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Traces, ideals, and arithmetic means

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved March 29, 2002 (received for review February 7, 2002)
Abstract
This article grew out of recent work of Dykema, Figiel, Weiss, and Wodzicki (Commutator structure of operator ideals) which inter alia characterizes commutator ideals in terms of arithmetic means. In this paper we study ideals that are arithmetically mean (am) stable, amclosed, amopen, softedged and softcomplemented. We show that many of the ideals in the literature possess such properties. We apply these notions to prove that for all the ideals considered, the linear codimension of their commutator space (the “number of traces on the ideal”) is either 0, 1, or ∞. We identify the largest ideal which supports a unique nonsingular trace as the intersection of certain Lorentz ideals. An application to elementary operators is given. We study properties of arithmetic mean operations on ideals, e.g., we prove that the amclosure of a sum of ideals is the sum of their amclosures. We obtain cancellation properties for arithmetic means: for principal ideals, a necessary and sufficient condition for first order cancellations is the regularity of the generator; for second order cancellations, sufficient conditions are that the generator satisfies the exponential Δ_{2}condition or is regular. We construct an example where second order cancellation fails, thus settling an open question. We also consider cancellation properties for inclusions. And we find and use lattice properties of ideals associated with the existence of “gaps.”
The algebra B(H) of bounded linear operators on a separable, infinitedimensional, complex Hilbert space has only one nonzero proper closed twosided ideal, the class of compact operators K(H). There is, however, a rich structure of nonclosed twosided ideals of B(H) (operator ideals). Their study was initiated by Calkin (1), who established a lattice isomorphism between ideals and characteristic sets, i.e., the hereditary (solid) positive cones Σ ⊂ c^{*}_{o} (the collection of monotone sequences decreasing to 0) that are invariant under ampliation: ξ → (ξ_{1}, ξ_{1}, ξ_{2}, ξ_{2}, ξ_{3}, ξ_{3}, …). Given an ideal I, call Σ(I) := {s(X)X ∈ I} the characteristic set of I where s(X) := 〈s_{n}(X)〉 is the sequence of snumbers of X, i.e., the eigenvalues of X counting multiplicities and arranged in decreasing order with infinitely many zeroes added in case X is finite rank. Conversely, if Σ is a characteristic set, the diagonal operators diag ξ with ξ ∈ Σ generate the (unique) ideal I such that Σ(I) = Σ.
For each ideal I, we denote by [I, B(H)] the commutator space for I (also known as the commutator ideal), i.e., the linear span of all the commutators XBBX where X ∈ I and B ∈ B(H). Commutator spaces are central to the theory of operator ideals. For instance, they play a key role in defining traces (see the next section). Starting with Halmos (2) and Pearcy and Topping (3), a great deal of effort has been devoted over the years to characterizing commutator spaces for various ideals (see ref. 4 for a comprehensive list of references). The line of research leading to this paper began with a result of one of the authors (5): for the trace class ideal Λ_{1} with the usual trace Tr, the commutator space [𝔏_{1}, B(H)] is strictly contained in ker Tr = {X ∈ 𝔏_{1}Tr X = 0}. The key test case was the diagonal operator X = diag (−1, d_{1}, d_{2}, …) where d_{n} ↓ 0 and ∑ d_{n} = 1. Then X ∈ [𝔏_{1}, B(H)] if and only if ∑ d_{n} log n < ∞. Notice that this condition is equivalent to asking that the arithmetic mean of the sequence λ := (−1, d_{1}, d_{2}, …) be itself summable. Kalton (6) characterized [𝔏_{1}, B(H)] this way in terms of arithmetic means of eigenvalue sequences. For an arbitrary sequence λ = 〈λ_{n}〉, denote by λ_{a} its arithmetic mean sequence, namely, λ_{a} := 〈(1/n) ∑ λ_{j}〉.
Dykema, Figiel, Weiss and Wodzicki proved the following (see ref. 4, Theorem 5.6, and Introduction formula 2).
Theorem 1.
For any proper ideal I, if X ∈ I is a normal operator with λ its sequence of eigenvalues counting multiplicities and ordered according to decreasing moduli, then X ∈ [I, B(H)] if and only if λ_{a} ∈ ∑(I).
This result, along with others in ref. 4, have consequences in the area of operator ideals and traces. Here we explore some of these consequences focusing on a number of questions.
How many traces can an ideal support? We found that for all the ideals in the literature that we considered, the answer is either 0, 1, or ∞; 0 can occur only when diag ω ∈ I (ω := 〈1/n〉 will denote the harmonic sequence), and 1 can occur if diag ω ∉ I. In the latter case, we determined the largest ideal possessing a unique trace. Our analysis here rests partly on the notions of softedged and softcomplemented ideals (see Definition 6), which include many of the ideals in the literature.
