Recent Results in the Theory of Infinite-Dimensional Banach Spaces

W. T. Gowers

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England

Many of the best known questions about separable infinite-dimensional Banach spaces are of at least one of the following forms.

1. If X is an arbitrary space, must it have a “nice” subspace?

2. If X is an arbitrary space, are there non-obvious examples of operators on X?

3. If certain operators are known to be defined on a space X, does this imply anything non-obvious about the structure of X?

A large number of such questions have been solved in the last three years, and surprising connections between them have been discovered. The purpose of this paper is to explain these developments. Unless otherwise stated, all spaces and subspaces mentioned will be infinite-dimensional separable Banach spaces.

We begin by discussing questions of the first kind above. It is clear straight away that not every space has a Hilbert space, the nicest space of all, as a subspace; the space ℓ_{1} is but one of many obvious counterexamples. However, if one asks whether every space contains c_{0} or ℓ_{p} for some 1 ≤ p < ∞, then one already has a very simple question to which the answer is not at all obvious. In fact, this question was not answered until the early seventies, when Tsirelson [T] used a clever inductive procedure to define a counterexample. The proof that his example does not contain c_{0} or ℓ_{p} is surprisingly short (this is even more true of the dual of his space as presented by Figiel and Johnson [FJ]) but the ideas he introduced have been at the heart of the recent progress.

There were two further weakenings of the notion of “nice” which left questions not answered by Tsirelson’s example. For the first, recall that a Schauder basis (we will often say simply basis) of a Banach space X is a sequence (x_{n})_{n=1}^{∞} such that every vector in X has a unique expression as a norm-convergent sum of the form ∑_{n=1}^{∞} a_{n}x_{n}. A simple result proved in the early thirties by Mazur (see [LT2]) is that every Banach space has a subspace with a basis. It was a famous open problem, answered negatively by Enflo [En] in 1973, whether every separable Banach space had a basis.

The definition of a Schauder basis is unlike that of an algebraic basis for a vector space in that the order of the x_{n} is important. A basis (x_{n})_{n=1}^{∞} with the property that (x_{π(n)})_{n=1}^{∞} is a basis for every permutation π of the positive integers is called an unconditional basis. It was shown by Mazur that (x_{n})_{n=1}^{∞} is an unconditional basis if and only if there is a constant C such that, for every sequence (a_{n})_{n=1}^{∞} of scalars and any sequence (ϵ_{n})_{n=1}^{∞} with each ϵ_{n} of modulus one, we have the inequality

∥∑_{n=1}^{∞} ϵ_{n}a_{n}x_{n}∥ ≤ C∥∑_{n=1}^{∞} a_{n}x_{n}∥ .

Such a basis is called C-unconditional. Notice that this implies that, for every subset A ⊂ ℕ, the projection ∑_{n∈ℕ} a_{n}x_{n} ↦ ∑_{n∈A} a_{n}x_{n} has norm at most C. Thus if X has an unconditional basis, then there are many non-trivial projections on X. A sequence that is an unconditional basis for its closed linear span is called an unconditional basic sequence.

We may now ask whether every space has a subspace with an unconditional basis, or, equivalently, contains an unconditional basic sequence (the unconditional basic sequence problem). This question was first asked as soon as unconditional bases were defined in the 1940s and appears in print in [BP2]. The lack of a proof after three or four decades led many people began to suspect that the answer was no, but there still seemed to be some chance of a positive answer to a yet weaker question: does every space contain c_{0}, ℓ_{1} or a reflexive subspace? The reason this is a weaker question is that a result of James [J1] states that a space with an unconditional basis not containing c_{0} or ℓ_{1} must itself be reflexive, so a positive answer would be implied by a positive answer to the first question.

In the summer of 1991, Maurey and I independently discovered spaces not containing unconditional basic sequences. There is not room to explain the constructions here, but we can make a few remarks. First, we introduce some notation. If X is a Banach space with a given basis (e_{n})_{n=1}^{∞}, then the support of a vector x = ∑_{n=1}^{∞} a_{n}e_{n} ∈ X is just the set of n for which a_{n} is non-zero. Given x, y ∈ X, we write x < y to mean that every element of the support of x is less than every element of the support of y. A sequence x_{1} < x_{2} < … of non-zero vectors is called a block basis, and a subspace generated by a block basis is called a block subspace. A simple but very useful result of Bessaga and Pelczynski [BP1] states that every subspace Y of a space X with a basis has a further subspace Z isomorphic to a block subspace of X. This reduces many problems about subspaces of Banach spaces to ones about block subspaces.

