim n →∞ xk n = lim n →∞ k n x = x for all x ∈ A . So, in this case, A has a two-sided approximate identity. In particular,S. J. Bhatt and V. Dedania pointed out in [3, Example 3.1] that the followingalgebras have orthogonal bases:(i) For the unit circle T , the Banach convolution algebra (Lebesgue space) L p ( T ), 1 < p < ∞ .(ii) The Banach sequence algebras c , ℓ p (1 ≤ p < ∞ ), with pointwise multi-plication.(iii) The Hardy spaces H p ( U ) (1 < p < ∞ ) on the open unit disk U .However, it is significantly more difficult to decide about the containment for thelatter two classes of Banach algebras in Theorem 1.1. To emphasize this, we revisitsome further examples which appear in [3]:(1) For a locally compact nondiscrete abelian group G , the convolution alge-bra L ( G ) is an example of a non-unital hermitian Banach ∗ -algebra withcontinuous involution. Moreover, L ( G ) has an approximate identity (seefor instance [11, p. 321]).(2) The subalgebras C ( T ) (continuous functions) and C m ( T ) ( C m -functions)of L ( T ) with respective norms k f k ∞ = sup t ∈ T | f ( t ) | and k f k m = sup t ∈ T m X j =0 (cid:12)(cid:12) f ( j ) ( t ) (cid:12)(cid:12) j ! , are examples of non-unital hermitian Banach ∗ -algebras with continuousinvolution. Moreover, these algebras are homogeneous on T . Consequently,it follows from [8, Theorem 2.11] that Fej´er’s kernel is an approximateidentity in both algebras.The authors of [3] did not explicitly give an example of a Banach algebra satisfyingcondition (ii) in Theorem 1.1. However, by the Gelfand-Naimark Theorem it canstraightforwardly be established that every non-unital commutative C ∗ -algebra A F. SCHULZ, R. BRITS, AND M. HASSE satisfies ∂A = M ( A ). Moreover, it is well known that every non-unital commutative C ∗ -algebra contains bounded approximate units and hence a bounded approximateidentity (see [1, Lemma 2.10.1] and [5, Proposition 2.9.14(ii)], respectively).In spite of the above, it turns out that each of the classes (ii) and (iii) in Theorem1.1 contains a Banach algebra without an approximate identity. We discuss theseexamples below: Example 1.9.
Let D := { z ∈ C : | z | < } , cl( D ) be the closure of D in C andlet I := [0 , B to be the so-called “tomato can algebra”; that is, B is theuniform algebra of all continuous complex-valued functions f on K := cl( D ) × I suchthat the function z f ( z,
1) from cl( D ) into C is analytic on D (and continuouson cl( D )). H. G. Dales and A. ¨Ulger have observed in [6, Example 4.8(ii)] that B is natural, that is, K can be identified with M ( B ) via the mapping x χ x , where χ x is the evaluation functional at x . Moreover, they showed that the closed ideal A := { f ∈ B : f (0 ,
1) = 0 } of B does not have an approximate identity. We now prove that ∂A = M ( A ).Recall that the hull of A viewed as an ideal of B is given by H ( A ) := { χ ∈ M ( B ) : χ ( A ) = { }} . Certainly, A contains the function( z, α ) z (( z, α ) ∈ K ) . Moreover, since K is a compact Hausdorff space, it follows that K is normal. Hence,by Urysohn’s Lemma, for each x ∈ K − (cl( D ) × { } ), there exists an f ∈ A suchthat f (cl( D ) × { } ) = { } and f ( x ) = 1. Consequently, since all the characters in M ( B ) are evaluation functionals, it readily follows that H ( A ) = (cid:8) χ (0 , (cid:9) . Now, by[5, Proposition 4.1.11] we may infer that the mapping χ χ | A from M ( B ) − H ( A )into M ( A ) is a homeomorphism. So M ( A ) consists of evaluation functionals at thepoints in K − { (0 , } . By the definition of a weak ∗ -open set in M ( A ), it is easy tosee that if x n → x as n → ∞ in K − { (0 , } , then χ x n → χ x as n → ∞ in M ( A ).Thus, in order to prove that ∂A = M ( A ), it will suffice to show that χ y ∈ ∂A foreach y ∈ K − (cl( D ) × { } ) (since ∂A is closed in M ( A )). Let y ∈ K − (cl( D ) × { } )be arbitrary. As above we choose f such that f (cl( D ) × { } ) = { } and f ( y ) = 1.Observe that K is metrizable and denote by d its metric. Next we consider thecontinuous function g : K → C defined by g ( x ) = 11 + d ( x, y ) ( x ∈ K ) . If we let h ( x ) = f ( x ) g ( x ) for each x ∈ K , then h ∈ A . Moreover, | h ( x ) | < x ∈ K − { y } and h ( y ) = 1. Hence, | χ ( h ) | < χ ∈ M ( A ) − { χ y } and | χ y ( h ) | = 1. It therefore follows that χ y ∈ ∂A for each y ∈ K − (cl( D ) × { } ),and so, ∂A = M ( A ) as advertised. This example shows that the hypotheses in part(ii) of Theorem 1.1 need not imply that A has an approximate identity. Example 1.10.
