aa r X i v : . [ m a t h . L O ] A p r INDISCERNIBLE SUBSPACES AND MINIMAL WIDE TYPES
JAMES HANSON
Abstract.
We develop the machinery of indiscernible subspaces in continu-ous theories of expansions of Banach spaces, showing that any such theory hasan indiscernible subspace and therefore an indiscernible set. We extend a re-sult of Shelah and Usvyatsov [14] by showing that a sequence of realizations ofa (possibly unstable) minimal wide type p is a Morley sequence in p if and onlyif it is the orthonormal basis of an indiscernible subspace in p . We also give anexample showing that minimal wide types do not generally have type-definableindiscernible subspaces (answering a question of Shelah and Usvyatsov [14]),as well as an example showing that our result fails for non-minimal wide types,even in ω -stable theories. Introduction
Continuous first-order logic is a generalization of first-order logic suitable forstudying metric structures, which are mathematical structures with an underlyingcomplete metric and with uniformly continuous functions and R -valued predicates.A rich class of metric structures arise from Banach spaces and expansions thereof.An active area of research in continuous logic is the characterization of inseparablycategorical continuous theories. For a general introduction to continuous logic, see[2].In the present work we will consider expansions of Banach spaces. We introducethe notion of an indiscernible subspace. An indiscernible subspace is a subspacein which types of tuples of elements only depend on their quantifier-free type inthe reduct consisting of only the metric and the constant . Similarly to indis-cernible sequences, indiscernible subspaces are always consistent with a Banachtheory (with no stability assumption, see Theorem 2.3), but are not always presentin every model. We will show that an indiscernible subspace always takes the formof an isometrically embedded real Hilbert space wherein the type of any tuple onlydepends on its quantifier-free type in the Hilbert space. The notion of an indis-cernible subspace is of independent interest in the model theory of Banach andHilbert structures, and in particular here we use it to improve the results of Shelahand Usvyatsov in the context of types in the full language (as opposed to ∆ -types).Specifically, in this context we give a shorter proof of Shelah and Usvyatsov’s mainresult [14, Prop. 4.13], we improve their result on the strong uniqueness of Morleysequences in minimal wide types [14, Prop. 4.12], and we expand on their com-mentary on the “induced structure” of the span of a Morley sequence in a minimalwide type [14, Rem. 5.6]. This more restricted case is what is relevant to insepara-bly categorical Banach theories, so our work is applicable to the problem of theircharacterization.Finally, we present some relevant counterexamples and in particular we resolve(in the negative) the question of Shelah and Usvyatsov presented at the end of Section 5 of [14], in which they ask whether or not the span of a Morley sequencein a minimal wide type is always a type-definable set.1.1.
Background.
For K ∈ { R , C } , we think of a K -Banach space X as being ametric structure X whose underlying set is the closed unit ball B ( X ) of X withmetric d ( x, y ) = k x − y k . This structure is taken to have for each tuple ¯ a ∈ K an | ¯ a | -ary predicate s ¯ a (¯ x ) = (cid:13)(cid:13)(cid:13)P i< | ¯ a | a i x i (cid:13)(cid:13)(cid:13) , although we will always write this inthe more standard form. Note that we evaluate this in X even if P i< | ¯ a | a i x i is notactually an element of the structure X . For convenience, we will also have a constantfor the zero vector, , and an n -ary function σ ¯ a (¯ x ) such that σ ¯ a (¯ x ) = P i< | ¯ a | a i x i if it is in B ( X ) and σ ¯ a (¯ x ) = P i< | ¯ a | a i x i k P i< | ¯ a | a i x i k otherwise. If | a | ≤ , we will write ax for σ a ( x ) . Note that while this is an uncountable language, it is interdefinable witha countable reduct of it (restricting attention to rational elements of K ). Thesestructures capture the typical meaning of the ultraproduct of Banach spaces. As iscommon, we will conflate X and the metric structure X in which we have encoded X . Definition 1.1. A Banach (or Hilbert) structure is a metric structure which isthe expansion of a Banach (or Hilbert) space. A
Banach (or Hilbert) theory is thetheory of such a structure. The adjectives real and complex refer to the scalar field K . C ∗ - and other Banach algebras are commonly studied examples of Banach struc-tures that are not just Banach spaces.A central problem in continuous logic is the characterization of inseparably cat-egorical countable theories, that is to say countable theories with a unique modelin each uncountable density character. The analog of Morley’s theorem was shownin continuous logic via related formalisms [1, 13], but no satisfactory analog of theBaldwin-Lachlan theorem or its precise structural characterization of uncountablycategorical discrete theories in terms of strongly minimal sets is known. Someprogress in the specific case of Banach theories has been made in [14], in whichShelah and Usvyatsov introduce the notion of a wide type and the notion of a min-imal wide type, which they argue is the correct analog of strongly minimal typesin the context of inseparably categorical Banach theories. Definition 1.2.