The unconditional basic sequence problem is no exception. It is straightforward to show that X contains no C-unconditional basic sequence if and only if every block subspace Y ⊂ X contains a sequence of vectors x_{1} < … < x_{n} such that

∥∑_{i=1}^{n} x_{i}∥ > C∥∑_{i=1}^{n}(−1)^{n}x_{i}∥ .

(This is not quite true if X has complex scalars. In that case, sequences satisfying the above inequality exist in every block subspace provided X contains no 2C-unconditional basic sequence.) Therefore X contains no unconditional basic sequence at all if and only if such a finite sequence can be found in every block subspace for every C. Notice that the condition on the supports of x_{1}, …, x_{n} is important, since otherwise a constant sequence would do.

Of great importance to us in constructing such a space was noticing that, for any fixed C, a space which had just been constructed by Schlumprecht [S1] could be renormed so that every block subspace did contain such a “C-conditional” sequence. However, Schlumprecht’s space has a 1-unconditional basis, so any renorming has a C-unconditional basis for some C. To find a space with no unconditional basic sequence at all one somehow had to use the ideas from the renorming of Schlumprecht’s space but produce a non-isomorphic space. This led to considerable conceptual and technical difficulties. Two other points are worth making. The first is that Schlumprecht’s space was, like Tsirelson’s space and indeed our spaces, constructed inductively (at first glance, the definition appears to be circular). The advantage of his space over Tsirelson’s was that it had stronger properties related to the so-called distortion of Banach spaces. Indeed, his space played an important part in his solution with Odell [OS1] of a famous problem known as the distortion problem. (In one formulation this asks: must every space isomorphic to a Hilbert space have a subspace almost isometric to a Hilbert space?) Again, there is not room to explain this connection (some indication can be found in [OS2]), but there is one and it is important. The second point is that the spaces that Maurey and I found were not in fact distinct, indicating that, however complicated, our approach was a natural one. The result, with some extensions which will be described in a moment, appears in a joint paper [GM1].

The first indication that the space X_{1} we constructed was also relevant to questions about operators was an observation of W. B. Johnson. He pointed out to us that we could alter our argument(s) and show that every continuous projection on X_{1} had finite rank or corank, and moreover that every subspace of X_{1} had the same property. This is equivalent to saying that no subspace Y of X_{1} can be written as a topological direct sum W ⊕ Z with W and Z infinite-dimensional. A space with this property he called hereditarily indecomposable. Another equivalent form of the property, which brings out its strangeness, is that, for any two infinitedimensional subspaces Y and Z of X_{1} and any ϵ > 0, there exist unit vectors y ∈ Y and z ∈ Z such that ∥y − z∥ < ϵ. In a certain sense, the angle between any two infinite-dimensional subspaces is zero.

Thus our space gave a strong answer to a question of Lindenstrauss [L2]: can every space be decomposed as a topological direct sum of two infinite-dimensional subspaces? (Such a space is simply called decomposable.) It showed that, for an arbitrary space X, one could not in general expect L(X), the space of operators on X, to contain interesting examples of projections.

Another important class of operators is isomorphisms onto proper subspaces, and the next step was the discovery that these also did not have to exist in general. I constructed a variant X_{U} of the space X_{1} above, which had an unconditional basis [G1]. However, it was not isomorphic to any proper subspace, and therefore solved the so-called hyperplane problem, which had its origins in Banach’s book [B]. This was the question of whether an arbitrary space is isomorphic to its closed subspaces of codimension one. (Note that if Y and Z are distinct such subspaces, then Y ≃ (Y ∩ Z) ⊕ ℂ ≃ Z, so any two subspaces of the same finite codimension are isomorphic.) Equivalently, X_{U} is not isomorphic to X_{U} ⊕ ℂ. (Actually, as Maurey recently pointed out to me, the precise question asked by Banach was whether every space X is isomorphic to a subspace of its hyperplanes, but this is also answered negatively by X_{U}.)