What are the implications for operator theory? Applications are given to elementary operators: Fuglede–Putnam type results and a question of Shulman.
As seen in Theorem 1 and throughout refs. 4 and 6, the arithmetic mean operation plays a critical role in the theory of commutators and of operator ideals. To make this role more transparent, given an ideal I, the associated arithmetic mean ideal I_{a} and, respectively, the prearithmetic mean ideal _{a}I are defined in ref. 4 as: So, for instance, a special case of Theorem 1 is that [I, B(H)]^{+} = (_{a}I)^{+} where (_{a}I)^{+} denotes the class of positive operators in the ideal _{a}I. The arithmetic mean (am) closure, respectively, aminterior, of an ideal I are defined in ref. 4 as I^{−} := _{a}(I_{a}) and I^{o} := (_{a}I)_{a} and play an important role in the theory. Indeed, many ideals in the literature are amclosed, i.e., I = I^{−}. This motivates us to investigate questions on amclosure, amclosed ideals, and their properties.
One question is whether the sum of amclosed ideals is amclosed. We prove that it is by showing that: (I + J)^{−} = I^{−} + J^{−}. The proof combines weak majorization theory, convexity, and stochastic matrices (extended to infinite sequences and to notions of infinite convexity).
Another set of questions relate to cancellation properties of arithmetic means: for which ideals I does J_{a} = I_{a}, J_{a} ⊂ I_{a}, and J_{a} ⊃ I_{a} imply, respectively, J = I, J ⊂ I, and J ⊃ I? And similarly for _{a}I? We answer these “first order cancellation” questions when I is principal. If X is a generator of I and π = s(X), denote I = (π). Then we prove (Theorem 11) that J_{a} = (π)_{a} (or _{a}J = _{a}(π)) implies J = (π) if and only if π is regular, i.e., π ≍ π_{a}, or equivalently, (π) = (π)_{a}. Here the equivalence of two sequences ξ ≍ η, means that both ξ = O(η) and η = O(ξ). Theorem 11 depends on properties of the lattice of principal ideals with regular generators. Harder, even for the principal ideal case, are second order cancellation questions: for which π does J_{a2} = (π)_{a2}, J_{a2} ⊂ (π)_{a2}, and J_{a2} ⊃ (π)_{a2} imply, respectively, J_{a} = (π)_{a}, J_{a} ⊂ (π)_{a}, and J_{a} ⊃ (π)_{a}? The first and second implications but not the third are true for π regular. Denoting h(π) := π_{a}/π, it turns out that sufficient conditions can be obtained in terms of the sequence h(π_{a}) = π_{a2}/π_{a}. If h(π_{a}) is equivalent to a monotone sequence, then the second implication holds (Proposition 13). Moreover, if σ(π) := 〈∑ π_{j}〉 satisfies an exponential Δ_{2}condition (see paragraph after Proposition 13) or equivalently, h(π_{a}) ≍ 〈log n〉, then all three cancellations hold (Proposition 14). However, in general, second order cancellation can fail: we construct a pair ξ, π ∈ c^{*}_{o} such that (ξ)_{a2} = (π)_{a2} but (ξ)_{a} ≠ (π)_{a}. This settles in the negative a question of M. Wodzicki.
Traces
The natural domain of the usual trace Tr on B(H) is the trace class 𝔏_{1}. However, ideals can support more exotic traces. Traces are unitarily invariant linear functionals on an ideal I. Equivalently, they are linear functionals vanishing on the commutator space [I, B(H)], i.e., elements of the linear dual to the quotient I/[I, B(H)]. In general, they are not assumed to be positive or faithful. A trace that vanishes on the ideal F of finite rank operators (a subspace of all nonzero ideals) is called singular, and nonsingular otherwise. The first example of a (positive) singular trace was given by Dixmier (7). Its natural domain 𝔖_{Ω} (although Dixmier's construction was somewhat more general) is the dual of the Macaev ideal 𝔖_{ω} (8), and is defined as the ideal with characteristic set {ξ ∈ c^{*}_{o}〈∑ ξ_{j}〉 = O(log n)}. 𝔖_{Ω} and 𝔖_{ω} are denoted by Connes in ref. 9 by 𝔏^{(1, ∞)} and 𝔏^{(∞, 1)}, respectively, and in ref. 4 by M(1/ω_{a}) and 𝔏(log), respectively. In the notation of ref. 4 and of this paper, 𝔖_{Ω} coincides with the amclosure (ω)^{−} of the principal ideal (ω) generated by diag ω, i.e., ∑(𝔖_{Ω}) = {ξ ∈ c^{*}_{o}ξ_{a} = O(ω_{a})}.