Denote by c the Banach algebra of all sequences of complexnumbers which converge to 0, equipped with the norm k α k ∞ = sup k ∈ N | α k | ( α = ( α k ) ∈ c ) . DENTITIES, APPROXIMATE IDENTITIES AND TDZ 7
For α = ( α k ) ∈ c , set p n ( α ) := 1 n n X k =1 k | α k +1 − α k | ( n ∈ N ) . Define A := { α ∈ c : sup n ∈ N p n ( α ) < ∞} and k α k := k α k ∞ + p ( α ) ( α ∈ A ) , where p ( α ) := sup n ∈ N p n ( α ). It can then be verified that ( A, k·k ) is a non-unitalcomplex Banach algebra. In fact, since termwise complex conjugation defines aninvolution on A and k α k = k α ∗ k for all α ∈ A , it readily follows that A is a non-unital hermitian Banach ∗ -algebra with continuous involution. This example is dueto J. Feinstein and is discussed in more rigorous detail in [5, Example 4.1.46]. Inparticular, it is shown there that A := { αβ : α ∈ A, β ∈ A } is separable, but that A is non-separable. Hence, A does not have an approximateidentity. So this example shows that the hypotheses in part (iii) of Theorem 1.1need not imply that A has an approximate identity.We now proceed to show that there is a large class of normed algebras each contain-ing a two-sided approximate identity, but whose completions are non-unital. Thefollowing results will be useful in this regard: Theorem 1.11.
Let A be a complex semisimple Banach algebra with a unit, andsuppose that the socle of A , denoted Soc( A ) , is nonzero. Then Soc( A ) has a two-sided approximate identity.Proof. Let x , . . . , x n ∈ Soc( A ). By [12, Theorem 3.13] there exists a subalgebra B of Soc( A ) such that x , . . . , x n ∈ B ∼ = M n ( C ) ⊕ · · · ⊕ M n k ( C ) , where the operations in the latter algebra are all pointwise. Let e be the unit of B . Then x j e = ex j = x j for each j ∈ { , . . . , n } . By the remarks in [7, §
1] this issufficient to infer the existence of a two-sided approximate identity in Soc( A ). (cid:3) Let A be a complex semisimple Banach algebra with a unit. By [13, Theorem 2.2] itfollows that Soc( A ) is finite-dimensional if and only if it is closed in A . Moreover, bythe remark after Theorem 2.2 in [13] it follows that if Soc( A ) is finite-dimensional,then Soc( A ) has the Wedderburn-Artin structure; that is, Soc( A ) is isomorphic asan algebra to M n ( C ) ⊕ · · · ⊕ M n k ( C ) (where the operations in the latter algebraare all pointwise). Proposition 1.12.
Let A be a complex semisimple Banach algebra with a unit.Then Soc( A ) is finite-dimensional if and only if the closure of Soc( A ) has an identityelement.Proof. By the paragraph preceding the proposition, the forward implication directlyfollows, and the reverse implication will be established if we can prove that if theclosure of Soc( A ) has an identity element, then Soc( A ) is closed. To this end, denoteby B the closure of Soc( A ) in A and let e be the identity element of B . Then, sinceSoc( A ) is an ideal, B = eAe . Hence, by [2, Lemma 2.5] it readily follows that B is a semisimple Banach algebra with identity e . But Soc( A ) is dense in B . Hence, F. SCHULZ, R. BRITS, AND M. HASSE there exists a sequence ( e n ) ⊆ Soc( A ) such that e n → e as n → ∞ . So, since e isthe identity of B , it follows that e n must be invertible in B for all n sufficientlylarge. Thus, since Soc( A ) is an ideal, it follows that e ∈ Soc( A ) and consequentlywe have B ⊆ Soc( A ). So Soc( A ) is closed which establishes the result. (cid:3) Theorem 1.13.
Let A be a complex semisimple Banach algebra with a unit. Then Soc( A ) is infinite-dimensional if and only if every element of Soc( A ) is a TDZ in Soc( A ) .Proof. This follows immediately from Theorem 1.11, Proposition 1.12 and Corollary1.5. (cid:3)
A recent paper [6] of H. G. Dales and A. ¨Ulger investigates (various notions of)approximate identities in function algebras. In Theorem 1.15 we show that everynon-unital commutative Banach algebra A with A = Z ( A ) generates a function(uniform) algebra which does not possess an approximate identity. We first needto establish Proposition 1.14.If A is any Banach algebra, then denote by A the standard unitization of A . Thisturns A into a unital Banach algebra via extension. Proposition 1.14.