It was a little strange that an unconditional basis was needed to make the above construction work, although it did mean that a question of the third kind was answered. The situation became clearer with the following result [GM1], which underlines the importance of hereditary indecomposability. Recall that an operator T : X → Y is Fredholm if the dimensions α(T) and β(T) of the kernel of T and Y/T X respectively are finite, and that the index of such an operator is defined to be α(T)−β(T). A strictly singular operator S : X → Y is one for which there is no subspace Z of X such that the restriction of S to Z is an isomorphism. Equivalently, for every Z ⊂ X and every ϵ > 0 there exists z ∈ Z with ∥Sz∥ < ϵ ∥z∥. The strictly singular operators share many of the smallness properties of the compact operators (indeed, for several spaces X, the strictly singular and compact operators in L(X) coincide). For example, if T is Fredholm and S is strictly singular then T + S is Fredholm with the same index as T.

Theorem 1. [GM1] Let X be a hereditarily indecomposable space with complex scalars. Then every operator T ∈ L(X) can be written in the form λI + S, where λ ∈ ℂ and S is strictly singular. In particular, every operator on X is either strictly singular or Fredholm with index zero.

It is easy to see that an isomorphism on to a proper subspace cannot be strictly singular or Fredholm with index zero, so the space X_{1} discussed earlier is another counterexample to Banach’s question. (In fact the space X_{1} can be real or complex. In the case of real scalars, we have a direct proof that it satisfies the conclusions of Theorem 1. In general, a real hereditarily indecomposable space is isomorphic to no proper subspace.) Thus, in a certain sense, L(X_{1}) is trivial and the answer to the second question with which we started is a simple no. On the other hand, it is not known whether there exists a space on which every operator is a compact perturbation of a multiple of the identity.

We have left behind the question of whether an arbitrary space must contain c_{0}, ℓ_{1} or a reflexive subspace. For this problem, unlike the unconditional basic sequence problem, it is not enough to find a space such that every (block) subspace contains a finite sequence of some kind. The problem is in this sense more genuinely infinite-dimensional, and for this reason it was not obvious whether the techniques used for constructing the hereditarily indecomposable space X could be extended to give a counterexample. In the end, however, it turned out to be possible [G2], and the resulting space X_{CLR} had a slightly stronger property. No subspace Y ⊂ X_{CLR} contains ℓ_{1} or has a separable dual. Note that it is not obvious that there even exists a space with non-separable dual not containing ℓ_{1}. This was a question asked by Banach and answered independently by examples of James [J2] and Lindenstrauss and Stegall [LS].

It would seem, then, that there is no reasonable sense of the word “nice” for which one can (truthfully) say that every space has a nice subspace. However, this conclusion, we shall see later, is premature.

Let us concentrate on questions of the third kind. We have already seen that the existence of plenty of projections on a space does not guarantee that there are non-trivial isomorphisms to subspaces. What about the reverse? If we are given a space which is isomorphic to a proper subspace, must there be non-trivial projections? There are many other questions of this general kind, and to answer some of them Maurey and I generalized the results of our first paper. We proved [GM2] that, under certain conditions on an algebra A, one can construct a space X = X(A) such that A embeds in an obvious way into L(X), and every operator on X is a small perturbation of (the image of) an element of A. Sometimes “small” in this statement simply means “strictly singular”, and sometimes a slight weakening which is nevertheless strong enough for applications.

To illustrate, let us consider what can be said about a space if it has a basis and the shift with respect to that basis is an isometry. From Theorem 1 we know straight away that the space is not hereditarily indecomposable, but this does not imply that there are non-trivial projections defined on the whole space. We also know that, writing S for the shift and L for the left shift, every operator of the form ∑_{n=0}^{∞} a_{n}S^{n} + ∑_{n=1}^{∞} b_{n}L^{n} is continuous if ∑_{n=0}^{∞} |a_{n}| + ∑_{n=1}^{∞} |b_{n}| < ∞. In other words, the algebra A of convolutions by absolutely summable sequences on ℤ embeds into the algebra of operators on the space. (This is not quite a homomorphism because LS ≠ SL, but finite-rank perturbations do not matter to us.) Our theorem gives a space X_{S} for which the shift is an isometry, such that every operator in L(X_{S}) is a strictly singular perturbation of an element of A. It is straightforward to deduce from this that every projection on X_{S} has finite rank or corank. Thus, if Y is a subspace of X_{S} and there is a continuous projection onto Y (such a subspace is called complemented), then Y is finite-codimensional. (Recall that subspaces are infinite-dimensional unless it is otherwise stated.) Since all subspaces of the same finite codimension are isomorphic, the existence of the shift guarantees that every complemented subspace of X_{S} is isomorphic to X_{S}. A space with this property is called prime. Before this example, the only known examples were c_{0} and ℓ_{p} for 1 ≤ p ≤ ∞. Apart from ℓ_{∞}, these were shown to be prime by Pelczynski [P]. Lindenstrauss [L1] proved it for ℓ_{∞}, the only known non-separable example.