A type p in a Banach theory is wide if its set of realizationsconsistently contain the unit sphere of an infinite dimensional real subspace.A type is minimal wide if it is wide and has a unique wide extension to everyset of parameters.In [14], Shelah and Usvyatsov were able to show that every Banach theory haswide complete types using the following classical concentration of measure resultsof Dvoretzky and Milman, which Shelah and Usvyatsov refer to as the Dvoretzky-Milman theorem. Fact 1.3 (Dvoretzky-Milman theorem) . Let ( X, k·k ) be an infinite dimensionalreal Banach space with unit sphere S and let f : S → R be a uniformly continuousfunction. For any k < ω and ε > , there exists a k -dimensional subspace Y ⊂ X For another equivalent approach, see [2], which encodes Banach structures as many-sortedmetric structures with balls of various radii as different sorts.
NDISCERNIBLE SUBSPACES AND MINIMAL WIDE TYPES 3 and a Euclidean norm |||·||| on Y such that for any a, b ∈ S ∩ Y , we have ||| a ||| ≤k a k ≤ (1 + ε ) ||| a ||| and | f ( a ) − f ( b ) | < ε . Shelah and Usvyatsov showed that in a stable Banach theory every wide typehas a minimal wide extension (possibly over a larger set of parameters) and thatevery Morley sequence in a minimal wide type is an orthonormal basis of a subspaceisometric to a real Hilbert space. Furthermore, they showed that in an inseparablycategorical Banach theory, every inseparable model is prime over a countable setof parameters and a Morley sequence in some minimal wide type, analogously tohow a model of a discrete uncountably categorical theory is always prime over somefinite set of parameters and a Morley sequence in some strongly minimal type.The key ingredient to our present work is the following result, due to Milman.It extends the Dvoretzky-Milman theorem in a manner analogous to the extensionof the pigeonhole principle by Ramsey’s theorem. Definition 1.4.
Let ( X, k·k ) be a Banach space. If a , a , . . . , a n − and b , b , . . . ,b n − are ordered n -tuples of elements of X , we say that ¯ a and ¯ b are congruent if k a i − a j k = k b i − b j k for all i, j ≤ n , where we take a n = b n = . We will writethis as ¯ a ∼ = ¯ b . Fact 1.5 ([12], Thm. 3) . Let S ∞ be the unit sphere of a separable infinite di-mensional real Hilbert space H and let f : ( S ∞ ) n → R be a uniformly continu-ous function. For any ε > and any k < ω there exists a k -dimensional sub-space V of H such that for any a , a , . . . , a n − , b , b , . . . , b n − ∈ S ∞ with ¯ a ∼ = ¯ b , | f (¯ a ) − f (¯ b ) | < ε . Note that the analogous result for inseparable Hilbert spaces follows immediately,by restricting attention to a separable infinite dimensional subspace. Also note thatby using Dvoretsky’s theorem and an easy compactness argument, Fact 1.5 can begeneralized to arbitrary infinite dimensional Banach spaces.1.2.