The amclosure (π)^{−} = _{a}((π)_{a}) of an arbitrary principal ideal (π) plays an important role in the theory of operator ideals. Denoted 𝔖_{Π} by Gohberg and Krein (10), (π)^{−} was shown to support the complete norm ∥X∥ := sup (s(X)_{a})_{n}/(π_{a})_{n}. Gohberg and Krein noticed that when π is regular, i.e., π ≍ π_{a}, then (π)^{−} = (π). Varga (11) proved that an ideal (π) supports a nontrivial positive trace precisely when (π)^{−} does, and this holds if and only if (π) ≠ (π)^{−} or equivalently, when π is irregular. Clearly, if π is regular then so is π_{a}. As a consequence of Varga's work, it turns out that if π_{a} is regular, so is π. Another proof of this fact is given in ref. 4 (Theorem 3.10). We found a quantitative version of the same result: for every n, there is an m > n (actually we can specify that m ≤ nh(π)_{n} where h(π) = π_{a}/π) such that ½ log h(π)_{n} ≤ h(π_{a})_{m}, and this inequality is sharp.
For general ideals, by Theorem 1, [I, B(H)]^{+} = (_{a}I)^{+} and thus _{a}I is the largest ideal contained in [I, B(H)]. Since I is also the smallest ideal containing [I, B(H)], _{a}I = [I, B(H)] if and only if _{a}I = I. From the chain of inclusions: we see that _{a}I = I if and only if I_{a} = I. An ideal I for which I = _{a}I = I_{a} is called arithmetically mean stable (stable for short). Thus stable ideals are precisely those with no nonzero traces (see ref. 4).
An important consequence of Theorem 1 is that an ideal I supports a nonsingular trace if and only if diag ω ∉ I, which condition is equivalent to [I, B(H)]^{+} = {0} and which in turn is equivalent to [I, B(H)] ∩ F ⊂ {X ∈ FTr X = 0}. This permits one to extend uniquely Tr from F to F + [I, B(H)] and then to I by a Hamel basis argument (nonuniquely when the containment F + [I, B(H)] ⊂ I is proper). Similarly, if I contains the trace class, Tr can be extended to I if and only if diag ω ∉ I. As a further consequence of Theorem 1 we obtain Proposition 2.
Proposition 2.
If diag ω ∉ I, then (𝔏_{1} + [I, B(H)])^{+} = 𝔏.
Notice that the converse is not true since [(ω), B(H)]^{+} = 𝔏.
Applications to Elementary Operators
Trace extensions find natural applications to questions on elementary operators. If A_{i}, B_{i} ∈ B(H), then the map B(H) ∋T → Δ(T) := ∑ A_{i}TB_{i} is called an elementary operator and the adjoint map is Δ*(T) := ∑ A^{*}_{i}TB^{*}_{i}. Elementary operators include commutators and intertwiners and hence their theory is connected to commutator spaces. The Fuglede–Putnam Theorem (12, 13) states that for the case Δ(T) = ATTB with A, B normal, Δ(T) = 0 implies Δ*(T) = 0. For n = 2, Weiss (14) generalized this to the case where {A_{i}} and {B_{i}} are separately commuting families of normal operators by proving that Δ(T) ∈ 𝔏_{2} implies Δ*(T) ∈ 𝔏_{2} and that ∥Δ(T)∥_{𝔏2} = ∥Δ*(T)∥_{𝔏2}. [This is also a consequence of Voiculescu's (15) Theorem 4.2 and Introduction to Section 4.] Shulman (16) proved that for n = 6, Δ(T) = 0 does not imply Δ*(T) ∈ 𝔏_{2}.