The general philosophy here is that, often, the existence of certain operators on a space X implies little more than that the algebra A generated by those operators embeds into L(X), which is obvious anyway. Two more examples are as follows. For every positive integer n ≥ 2 there exists a space X_{n} such that two finite-codimensional subspaces of X_{n} are isomorphic if and only if they have the same codimension modulo n, and there exists a space Z_{n} such that the product spaces Z_{n}^{r} and Z_{n}^{s} are isomorphic if and only if r and s are equal modulo n − 1. For example, there is a space isomorphic to its subspaces of codimension two but not to its hyperplanes, and there is a space isomorphic to its cube but not to its square. For the second class of examples, one obtains algebras A_{n} which resemble the C^{∗}-algebras O_{n}, which were analysed using K-theory by Cuntz [C]. Our proof that the algebras A_{n} do not contain isomorphisms between Z_{n}^{r} and Z_{n}^{s} when r and s are not equal modulo n − 1 was a modification of his argument. Notice that if X ≃ X^{3} but X ≄ X^{2}, then we have an example of two non-isomorphic spaces X and Y such that either embeds into the other as a complemented subspace. This gives a negative answer to the so-called Schroeder-Bernstein problem for Banach spaces, originally solved in [G3]. We have another example if we take the space X_{2} above, which is isomorphic to X_{2} ⊕ ℂ^{2} but not to X_{2} ⊕ ℂ.

The theorem proved in [GM2] gives fairly general circumstances under which the third kind of question from the beginning of this paper has a negative answer. However, there are certain very strong assumptions one can make about a space where our theorem has nothing to say. For example, a famous result of Lindenstrauss and Tzafriri [LT1], solving a problem known as the complemented subspaces problem, states that if every closed subspace of a Banach space is complemented, then the space must be isomorphic to a Hilbert space. If we consider isomorphisms instead of projections, we get the following question of Banach [B]: if X is a space isomorphic to all its (closed infinite-dimensional) subspaces, must X be isomorphic to a Hilbert space? A space isomorphic to all its subspaces is nowadays called homogeneous. Are there examples other than ℓ_{2}?

Szankowski [Sz] generalized Enflo’s solution to the basis problem, by showing that, unless a space is very close to a Hilbert space in a certain technical sense, then it has a subspace without a basis (or even the approximation property). Since we have Mazur’s result that every space has a subspace with a basis, it follows that a homogeneous space must be close to a Hilbert space. On the other hand, Johnson [Jo1] showed that a variant of Tsirelson’s space (the 2-convexified version) has the property that every quotient of every subspace has a basis, but the space does not contain ℓ_{2}.

Until recently, the most powerful result in the positive direction was also due to Johnson [Jo2]. He showed that if X is a homogeneous space, and X and X^{∗} have what is known as the GL-property, a property related to, but much weaker than, having an unconditional basis, then X is isomorphic to ℓ_{2}. Then, early in 1993, Komorowski and Tomczak-Jaegermann proved the following result.

Theorem 2. [KT] Let X be a Banach space of cotype q for some q < ∞. Then either X contains ℓ_{2} or X contains a subspace without an unconditional basis.

It does not matter too much here what it means to be of cotype q. Suffice it to say that, by Szankowski’s result, a homogeneous space is too close to ℓ_{2} to fail the condition of Theorem 2. It follows that a homogeneous space not isomorphic to (and therefore not containing) ℓ_{2} must have a subspace without an unconditional basis. But since the space is homogeneous, it does not even contain an unconditional basic sequence. Another way of saying this is that a homogeneous space with an unconditional basis must be ℓ_{2}. Theorem 2 therefore uses a stronger assumption on the space than Johnson, but the assumption about the dual space is no longer necessary. This theorem of Komorowski and Tomczak-Jaegermann provided a remarkable link between Banach’s homogeneous spaces problem and the unconditional basic sequence problem. Moreover, it was potentially very useful, because the examples that had been constructed of spaces not containing unconditional basic sequences had, as we have seen in some cases, nowhere near enough operators to be homogeneous - quite the reverse!