Connection to Extreme Amenability.
A modern proof of Fact 1.5 wouldgo through the extreme amenability of the unitary group of an infinite dimensionalHilbert space endowed with the strong operator topology, or in other words thefact that any continuous action of this group on a compact Hausdorff space has afixed point, which was originally shown in [5]. This connection is unsurprising. It iswell known that the extreme amenability of
Aut( Q ) (endowed with the topology ofpointwise convergence) can be understood as a restatement of Ramsey’s theorem.It is possible to use this to give a high brow proof of the existence of indiscerniblesequences in any first-order theory T : Proof.
Fix a first-order theory T . Let Q be a family of variables indexed by therational numbers. The natural action of Aut( Q ) on S Q ( T ) , the Stone space of typesover T in the variables Q , is continuous and so by extreme amenability has a fixedpoint. A fixed point of this action is precisely the same thing as the type of a A norm |||·||| is Euclidean if it satisfies the parallelogram law, ||| a ||| + 2 ||| b ||| = ||| a + b ||| + ||| a − b ||| , or equivalently if it is induced by an inner product. Fact 1.3 without f is (a form of) Dvoretsky’s theorem. The original Dvoretzky-Milman result is often compared to Ramsey’s theorem, such as whenGromov coined the term the Ramsey-Dvoretzky-Milman phenomenon [4], but in the context ofFact 1.5 it is hard not to think of the n = 1 case as being analogous to the pigeonhole principleand the n > cases as being analogous to Ramsey’s theorem. JAMES HANSON Q -indexed indiscernible sequence over T , and so we get that there are models of T with indiscernible sequences. (cid:3) A similar proof of the existence of indiscernible subspaces in Banach theories(Theorem 2.3) is possible, but requires an argument that the analog of S Q ( T ) is non-empty (which follows from Dvoretzky’s theorem) and also requires moredelicate bookkeeping to define the analog of S Q ( T ) and to show that the actionof the unitary group of a separable Hilbert space is continuous. In the end this ismore technical than a proof using Fact 1.5 directly.2. Indiscernible Subspaces
Definition 2.1.
Let T be a Banach theory. Let M | = T and let A ⊆ M be someset of parameters. An indiscernible subspace over A is a real subspace V of M suchthat for any n < ω and any n -tuples ¯ b, ¯ c ∈ V , ¯ b ≡ A ¯ c if and only if ¯ b ∼ = ¯ c .If p is a type over A , then V is an indiscernible subspace in p (over A ) if it is anindiscernible subspace over A and b | = p for all b ∈ V with k b k = 1 .Note that an indiscernible subspace is a real subspace even if T is a complex Ba-nach theory. Also note that an indiscernible subspace in p is not literally containedin the realizations of p , but rather has its unit sphere contained in the realizationsof p . It might be more accurate to talk about “indiscernible spheres,” but we findthe subspace terminology more familiar.Indiscernible subspaces are very metrically regular. Proposition 2.2.