If we impose some additional idealtype conditions on the elementary operator Δ and/or on T, we can extend these implications to arbitrary n past the obstruction found by Shulman and the limitations of ref. 15. Assume there is an ideal I not containing diag ω but containing 𝔏_{1} such that the product (A_{i}T)(B_{i}) of the principal ideals (A_{i}T) and (B_{i}) is contained in I^{1/2} for all i (resp., (A_{i})(TB_{i}) ⊂ I^{1/2} for all i). This includes, for instance, the cases where for each i, at least one of the operators A_{i}, B_{i}, T is in I^{1/2}; or at least two are in I^{1/4}; or all three are in I^{1/6}. The usefulness of these conditions lies in the fact that the ideal I can be “much larger” than 𝔏_{1}. Then by using the general identity [I, J] = [IJ, B(H)] from ref. 4 (Theorem 5.10), we obtain that Δ*(T)^{2} − Δ(T)^{2} ∈ [I, B(H)] (resp., (Δ*(T))*^{2} − (Δ(T))*^{2} ∈ [I, B(H)]). Thus, if Δ(T) ∈ 𝔏_{2}, it follows that Δ*(T)^{2} is in (𝔏_{1} + [I, B(H)])^{+} and, by Proposition 2, that Δ*(T) ∈ 𝔏_{2}. Moreover, since diag ω ∉ I, Tr can be extended to I and thus its extension must vanish on [I, B(H)]. Therefore ∥Δ(T)∥_{𝔏2} = ∥Δ*(T)∥_{𝔏2} and, in particular, Δ(T) = 0 implies Δ*(T) = 0.
A further application is to a problem considered by Shulman. Here the operators {A_{i}} and {B_{i}} are not necessarily commuting nor normal. He showed that Δ*Δ(T) = 0 does not imply Δ(T) = 0 and conjectured that this implication holds under the additional assumption that Δ(T) ∈ 𝔏_{2}. Reasoning as in the previous case, if there is an ideal I not containing diag ω such that (A_{i}T)(B_{i}) ⊂ I^{1/2} for all i (resp., (A_{i})(TB_{i}) ⊂ I^{1/2} for all i), then Δ(T)^{2} − T*Δ*Δ(T) ∈ [I, B(H)] (resp., (Δ(T))*^{2} − T(Δ*Δ(T))* ∈ [I, B(H)]). Hence, if Δ*Δ(T) ∈ 𝔏_{1}, we obtain that Δ(T) ∈ 𝔏_{2} and that ∥Δ(T)∥𝔏 = Tr T*Δ*Δ(T). In particular, if Δ*Δ(T) = 0 it follows that Δ(T) = 0.
Uniqueness of Traces
For all 0 ≠ X ∈ 𝔏_{1}, s(X)_{a} ≍ ω. So, for X ∈ 𝔏, instead of the arithmetic mean, the relevant operation is the arithmetic mean at infinity X_{a∞} := diag 〈(1/n) ∑ s_{j}(X)〉 (see ref. 4, formula 17 and Theorem 5.11). A special case of Theorem 5.11 (iii) is that if I ⊂ 𝔏_{1}, then [I, B(H)] contains all the operators in I with zero trace if and only if I is invariant under the arithmetic mean at infinity. Further using Theorem 1, we obtain Proposition 3.
Proposition 3.
If I is an ideal not containing diag ω, then (F + [I, B(H)])^{+} = {X ∈ 𝔏X_{a∞} ∈ I}.
As a consequence, (F + [I, B(H)])^{+} is always hereditary (solid). Notice that an ideal I not containing diag ω has a unique trace (up to scalar multiplication), i.e., dim I/[I, B(H)] = 1, if and only if I = F + [I, B(H)]. It is easy to verify that X ∈ (F + [𝔏_{1}, B(H)])^{+}, i.e., X_{a∞} ∈ 𝔏_{1}, if and only if X ∈ 𝔏(σ(log))^{+}. Here 𝔏(σ(log)) is the Lorentz ideal with characteristic set {ξ ∈ c^{*}_{o}∑ ξ_{n} log n < ∞} and σ(λ) := 〈∑ λ_{j}〉 denotes the initial partial sum sequence of λ (see ref. 4, 2.25). Similarly, one obtains X_{a∞} ∈ 𝔏(σ(log^{p})) (the Lorentz ideal with characteristic set {ξ ∈ c^{*}_{o}∑ ξ_{n} log^{p}n < ∞} if and only if X ∈ 𝔏(σ(log^{p+1}))^{+}. Notice that from Proposition 3 it follows that 𝔏(σ(log))^{+} = (F + [𝔏_{1}, B(H)])^{+}. Another useful consequence of Proposition 3 is that if diag ω ∉ I, then I has a unique trace, i.e., I = F + [I, B(H)], if and only if X_{a∞} ∈ I for every X ∈ I^{+}. Consequently all such I are contained in ∩ 𝔏(σ(log^{p})). Thus, we obtain Theorem 4.
Theorem 4.
∩ 𝔏(σ(log^{p})) is the largest ideal (not containing diag ω) with a unique trace, and that trace is Tr.
