HILBERT SPACES WITH GENERIC PREDICATES
ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
Abstract. We study themodel theory of expansions of Hilbert spaces by generic
predicates. We first prove the existence of model companions for generic ex-
pansions of Hilbert spaces in the form first of a distance function to a random
substructure, then a distance to a random subset. The theory obtained with the
random substructure isω-stable, while the one obtained with the distance to a
random subset is TP2 and NSOP1. That example is the first continuous struc-
ture in that class.
1. Introduction
This paper deals with Hilbert spaces expanded with random predicates in the
framework of continuous logic as developed in [2]. The model theory of Hilbert
spaces is very well understood, see [2, Chapter 15] or [5]. However, we briefly
review some of its properties at the end of this section.
In this paper we build several new expansions, by various kinds of random
predicates (random substructure and the distance to a random subset) of Hilbert
spaces, and study them within the framework of continuous logic. While our
constructions are not exactlymetric Fraïssé (failing the hereditaryproperty), some
of them are indeed amalgamation classes and we study the model theory of their
limits.
Several papers deal with generic expansions of Hilbert spaces. Ben Yaacov,
Usvyatsov and Zadka [3] studied the expansion of a Hilbert space with a generic
This workwas partially supported by Colciencias grantMétodos de Estabilidad en Clases No Esta-
bles. The second and third author were also sponsored by Catalonia’s Centre de Recerca Matemàtica
(Intensive Research Program in Strong Logics) and by the University of Helsinki in 2016 for part
of this work. The third author was also partially sponsored by Colciencias (Proy. 1101-05-13605
CT-210-2003).
http://arxiv.org/abs/0704.1633v2
2 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
automorphism. The models of this theory are expansions of Hilbert spaces with a
unitary map whose spectrum is S1. A model of this theory can be constructed by
amalgamating together the collection ofn-dimensional Hilbert spaces with a uni-
tary map whose eigenvalues are the n-th roots of unity as n varies in the positive
integers. More work on generic automorphisms can be found in [4], where the
first author of this paper studied Hilbert spaces expanded with a random group
of automorphisms G.
There are also several papers about expansions of Hilbert spaces with random
subspaces. In [5] the first author and Buechler identified the saturated models of
the theory of beautiful pairs of a Hilbert space. An analysis of lovely pairs (the
generalization of beautiful pairs (belles paires) to simple theories) in the setting of
compact abstract theories is carried out in [1]. In the (very short) second section of
this paper we build the beautiful pairs of Hilbert spaces as the model companion
of the theory of Hilbert spaces with an orthonormal projection. We provide an
axiomatization for this class and we show that the projection operator into the
subspace is interdefinable with a predicate for the distance to the subspace. We
also prove that the theory of beautiful pairs of Hilbert spaces is ω-stable. Many
of the properties of beautiful pairs of Hilbert spaces are known from the literature
or folklore, so this section is mostly a compilation of results.
In the third section we add a predicate for the distance to a random subset. This
construction was inspired by the idea of finding an analogue to the first order
generic predicates studied by Chatzidakis and Pillay in [7]. The axiomatization
we found for the model companion was inspired in the ideas of [7] together with
the following observation: in Hilbert spaces there is a definable function that
measures the distance between a point and a model. We prove that the theory
of Hilbert spaces with a generic predicate is unstable. We also study a natural
notion of independence in a monster model of this theory and prove some of its
properties. Several natural independence notions have various good properties,
but the theory fails to be simple and even fails to be NTP2.