Suppose V is an indiscernible subspace in some Banach struc-ture. Then V is isometric to a real Hilbert space.In particular, a real subspace V of a Banach structure is indiscernible over A ifand only if it is isometric to a real Hilbert space and for every n < ω and everypair of n -tuples ¯ b, ¯ c ∈ V , ¯ b ≡ A ¯ c if and only if for all i, j < n h b i , b j i = h c i , c j i .Proof. For any real Banach space W , if dim W ≤ , then W is necessarily isometricto a real Hilbert space. If dim V ≥ , let V be a -dimensional subspace of V . Asubspace of an indiscernible subspace is automatically an indiscernible subspace, so V is indiscernible. For any two distinct unit vectors a and b , indiscernibility impliesthat for any r, s ∈ R , k ra + sb k = k sa + rb k , hence the unique linear map thatswitches a and b fixes k·k . This implies that the automorphism group of ( V , k·k ) istransitive on the k·k -unit circle. By John’s theorem on maximal ellipsoids [8], theunit ball of k·k must be an ellipse, so k·k is a Euclidean norm.Thus every -dimensional real subspace of V is Euclidean and so ( V, k·k ) satisfiesthe parallelogram law and is therefore a real Hilbert space.The ‘in particular’ statement follows from the fact that in a real Hilbert subspaceof a Banach space, the polarization identity [3, Prop. 14.1.2] defines the innerproduct in terms of a particular quantifier-free formula: h x, y i = 14 (cid:16) k x + y k − k x − y k (cid:17) . (cid:3) There is also a polarization identity for the complex inner product: h x, y i C = 14 (cid:16) k x + y k − k x − y k + i k x − iy k − i k x + iy k (cid:17) . NDISCERNIBLE SUBSPACES AND MINIMAL WIDE TYPES 5
Existence of Indiscernible Subspaces.
As mentioned in [14, Cor. 3.9], itfollows from Dvoretzky’s theorem that if p is a wide type and M is a sufficientlysaturated model, then p ( M ) contains the unit sphere of an infinite dimensionalsubspace isometric to a Hilbert space. We refine this by showing that, in fact, anindiscernible subspace can be found. Theorem 2.3.
Let A be a set of parameters in a Banach theory T and let p bea wide type over A . For any κ , there is M | = T and a subspace V ⊆ M ofdimension κ such that V is an indiscernible subspace in p over A . In particular,any ℵ + κ + | A | -saturated M will have such a subspace.Proof. For any set ∆ of A -formulas, call a subspace V of a model N of T A ∆ -indiscernible in p if every unit vector in V models p and for any n < ω and anyformula ϕ ∈ ∆ of arity n and any n -tuples ¯ b, ¯ c ∈ V with ¯ b ∼ = ¯ c , we have N | = ϕ (¯ b ) = ϕ (¯ c ) .Since p is wide, there is a model N | = T containing an infinite dimensionalsubspace W isometric to a real Hilbert space such that for all b ∈ W with k b k = 1 , b | = p . This is an infinite dimensional ∅ -indiscernible subspace in p .Now for any finite set of A -formulas ∆ and formula ϕ , assume that we’ve shownthat there is a model N | = T containing an infinite dimensional ∆ -indiscerniblesubspace V in p over A . We want to show that there is a ∆ ∪ { φ } -indiscerniblesubspace in V . By Fact 1.5, for every k < ω there is a k -dimensional subspace W k ⊆ V such that for any unit vectors b , . . . , b ℓ − , c , . . . , c ℓ − in W k with ¯ b ∼ = ¯ c ,we have that | ϕ N (¯ b ) − ϕ N (¯ c ) | < − k . If we let N k = ( N k , W k ) where we’ve expandedthe language by a fresh predicate symbol D such that D N k ( x ) = d ( x, W k ) , then anultraproduct of the sequence N k will be a structure ( N ω , W ω ) in which W ω is aninfinite dimensional Hilbert space. Claim: W ω is ∆ ∪ { ϕ } -indiscernible in p . Proof of claim.