In particular, 𝔏_{1} has no unique trace, not even a unique nonsingular trace. If I is the principal ideal generated by an operator X ∈ 𝔏_{1}, the conditions of Theorem 1 are satisfied if and only if for some c > 0, X_{a∞} ≤ c diag s(X). In ref. 18, Corollary 7, Kalton proves that an operator X ∈ 𝔏_{1} is uniquely traceable, i.e., I supports a unique separately continuous trace (see ref. 18 for the definition) if and only if there is a p > 1 and c > 0 such that s_{m}(X) ≤ c(m/n)^{−p} s_{n}(X). As this condition clearly implies X_{a∞} ≤ c′ diag s(X) for some c′ > 0, we see that I supports only one separately continuous trace precisely when it supports only one trace, namely, the usual trace Tr.
Dimension of I/[I, B(H)]
Beyond the question of uniqueness of traces, it is natural to ask “how many” traces are supported by an ideal I or, equivalently, what are the possible values of dim I/[I, B(H)]?
As mentioned in ref. 4 (5.27Remark 1), Dixmier's method in constructing nonsingular traces can be used to prove that dim(π)^{−}/[(π)^{−}, B(H)] = ∞ whenever π = o(π_{a}). The condition s(Y) = o(s(X)) for X, Y ∈ K(H) (equivalently, Y = KX for some K ∈ K(H)) also plays an important role in Varga's treatment of traces on principal ideals and their amclosures. When s(Y) = o(s(X)), we can interpolate between Y and X any number of operators. This leads to:
Lemma 5.
If there exist X ∈ I and K ∈ K(H) such that 0 ≤ KX ∉ F + [I, B(H)], then I/(F + [I, B(H)]) has uncountable dimension.
Considering for which kind of ideals the existence of a positive X ∉ F + [I, B(H)] guarantees that we can always find a K ∈ K(H) for which we also have 0 ≤ KX ∉ F + [I, B(H)] leads us to the definitions:
Definition 6.
We call an ideal I softedged if I = K(H)I and softcomplemented if, for any ideal J, the condition K(H)J ⊂ I implies J ⊂ I.
Thus, an ideal I is softedged if for every π ∈ ∑(I) there is a ξ ∈ ∑(I) such that π = o(ξ). An ideal I is softcomplemented if for every c^{*}_{o} ∋π ∉ ∑(I) there is a c^{*}_{o} ∋ξ ∉ ∑(I) such that ξ = o(π).
The main ideals in the literature are all either softedged or softcomplemented. We prove that countably generated ideals are softcomplemented. They are not necessarily softedged. Indeed, if π satisfies the Δ_{2}condition, i.e., sup π_{n}/π_{2n} < ∞, then the principal ideal (π) is not softedged. Lorentz ideals are both softedged and softcomplemented. If an ideal I is softcomplemented, its prearithmetic mean ideal _{a}I is also softcomplemented. Thus all Marcinkiewicz ideals (the prearithmetic means of principal ideals) are softcomplemented. If M is a nondecreasing function, the associated Orlicz ideal 𝔏_{M} (respectively, small Orlicz ideal 𝔏) are the ideals with characteristic set {ξ ∈ c^{*}_{o} ∑_{n} M(tξ_{n}) < ∞ for some t > 0} (respectively, {ξ ∈ c^{*}_{o}∑_{n} M(tξ_{n}) < ∞ for all t > 0}) (see ref. 4, 2.37 and 4.7). We show that small Orlicz ideals are softedged and Orlicz ideals are softcomplemented. The Gohberg and Krein ideals 𝔖_{φ} generated by symmetric norming functions φ are always softcomplemented and they are softedged if and only if φ is mononormalizing (see ref. 10 for the definitions).
Theorem 7.
If I is softedged or softcomplemented then dim I/(F + [I, B(H)]) is either 0 or ∞.
When this dimension is 0, dim I/[I, B(H)] is either 1 or 0 according to whether diag ω ∉ I or diag ω ∈ I. A further consequence of Lemma 5 is that if diag ω ∉ I and I ⊄ ∩ 𝔏(σ(log^{p})) (so, in particular, if I ⊄ 𝔏_{1}) then dim I/(F + [I, B(H)]) = ∞. The key fact in this argument is that the ideals 𝔏(σ(log^{p})) are softcomplemented.
Of course not all ideals are either softedged or softcomplemented. For instance, if I is softcomplemented but not softedged, then any ideal J strictly between K(H)I and I is neither softedged nor softcomplemented. The simplest example of such a situation is when I = (π) is the principal ideal for some π satisfying the Δ_{2}condition. Then ∑(K(H)I) = o(π) ∩ c^{*}_{o}, ∑(I) = O(π) ∩ c^{*}_{o}, and there are infinitely many ideals lying between K(H)I and I. This example leads us to Definition 8.