Let A be a non-unital Banach algebra, and let a ∈ A . Then a is a TDZ in A if and only if a is a TDZ in A .Proof. The forward implication is obvious. For the reverse implication assume a isa TDZ in A . Then we can find, without loss of generality, a sequence ( z n ) in A ,and a sequence of complex numbers ( λ n ) such that k z n k + | λ n | = 1 for all n ∈ N and, lim a ( z n + λ n ) = 0 . So, the sequence ( λ n ) is bounded, and, by the Bolzano-Weierstrass Principle, wecan assume without loss of generality that it converges. Say lim λ n = λ . Thuslim az n = − λa . If λ = 0, then lim az n / k z n k = 0, and the proof is complete.Suppose λ = 0. Then there is a sequence, say ( y n ), in A such that lim ay n = a .If ( y n ) does not converge then it is not Cauchy in A , and similar to the argumentused in the proof of Theorem 1.6 we may conclude that a is a TDZ in A . Assumetherefore that ( y n ) converges; if ( y n ) has limit, say y ∈ A , then a ( y − y ) = 0 fromwhich it follows (as in the proof of Theorem 1.6) that a is a TDZ in A , unless y isan idempotent of A commuting with a . But if the latter case prevails then, since A is non-unital, and y ∈ A , we can again argue as in the proof of Theorem 1.6 toconclude that a is a TDZ in A . (cid:3) For a commutative Banach algebra A we denote by x ˆ x the Gelfand transformof A , and write ˆ A := { ˆ a : a ∈ A } . ˆ A is a normed algebra under the spectral radius ρ A ( · ) for elements of A . That is, if ˆ a ∈ ˆ A , then k ˆ a k = ρ A ( a ) defines a (possiblyincomplete) norm on ˆ A . Theorem 1.15.
Let A be a non-unital commutative Banach algebra. If some a ∈ A is not a TDZ in A then cl( ˆ A ) does not have an approximate identity.Proof. If a is not a TDZ in A , then, by Proposition 1.14, a is not a TDZ in A . Aresult of Arens’ (see for instance [9, p. 48]) then says that a is invertible in someBanach superalgebra, say B , of A . Thus ˆ a cannot be a TDZ in ˆ A . Observe that, DENTITIES, APPROXIMATE IDENTITIES AND TDZ 9 since ˆ A ∼ = ( ˆ A ) , ˆ a is not a TDZ in ˆ A , and thus also not a TDZ in cl( ˆ A ). So cl( ˆ A )cannot have an approximate identity. (cid:3) References
1. W. Arveson,
A Short Course on Spectral Theory , Graduate Texts in Mathematics, U.S. Gov-ernment Printing Office, 2002.2. B. Aupetit,
Spectrum Preserving Linear Mappings between Banach Algebras or Jordan Ba-nach Algebras , J. London Math. Soc. (2000), 917–924.3. S. J. Bhatt and H. V. Dedania, Banach Algebras in which every element is a TopologicalDivisor of Zero , Proc. Amer. Math. Soc. (1995), 735–737.4. M. Cabrera and J. Mart´ınez,
Inner Derivations on Ultraprime Normed Algebras , Proc. Amer.Math. Soc. (1997), 2033–2039.5. H. G. Dales,
Banach Algebras and Automatic Continuity , London Mathematical SocietyMonographs 24, Claredon Press, Oxford, 2000.6. H. G. Dales and A. Ulger,
Approximate Identities in Banach Function Algebras , Studia Math. (2015), 155–187.7. P. G. Dixon,
Unbounded Approximate Identities in Normed Algebras , Glasgow Math. J. (1992), 189–192.8. Y. Katznelson, An introduction to harmonic analysis , Cambridge Mathematical Library, Cam-bridge University Press, 2004.9. R. Larsen,
Banach algebras: An introduction , Pure and Applied Mathematics Series, MarcelDekker Incorporated, 1973.10. R. J. Loy,
Identities in Tensor Products of Banach Algebras , Bull. Aust. Math. Soc. (1970),253–260.11. C.E. Rickart, General Theory of Banach Algebras , University series in higher mathematics,Van Nostrand, 1960.12. F. Schulz and R. Brits,
Commutators, Commutativity and Dimension in the Socle of a BanachAlgebra: A generalized Wedderburn-Artin and Shoda’s Theorem , Linear Algebra Appl. (2015), 175–198.13. F. Schulz, R. Brits, and G. Braatvedt,
Trace Characterizations and Socle Identifications inBanach Algebras , Linear Algebra Appl. (2015), 151–166.14. J. Wichmann,
Bounded Approximate Units and Bounded Approximate Identities , Proc. Amer.Math. Soc. (1973), 547–550.15. W. ˙Zelazko, On a certain class of Non-removable Ideals in Banach Algebras , Stud. Math. (1972), 87–92. Department of Pure and Applied Mathematics, University of Johannesburg, SouthAfrica
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