Fix an m -ary formula ψ ∈ ∆ ∪ { ϕ } and let f ( k ) = 0 if ψ ∈ ∆ and f ( k ) = 2 − k if ψ = ϕ . For any k ≥ m , fix b , . . . , b m − , c , . . . , c m − in theunit ball of W k , there is a m dimensional subspace W ′ ⊆ W k containing ¯ b, ¯ c .By compactness of B ( W ′ ) m (where B ( X ) is the unit ball of X ), we have thatfor any ε > there is a δ ( ε ) > such that if | h b i , b j i − h c i , c j i | < δ ( ε ) for all i, j < m then | ψ N (¯ b ) − ψ N (¯ c ) | ≤ f ( k ) + ε . Note that we can take the function δ to only depend on ψ , specifically its arity and modulus of continuity, and not on k , since B ( W ′ ) m is always isometric to B ( R m ) m . Therefore, in the ultraproductwe will have ( ∀ i, j < m ) | h b i , b j i − h c i , c j i | < δ ( ε ) ⇒ | ψ N (¯ b ) − ψ N (¯ c ) | ≤ ε and thus ¯ b ∼ = ¯ c ⇒ ψ N ω (¯ b ) = ψ N ω (¯ c ) , as required. (cid:3) Claim
Now for each finite set of A -formulas we’ve shown that there’s a structure ( M ∆ , V ∆ ) (where, again, V ∆ is the set defined by the new predicate symbol D )such that M ∆ | = T A and V ∆ is an infinite dimensional ∆ -indiscernible subspacein p . By taking an ultraproduct with an appropriate ultrafilter we get a struc-ture ( M , V ) where M | = T A and V is an infinite dimensional subspace. V is anindiscernible subspace in p over A by the same argument as in the claim.Finally note that by compactness we can take V to have arbitrarily large di-mension and that any subspace of an indiscernible subspace in p over A is anindiscernible subspace in p over A , so we get the required result. (cid:3) Together with the fact that wide types always exist in Banach theories withinfinite dimensional models [14, Thm. 3.7], we get a corollary.
JAMES HANSON
Corollary 2.4.
Every Banach theory with infinite dimensional models has an in-finite dimensional indiscernible subspace in some model. In particular, every suchtheory has an infinite indiscernible set, namely any orthonormal basis of an infinitedimensional indiscernible subspace. Minimal Wide Types
Compare the following Theorem 3.1 with this fact in discrete logic: If p is aminimal type (i.e. p has a unique global non-algebraic extension), then an infi-nite sequence of realizations of p is a Morley sequence in p if and only if it is anindiscernible sequence.Here we are using the definition of Morley sequence for (possibly unstable) A -invariant types: Let p be a global A -invariant type, and let B ⊇ A be some set ofparameters. A sequence { c i } i<κ is a Morley sequence in p over B if for all i < κ , tp( c i /Bc
Let p be a minimal wide type over the set A . For κ ≥ ℵ , a set ofrealizations { b i } i<κ of p is a Morley sequence in (the unique global minimal wideextension of ) p if and only if it is an orthonormal basis of an indiscernible subspacein p over A .Proof. All we need to show is that an orthonormal basis of an indiscernible subspacein p over A is a Morley sequence in p . The converse will follow from the fact that allMorley sequences in a fixed invariant type of the same length have the same typealong with the fact that minimal wide types have a unique global wide extension,which is therefore invariant.Let V be an indiscernible subspace in p over A . Let { e i } i<κ be an orthonor-mal basis of V . By construction, tp( e /A ) = p . Let q be the global minimalwide extension of p . Assume that for some j < κ we’ve shown for all i < j that tp( e i /Ae
Question 3.2. If p is a minimal wide type over the set A , is it stable in the senseof [14, Def. 4.1] ? In other words, is every type q extending p over a model M ⊇ A a definable type? Counterexamples
Here we collect some counterexamples that may be relevant to any model theo-retic development of the ideas presented in this paper.4.1.
No Infinitary Ramsey-Dvoretzky-Milman Phenomena in General.
Unfortunately some elements of the analogy between the Ramsey-Dvoretzky-MilmanPhenomenon and discrete Ramsey theory do not work. In particular, there is noextension of Dvoretzky’s theorem, and therefore Fact 1.3, to k ≥ ω , even for a fixed ε > . Recall that a linear map T : X → Y between Banach spaces is an isomor-phism if it is a continuous bijection. This is enough to imply that T is invertible and NDISCERNIBLE SUBSPACES AND MINIMAL WIDE TYPES 7 that both T and T − are Lipschitz. An analog of Dvoretzky’s theorem for k ≥ ω would imply that every sufficiently large Banach space has an infinite dimensionalsubspace isomorphic to Hilbert space, which is known to be false. Here we will seeas specific example of this.The following is a well known result in Banach space theory (for a proof see thecomment after Proposition 2.a.2 in [10]). Fact 4.1.