Let us consider the prime space X_{S} mentioned earlier. It is an easy consequence of the results of [GM2] that it contains no unconditional basic sequence. We also know that it is not hereditarily indecomposable. However, it can be shown that X_{S} has a hereditarily indecomposable subspace. The same is true of the spaces X_{n} and Z_{n}. Could this be a general phenomenon? The answer is yes.

Theorem 3. [G4] Every Banach space X has a subspace Y which either has an unconditional basis or is hereditarily indecomposable.

The solution to Banach’s question is now easy. We have seen that a homogeneous space X not isomorphic to ℓ_{2} contains no unconditional basic sequence. Therefore, by Theorem 3, it contains a hereditarily indecomposable subspace. Hence, as X is homogeneous, X is itself hereditarily indecomposable. But this, in the light of Theorem 1, is a very strong contradiction.

Before we move on, it is worth pointing out that there is no connection between the proofs of Theorem 2 and Theorem 3. It was a historical accident that they were proved in the order that they were. Similarly, Theorem 3 is completely independent of the actual existence of hereditarily indecomposable spaces, except that nobody thought to look at the notion of hereditary indecomposability until an example of a space exhibiting it was produced.

We now return to the word “nice”. The reason one would like to find “nice” subspaces is that one can say more about them than about general spaces. But if we take this as a vague definition of niceness, then Theorem 3 does give us a nice subspace. For, in a sense, we know everything there is to know about a space if we know that it is hereditarily indecomposable. If, on the other hand, it has an unconditional basis, we also have a lot of information about it. This is not just a quibble about words either, since we have given an example of a problem where this kind of niceness gave exactly the sort of control that was needed. From this point of view one would say that the questions that were traditionally asked about nice subspaces were of the wrong kind (although more examples of Banach spaces were needed to make this clear). It is more fruitful to look for theorems such as the last two, where one obtains subspaces with one of two or more very different properties.

Indeed, such theorems already exist, Rosenthal’s ℓ_{1}-theorem [R1] being the most famous example (see also [R2]). It is interesting that infinite Ramsey theory, which can be used to prove Rosenthal’s theorem [F], was also important in the proof of Theorem 3. We now briefly outline a very general and essentially combinatorial result, of which Theorem 3 is an easy consequence. We shall need a small amount of notation.

Given a space X with a fixed basis, let Σ = Σ(X) denote the set of all finite sequences x_{1}, …, x_{n} of vectors of norm at most 1, such that x_{1} < … < x_{n}. (The meaning of “<” was given earlier.) Given a subset σ ⊂ Σ, let us say that it is large if every block subspace Y (still infinite-dimensional) contains some sequence in σ.

Every set σ ⊂ Σ defines a two-player game as follows. The first player, S, chooses a block subspace X_{1} of X, then the second player, P, chooses a point x_{1} ∈ X_{1} of finite support and norm at most 1. Then S chooses X_{2} and P chooses x_{2} ∈ X_{2} of norm at most 1 such that x_{1} < x_{2}. They continue in this way, alternately picking subspaces and points in them. The game is won by P if at some stage the sequence (x_{1}, …, x_{n}) is in σ. If it goes on for ever without this happening, then S wins. Clearly, if P is to have any chance of winning, then σ must at least be large, since otherwise S could repeatedly choose the same subspace containing no sequence in σ. In fact, saying that P has a winning strategy for the set σ is a much stronger statement than saying that σ is large, and that is the point of the next theorem. If Δ = (δ_{1}, δ_{2}, …) is a sequence of positive reals and σ ⊂ Σ, we will write σ_{Δ} for the set of sequences (x_{1}, …, x_{n}) ∈ Σ “within Δ of σ”; that is, such that there exists (y_{1}, …, y_{n}) ∈ σ with ∥x_{i} − y_{i}∥ ≤ δ_{i} for 1 ≤ i ≤ n.