Definition 8.
Given an ideal I, let se(I) = K(H)I and let sc(I) be the ideal with characteristic set {ξ ∈ c^{*}_{o}o(ξ) ∩ c^{*}_{o} ⊂ ∑(I)}.
Then, se(I) is the largest softedged ideal contained in I and sc(I) is the smallest softcomplemented ideal containing I or, equivalently, it is the largest ideal J for which K(H)J ⊂ I.
It follows that sc(se(I)) = sc(I) and se(sc(I)) = se(I). As a consequence, if se(I) ⊂ J ⊂ sc(I), then se(J) = se(I) and sc(J) = sc(I). This leads us to Theorem 9.
Theorem 9.
If either se(I)/[se(I), B(H)] or sc(I)/[sc(I), B(H)] has infinite dimension, then both have infinite dimension. In this case, I/[I, B(H)] has infinite dimension.
In particular, se(I) is stable, i.e., se(I) = [se(I), B(H)], if and only if sc(I) is stable. Applying these ideas to Orlicz ideals, one first shows that se(𝔏_{M}) = 𝔏 and that sc(𝔏) = 𝔏_{M}. Then if 𝔏_{M} ≠ 𝔏 and if either of the two ideals is not stable, then the other one is not stable (see ref. 4, Theorem 5.26), in which case, for every ideal I between 𝔏 and 𝔏_{M}, it follows that dim I/[I, B(H)] = ∞.
Amclosure, AmInterior, and Related Ideals
We have seen that the notion of amclosure plays a critical role for traces and norms on principal ideals. This notion was studied in ref. 4 for general ideals where it turned out to be relevant also in the analysis of single commutators (see ref. 4, Theorem 7.3 and Corollary 7.10).
The Gohberg and Krein ideals 𝔖_{φ} generated by symmetric norming functions φ are amclosed, and they include many ideals in the literature such as Lorentz ideals for concave functions, Marcinkiewz ideals, and Orlicz ideals for convex functions. (See ref. 4, 4.7 and 4.9.)
Majorization theory is useful in investigating amclosed ideals. By definition, an ideal is amclosed, if and only if for ξ ∈ c^{*}_{o}, ξ_{a} ≤ η_{a} for some η ∈ ∑(I) implies ξ ∈ ∑(I). In other words, the characteristic set ∑(I) of an amclosed ideal I is hereditary under weak majorization (the Hardy–Littlewood–Polya–Schur order). We prove that this is equivalent to the conditions that ∑(I) contains all summable monotone sequences and is invariant under direct sums of block doubly stochastic finite matrices followed by monotonizing. In other words, if ξ ∈ ∑(I) and P = ∑ ⊕ P_{k} is a direct sum of doubly stochastic finite matrices, i.e., matrices with nonnegative entries with rows and columns summing to 1, then the monotonization (Pξ)* of Pξ must be in ∑(I). This implies but is not equivalent to the conditions that ∑(I) contains all summable monotone sequences and that ∑(I) is invariant under infinite convex combinations of infinite permutation matrices followed by monotonizing.
A useful application of this result is that: (I + J)^{−} = I^{−} + J^{−} for arbitrary pairs of ideals. Since directed unions of amclosed ideals are amclosed, this identity shows that every ideal I contains a largest amclosed ideal which we denote by I_{−}. We obtain analogous results for amopen ideals, i.e., that every ideal I is contained in a smallest amopen ideal I^{oo} and that (I + J)^{oo} = I^{oo} + J^{oo}. Notice that (I + J)^{o} ⊃ I^{o} + J^{o} and (I + J)_{−} ⊃ I_{−} + J_{−} but both inclusions can be proper.
For a principal ideal I = (π), both I^{o} and I^{oo} are principal and have generators diag π^{o} and diag π^{oo}, respectively. Here π^{o} is (up to equivalence) the largest average ξ_{a} ≤ π, and π^{oo} is the smallest average ξ_{a} ≥ π (no equivalence is necessary here). To identify (π)_{−}, we prove that if ξ ∈ c^{*}_{o} and ζ_{a} ≤ ξ_{a} for ζ ∈ c^{*}_{o} implies ζ ≤ π, then ξ_{a} ≤ π. In other words, (ξ)^{−} ⊂ (π) implies (ξ)_{a} ⊂ (π) and hence (ξ)^{−} ⊂ _{a}(π). As _{a}(π) is amclosed, this yields:
Theorem 10.
If I is a principal ideal, then I_{−} = _{a}I.