For any distinct
X, Y ∈ { ℓ p : 1 ≤ p < ∞} ∪ { c } , no subspace of X isisomorphic to Y . Note that, whereas Corollary 2.4 says that every Banach theory is consistent withthe partial type of an indiscernible subspace, the following corollary says that thistype can sometimes be omitted in arbitrarily large models (contrast this with thefact that the existence of an Erdös cardinal implies that you can find indiscerniblesequences in any sufficiently large structure in a countable language [9, Thm. 9.3]).
Corollary 4.2.
For p ∈ [1 , ∞ ) \ { } , there are arbitrarily large models of Th( ℓ p ) that do not contain any infinite dimensional subspaces isomorphic to a Hilbert space.Proof. Fix p ∈ [1 , ∞ ) \ { } and κ ≥ ℵ . Let ℓ p ( κ ) be the Banach space of functions f : κ → R such that P i<κ | f ( i ) | p < ∞ . Note that ℓ p ( κ ) ≡ ℓ p . Pick a subspace V ⊆ ℓ p ( κ ) . If V is isomorphic to a Hilbert space, then any separable V ⊆ V willalso be isomorphic to a Hilbert space. There exists a countable set A ⊆ κ such that V ⊆ ℓ p ( A ) ⊆ ℓ p ( κ ) . By Fact 4.1, V is not isomorphic to a Hilbert space, which isa contradiction. Thus no such V can exist. (cid:3) Even assuming we start with a Hilbert space we do not get an analog of theinfinitary pigeonhole principle (i.e. a generalization of Fact 1.3). The discussion byHájeck and Matěj in [6, after Thm. 1] of a result of Maurey [11] implies that thereis a Hilbert theory T with a unary predicate P such that for some ε > thereare arbitrarily large models M of T such that for any infinite dimensional subspace V ⊆ M there are unit vectors a, b ∈ V with | P M ( a ) − P M ( b ) | ≥ ε .Stability of a theory often has the effect of making Ramsey phenomena moreprevalent in its models, so there is a natural question as to whether anything similarwill happen here. Recall that a function f : S ( X ) → R on the unit sphere S ( X ) of a Banach space X is oscillation stable if for every infinite dimensional subspace Y ⊆ X and every ε > there is an infinite dimensional subspace Z ⊆ Y such thatfor any a, b ∈ S ( Z ) , | f ( a ) − f ( b ) | ≤ ε . Question 4.3.
Does (model theoretic) stability imply oscillation stability? That isto say, if T is a stable Banach theory, is every unary formula oscillation stable onmodels of T ? The (Type-)Definability of Indiscernible Subspaces and Complex Ba-nach Structures.
A central question in the study of inseparably categorical Ba-nach space theories is the degree of definability of the ‘minimal Hilbert space’ that To see this, we can find an elementary sub-structure of ℓ p ( κ ) that is isomorphic to ℓ p : Let L be a separable elementary sub-structure of ℓ p ( κ ) . For each i < ω , given L i , let B i be the setof all f ∈ ℓ p ( κ ) that are the indicator function of a singleton { i } for some i in the support ofsome element of L i . B i is countable. Let L i +1 be a separable elementary sub-structure of ℓ p ( κ ) containing L i ∪ B i . S i<ω L i +1 is equal to the span of S i<ω B i and so is a separable elementarysub-structure of ℓ p ( κ ) isomorphic to ℓ p . JAMES HANSON controls a given inseparable model of the theory. Results of Henson and Raynaud in[7] imply that in general the Hilbert space may not be definable. In [14], Shelah andUsvyatsov ask whether or not the Hilbert space can be taken to be type-definableor a zeroset. In Example 4.5 we present a simple, but hopefully clarifying, exampleshowing that this is slightly too much to ask.It is somewhat uncomfortable that even in complex Hilbert structures we areonly thinking about real indiscernible subspaces rather than complex indiscerniblesubspaces. One problem is that Ramsey-Dvoretzky-Milman phenomena only dealwith real subspaces in general. The other problem is that Definition 1.4 is incom-patible with complex structure:
Proposition 4.4.