Theorem 4. [G4] Let X be a Banach space with a basis and let σ be a large subset of Σ(X). Then, for every positive sequence Δ = (δ_{1}, δ_{2}, …) there exists a block subspace Y ⊂ X such that, if S and P play the above game in the space Y, then P has a winning strategy for obtaining sequences in σ_{Δ}.

Ignoring perturbations, this says that if σ is large, then P has a winning strategy in some subspace. The proof of the theorem (and of the main result of [G5] which we shall mention in a moment) is related to arguments due to Galvin and Prikry [GP] and Ellentuck [E] concerning infinite versions of Ramsey’s theorem.

Now one of the remarks earlier was that if X contains no unconditional basic sequence, then for every C the set σ of sequences x_{1} < … < x_{n} with the property that ∥∑_{i=1}^{n} x_{i}∥ > C ∥∑_{i=1}^{n}(−1)^{i}x_{i}∥ is large. Theorem 4 allows us to drop to a subspace Y in which P has a winning strategy for finding such sequences (say with C replaced by C/2). Suppose S plays a strategy which simply involves alternating between two subspaces W and Z of Y. Then P’s strategy guarantees the existence of one of these sequences with the odd-numbered x_{i}s in W and the even-numbered ones in Z. Letting w and z be the sums of the odd-numbered and even numbered vectors respectively, we have the inequality ∥z + w∥ > C ∥z − w∥. If C is large, it follows easily that z/ ∥z∥ and w/ ∥w∥ are close.

We have not quite managed to find arbitrarily close unit vectors in Z and W, since we made a fixed choice of C at the beginning of the argument above. However, one can repeat it for a sequence of C_{n} tending to infinity and obtain a nested sequence Y_{1} ⊃ Y_{2} ⊃ … of subspaces such that for each n the argument works for C_{n} in the subspace Y_{n}. It is then easy to check that a “diagonal” subspace generated by a block basis y_{1}, y_{2}, … with y_{n} ∈ Y_{n} is hereditarily indecomposable. Thus, Theorem 3 follows from Theorem 4. An adaptation of this proof which is in some ways more direct was found by Maurey and appears in [M].

It is noticeable that, from the point of view of the general theory of Banach spaces, certain examples of spaces appear to be more natural than others. The property that distinguishes one of these natural examples is that every subspace has a further subspace which is not interestingly different from the whole space. For example, every block subspace of Tsirelson’s space mentioned earlier turns out to be a “Tsirelson-type” space (this statement can be made quite precise), Schlumprecht’s space has the very strong property that every subspace has a further subspace not only isomorphic to the whole space but complemented inside it (see [S2]) and every subspace of the hereditarily indecomposable space X_{1} has a further subspace which, although definitely not isomorphic to X_{1}, can be described in an almost identical way. Of the classical spaces, only c_{0} and the ℓ_{p}-spaces are natural in this sense, since every classical space contains one of them. The spaces X_{S}, X_{n} and Z_{n} (for n ≥ 2) are good examples of unnatural spaces because they lose what little structure they have when one passes to an appropriate subspace (recall that they have hereditarily indecomposable subspaces).

It is possible that a precise definition of “natural” will emerge, but even without it, one can attempt to classify the natural spaces. Theorem 3 shows that they either have an unconditional basis or are hereditarily indecomposable. In [G5], a generalization is presented of Theorem 4 to analytic sets of infinite sequences (in a certain sensible topology), and more information is given about natural spaces with an unconditional basis. Loosely speaking, there is a theorem which states that such a space either has many isomorphisms between its subspaces or has none that do not follow trivially from the existence of the unconditional basis. The space X_{U} mentioned earlier is an example where the second possibility holds.

From another point of view the non-classical “natural” spaces mentioned above are extremely unnatural. They all have Tsirelson-type inductive definitions, as opposed to formulae for their norms. Given a vector with support of size n, it does not even seem to be possible to calculate its norm in one of these spaces in a time polynomial in n. It is tempting to ask whether there is a meta-theorem which states that a norm which is in some sense directly defined must give a space which has some ℓ_{p}-space or c_{0} as a subspace. At the moment there is not even a precise conjecture along these lines, but it could be that there is, waiting to be discovered, a theory of “easily described” Banach spaces and linear operators very different indeed from the theory of general spaces as outlined in this paper.

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