Theorem 10 has several consequences. First of all, it yields a new proof of the fact that a principal ideal is amclosed if and only if it is stable (11) (see also ref. 4, Theorem 5.20). Notice that the amclosure of an ideal is principal if and only if the ideal is principal. This follows from the fact mentioned earlier that for any π ∈ c^{*}_{o}, π is regular if and only if π_{a} is regular.
Further consequences of Theorem 10 are obtained by exploiting lattice properties of ideals, in particular, of some classes of principal ideals. Blass and Weiss (19) proved that K(H) is the sum of two proper ideals (neither of which can be countably generated) and in general, every ideal that properly contains F is the sum of two proper ideals. Here we obtain that with respect to the inclusion order, the lattice of principal ideals has no “gaps”, that is, between any two principal ideals lies another one. Notice this is not true in general, e.g., below every stable principal ideal (π) there is a gap between (π) and a maximal ideal in (π) not containing diag π. Such a maximal ideal must necessarily also be stable but cannot be principal.
A principal ideal has a unique generator up to ssequence equivalence if and only if any (and hence all) of its generators have their ssequence satisfying the Δ_{2}condition, in short, a Δ_{2}generator. We obtain that between an ideal with a Δ_{2}generator and another comparable principal ideal (whether contained in it or containing it) lies a principal ideal with a Δ_{2}generator. The same holds replacing Δ_{2}generators with regular generators: between two comparable principal ideals, one of which has a regular generator, i.e., is stable, lies another principal stable ideal.
Cancellation Properties for Arithmetic MeansFirst Order
In studying the arithmetic mean operations on ideals it is natural to consider cancellation properties of the kind: for which ideals I does I_{a} = J_{a} imply I = J? And similarly, when does _{a}I = _{a}J imply I = J? Notice first that I_{a} = (I^{−})_{a}. So for the first question, a necessary condition is that I be amclosed (though not sufficient since 𝔏_{1} is amclosed and (𝔏_{1})_{a} = (F)_{a}). As the examples following Theorem 11 will show, the general question has no simple answer, but we can settle the case when I is principal. The case J_{a} = (π)_{a} is simpler. As noticed above, a necessary condition is that (π) is amclosed and this requires π to be regular, that is, (π) to be stable. The condition is also sufficient. Indeed, if J_{a} = (π)_{a} = (π), then J^{−} = (π) is principal and hence J too is principal and so it must coincide with (π). The _{a}J = _{a}(π) case has the same answer but its proof requires the use of lattice properties of principal ideals and Theorem 10. In summary:
Theorem 11.
(i) J_{a} = (π)_{a} implies J = (π) if and only if (π) is stable.
(ii) _{a}J = _{a}(π) implies J = (π) if and only if (π) is stable.
For general ideals, the stability of I is no longer sufficient in either case and we find the counterexamples interesting. For case (i) we construct an ideal L which is not stable but whose arithmetic mean L_{a} is stable. L is generated by sequences π(n) chosen so that for each n, ω_{an} ≤ π(n)_{a} ≤ ω_{a2n}, but such that diag ω ∉ L. Then L_{a} = ∪(ω)_{an} is the amstabilizer of (ω) and hence it is stable, but diag ω ∈ L_{a}∖ L so L is not. For case (ii), we take I = ∪(ω)_{an} and set J = I + (π), where π is chosen so that diag π ∉ I and hence J ≠ I, but for each n, (ω_{an}) = (ω_{an} + π)^{o} so _{a}J = _{a}I.
Directly from the definition of amclosure (respectively, aminterior) it follows that I is amclosed (respectively, amopen) precisely when J_{a} ⊂ I_{a} implies J ⊂ I. (respectively, _{a}J ⊃ _{a}I implies J ⊃ I). The reverse direction is subtler. It turns out that (ω^{1/2})_{a} ⊂ I_{a} does not imply that (ω^{1/2}) ⊂ I but only that (ω) ⊂ I. In fact, (ω) is the largest ideal with this property. More generally, if 0 < p < 1 and if 1/p − 1/p′ = 1, then (ω^{p′}) is the largest ideal J such that (ω^{p}) ⊂ I_{a} implies J ⊂ I. We can generalize this result to all principal ideals (π). If π is summable then for every proper ideal I, I_{a} ⊃ (ω) = (π)_{a}. Therefore ∩ {JJ_{a} ⊃ (π)_{a}} is the ideal F of finite rank operators which is principal and is generated by π^{∼} := (1, 0, 0, …). If π is nonsummable, set φ_{n} := min {k∑ π_{i} ≥ n} and define π := (π_{a})_{φn}. Then we prove that (π)_{a} ⊂ I_{a} implies (π^{∼}) ⊂ I and that (π^{∼}) is the largest ideal with that property, namely:
Proposition 12.