Let T be a complex Banach theory. Let V be an indiscerniblesubspace in some model of T . For any non-zero a ∈ V and λ ∈ C \ { } , if λa ∈ V ,then λ ∈ R .Proof. Assume that for some non-zero vector a , both a and ia are in V . We havethat ( a, ia ) ≡ ( ia, a ) , but ( a, ia ) | = d ( ix, y ) = 0 and ( ia, a ) = d ( ix, y ) = 0 , whichcontradicts indiscernibility. Therefore we cannot have that both a and ia are in V . The same statement for a and λa with λ ∈ C \ R follows immediately, since a, λa ∈ V ⇒ ia ∈ V . (cid:3) In the case of complex Hilbert space and other Hilbert spaces with a unitaryLie group action, this is the reason that indiscernible subspaces can fail to be type-definable. We will explicitly give the simplest example of this.
Example 4.5.
Let T be the theory of an infinite dimensional complex Hilbert spaceand let C be the monster model of T . T is inseparably categorical, but for any partialtype Σ over any small set of parameters A , Σ( C ) is not an infinite dimensionalindiscernible subspace (over ∅ ).Proof. T is clearly inseparably categorical by the same reasoning that the theoryof real infinite dimensional Hilbert spaces is inseparably categorical (being an infi-nite dimensional complex Hilbert space is first-order and there is a unique infinitedimensional complex Hilbert space of each infinite density character).If Σ( C ) is not an infinite dimensional subspace of C , then we are done, so as-sume that Σ( C ) is an infinite dimensional subspace of C . Let N be a small modelcontaining A . Since N is a subspace of C , Σ( N ) = Σ( C ) ∩ N is a subspace of N .Let v ∈ Σ( C ) \ Σ( N ) . This implies that v ∈ C \ N , so we can write v as v k + v ⊥ ,where v k is the orthogonal projection of v onto N and v ⊥ is complex orthogonalto N . Necessarily we have that v ⊥ = 0 . Let N ⊥ be the orthocomplement of N in C . If we write elements of C as ( x, y ) with x ∈ N and y ∈ N ⊥ , then the maps ( x, y ) ( x, − y ) , ( x, y ) ( x, iy ) , and ( x, y ) ( x, − iy ) are automorphisms of C fixing N . Therefore ( v k + v ⊥ ) ≡ N ( v k − v ⊥ ) ≡ N ( v k + iv ⊥ ) ≡ N ( v k − iv ⊥ ) , so wemust have that ( v k − v ⊥ ) , ( v k + iv ⊥ ) , ( v k − iv ⊥ ) ∈ Σ( C ) as well. Since Σ( C ) is asubspace, we have that b ⊥ ∈ Σ( C ) and ib ⊥ ∈ Σ( C ) . Thus by Proposition 4.4 Σ( C ) is not an indiscernible subspace over ∅ . (cid:3) This example is a special case of this more general construction: If G is a compactLie group with an irreducible unitary representation on R n for some n (i.e. thegroup action is transitive on the unit sphere), then we can extend this action to ℓ by taking the Hilbert space direct sum of countably many copies of the irreducibleunitary representation of G , and we can think of this as a structure by adding NDISCERNIBLE SUBSPACES AND MINIMAL WIDE TYPES 9 function symbols for the elements of G . The theory of this structure will be totallycategorical and satisfy the conclusion of Example 4.5.Example 4.5 is analogous to the fact that in many strongly minimal theoriesthe set of generic elements in a model is not itself a basis/Morley sequence. Theimmediate response would be to ask the question of whether or not the unit sphereof the complex linear span (or more generally the ‘ G -linear span,’ i.e. the linearspan of G · V ) of the indiscernible subspace in a minimal wide type agrees with theset of realizations of that minimal wide type, but this can overshoot: Example 4.6.
Consider the structure whose universe is (the unit ball of ) ℓ ⊕ ℓ (where we are taking ℓ as a real Hilbert space), with a complex action ( x, y ) ( − y, x ) and orthogonal projections P and P for the sets ℓ ⊕ { } and { } ⊕ ℓ ,respectively. Let T be the theory of this structure. This is a totally categoricalcomplex Hilbert structure, but for any complete type p and M | = T , p ( M ) does notcontain the unit sphere of a non-trivial complex subspace.Proof. T is bi-interpretable with a real Hilbert space, so it is totally categorical.For any complete type p , there are unique values of k P ( x ) k and k P ( x ) k thatare consistent with p , so the set of realizations of p in any model cannot contain { λa } λ ∈ U(1) for a , a unit vector, and U(1) ⊂ C , the set of unit complex numbers. (cid:3) The issue, of course, being that, while we declared by fiat that this is a complexHilbert structure, the expanded structure does not respect the complex structure.So, on the one hand, Example 4.6 shows that in general the unit sphere of thecomplex span won’t be contained in the minimal wide type. On the other hand,a priori the set of realizations of the minimal wide type could contain more thanjust the unit sphere of the complex span, such as if we have an
SU( n ) action. Thecomplex (or G -linear) span of a set is of course part of the algebraic closure of theset in question, so this suggests a small refinement of the original question of Shelahand Usvyatsov: Question 4.7. If T is an inseparably categorical Banach theory, p is a minimalwide type, and M is a model of T which is prime over an indiscernible subspace V in p , does it follow that p ( M ) is the unit sphere of a subspace contained in thealgebraic closure of V ? This would be analogous to the statement that if p is a strongly minimal type inan uncountably categorical discrete theory and M is a model prime over a Morleysequence I in p , then p ( M ) ⊆ acl( I ) .4.3. Non-Minimal Wide Types.
The following example shows, unsurprisingly,that Theorem 3.1 does not hold for non-minimal wide types.
Example 4.8.
Let T be the theory of (the unit ball of ) the infinite Hilbert spacesum ℓ ⊕ ℓ ⊕ . . . , where we add a predicate D that is the distance to S ∞ ⊔ S ∞ ⊔ . . . ,where S ∞ is the unit sphere of the corresponding copy of ℓ . This theory is ω -stable.The partial type { D = 0 } has a unique global non-forking extension p that is wide,but the unit sphere of the linear span of any Morley sequence in p is not containedin p ( C ) .Proof. This follows from the fact that on D the equivalence relation ‘ x and y arecontained in a common unit sphere’ is definable by a formula, namely E ( x, y ) = inf z,w ∈ D ( d ( x, z ) ·−
1) + ( d ( z, w ) ·−
1) + ( d ( w, y ) ·− , where a ·− b = max { a − b, } . If x, y are in the same sphere, then let S be agreat circle passing through x and y and choose z and w evenly spaced along theshorter path of S . It will always hold that d ( x, z ) , d ( z, w ) , d ( w, y ) ≤ , so we willhave E ( x, y ) = 0 . On the other hand, if x and y are in different spheres, then E ( x, y ) = √ − .Therefore a Morley sequence in p is just any sequence of elements of D whichare pairwise non- E -equivalent and the unit sphere of the span of any such set isclearly not contained in D . (cid:3) References [1] I. Ben Yaacov. Uncountable dense categoricity in cats.
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