For all π ∈ c^{*}_{o}, (π^{∼}) = ∩{JJ_{a} ⊃ (π)_{a}}.
In particular, J_{a} ⊃ (π)_{a} implies J ⊃ (π) if and only if (π) = (π^{∼}). We have an example of a regular sequence π satisfying the latter condition, but as mentioned above, for 0 < p < 1, the regular sequences ω^{p} do not.
The situation for the prearithmetic mean ideal _{a}I is different: for every π there exists ξ such that _{a}(ξ) ⊂ _{a}(π) but (ξ) ⊄ (π).
Second Order Cancellation
Wodzicki asked whether or not (ξ)_{a2} = (π)_{a2} implies (ξ)_{a} = (π)_{a}. If π is regular, then the answer is clearly affirmative. We found that in general the answer is negative and then we investigated the properties of π which guarantee that this cancellation holds. This led to properties of the ratio h(π_{a}) = π_{a2}/π_{a}. The first step is Proposition 13.
Proposition 13.
Given an ideal I, then J_{a2} ⊂ I_{a2} implies J_{a} ⊂ I_{a} if and only if I_{a} = (I_{a})^{−o}. A sufficient condition is that each π ∈ ∑(I) is dominated by some ξ ∈ ∑(I) such that h(ξ_{a}) is equivalent to a monotone nondecreasing sequence.
Notice that for all ξ ∈ c^{*}_{o}, 1 ≤ h(ξ_{a}) ≤ log and that log − h(ξ_{a}) is strictly increasing (to infinity if ξ is nonsummable). So we have two extremal cases: when h(ξ_{a}) ≍ 1, i.e., ξ_{a} and hence ξ are regular, and when h(ξ_{a}) ≍ log. Interestingly, it turns out that the latter case is equivalent to σ(ξ) satisfying the exponential Δ_{2}condition, i.e., for some c > 0, ∑ ξ_{i} ≤ c ∑ ξ_{i} for all n. This condition is also sufficient for second order cancellation for the reverse inclusions.
Proposition 14.
Let I be an ideal such that every π ∈ ∑(I) is dominated by some ξ ∈ ∑(I), such that σ(ξ) satisfies the exponential Δ_{2}condition. Then J_{a2} ⊃ I_{a2} implies J_{a} ⊃ I_{a}.
Notice that this cancellation can fail even for I principal and stable, e.g., for I = (ω^{1/2}). Referring back to Wodzicki's question we see that (ξ)_{a2} = (π)_{a2} implies (ξ)_{a} = (π)_{a} in the cases when π is regular or when σ(π) satisfies the exponential Δ_{2}condition (the two extreme cases). But in general the answer is negative. The counterexample outlined below illustrates some of the features of the theory.
There is an increasing sequence n_{k} and two c^{*}_{o}sequences η ≥ ξ defined by ξ_{j} := e^{−k}η_{nk} for n_{k} < j ≤ n_{k+1} and η_{j} := e^{−k}η_{nk} for n_{k} < j ≤ e^{k}n_{k} and η_{j} = ξ_{j} for e^{k}n_{k} < j ≤ n_{k+1}. The n_{k} are taken sufficiently large to achieve the asymptotics (ξ)_{nk} ∼ (ξ_{a})_{nk} ∼ (ξ_{a2})_{nk} ∼ η_{nk} ∼ (η_{a})_{nk} ∼ (η_{a2})_{nk} (a “resetting of the clock” process). The ratio η_{a}/ξ_{a} increases throughout the interval n_{k} < j ≤ e^{k}n_{k}, becomes “large” for j ≈ e^{k}n_{k}, and then decreases. The ratio η_{a2}/ξ_{a2} is more “resistant to change.” It increases more slowly but continues to increase past e^{k}n_{k} and reaches a maximum on e^{k}n_{k} < j ≤ n_{k+1}. Finetuning of the growth constants so to make the ratio η_{a2}/ξ_{a2} close to 1 for j ≈ e^{k}n_{k} guarantees that this maximum can be kept uniformly bounded. Consequently, η_{a} ξ_{a} but η_{a2} ≍ ξ_{a2}.
Footnotes

↵† To whom reprint requests should be addressed. Email: kaftal{at}math.uc.edu.

This paper was submitted directly (Track II) to the PNAS office.
 Received February 7, 2002.
 Copyright © 2002, The National Academy of Sciences
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