1.1. Model theory of Hilbert spaces (quick review).
HILBERT SPACES WITH GENERIC PREDICATES 3
1.1.1. Hilbert spaces. We follow [2] in our study of the model theory of a real
Hilbert space and its expansions. We assume the reader is familiar with the basic
concepts of continuous logic as presented in [2]. A Hilbert spaceH can be seen as
amulti-sorted structure (Bn(H), 0,+, 〈, 〉, {λr : r ∈ R})0<n<ω, whereBn(H) is the
ball of radiusn,+ stands for addition of vectors (defined fromBn(H)×Bn(H) into
B2n(H)), 〈, 〉 : Bn(H) × Bn(H) → [−n2, n2] is the inner product, 0 is a constant
for the zero vector and λr : Bn(H) → Bn(⌈|r|⌉)H is the multiplication by r ∈ R.
We denote by L the language of Hilbert spaces and by T the theory of Hilbert
spaces.
By a universal domainH of T we mean a Hilbert spaceH which is κ-saturated
and κ-strongly homogeneous with respect to types in the language L, where κ is
a cardinal larger than 2ℵ0 . Constructing such a structure is straightforward —just
consider a Hilbert space with an orthonormal basis of cardinality at least κ.
We will assume that the reader is familiar with the metric versions of definable
closure and non-dividing. The reader can check [2, 5] for the definitions.
1.1. Notation. Let dcl stand for the definable closure and acl stand for the alge-
braic closure in the language L.
1.2. Fact. Let A ⊂ H be small. Then dcl(A) = acl(A) = the smallest Hilbert
subspace ofH containing A.
Proof. See Lemma 3 in [5, p. 80] �
Recall a characterization of non-dividing in pure Hilbert spaces (that will be
useful in the more sophisticated constructions in forthcoming sections):
1.3. Proposition. Let B,C ⊂ H be small, let (a1, . . . , an) ∈ Hn and assume that
C = dcl(C), soC is a Hilbert subspace ofH. Denote by PC the projection onC. Then
tp(a1, . . . , an/C ∪ B) does not divide over C if and only if for all i ≤ n and all
b ∈ B, ai − PC(ai) ⊥ b− PC(b).
Proof. Proved as Corollary 2 and Lemma 8 of [5, pp. 81–82]. �
4 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
For A,B,C ⊂ H small, we say that A is independent from B over C if for all
n ≥ 1 and ā ∈ An, tp(ā/C ∪ B) does not divide over C.
Under non-dividing independence, types over sets are stationary. In particular,
the independence theorem holds over sets, and we may refer to this property as
3-existence. It is also important to point out that non-dividing is trivial, that is,
for all sets B,C and tuples (a1, . . . , an) from H, tp(a1, . . . , an/C ∪ B) does not
divide over C if and only if tp(ai/B ∪ C) does not divide over C for i ≤ n.
2. Random subspaces and beautiful pairs
We now deal with the easiest situation: a Hilbert space with an orthonormal
projection operator onto a subspace. Let Lp = L ∪ {P} where P is a new unary
function andwe consider structures of the form (H, P), whereP : H→ H is a pro-
jection into a subspace. Note that P : Bn(H) → Bn(H) and that P is determined
by its action onB1(H). Recall that projections are bounded linear operators, char-
acterized by two properties:
(1) P2 = P
(2) P∗ = P
The second condition means that for any u, v ∈ H, 〈P(u), v〉 = 〈u, P(v)〉.
A projection also satisfies, for any u, v ∈ H, ‖P(u) − P(v)‖ ≤ ‖u − v‖. In
particular, it is a uniformly continuous map and its modulus of uniform continuity
is ∆P(ǫ) = ǫ.
We start by showing that the class of Hilbert spaces with projections has the
free amalgamation property:
2.1. Lemma. Let (H0, P0) ⊂ (Hi, Pi) where i = 1, 2 and H1 |
H2 be (possibly
finite dimensional) Hilbert spaces with projections. Then H3 = span{H1, H2} is a
Hilbert space and P3(v3) = P1(v1) + P2(v2) is a well defined projection, where
v3 = v1 + v2 and v1 ∈ H1, v2 ∈ H2.
HILBERT SPACES WITH GENERIC PREDICATES 5
Proof. It is clear that H3 = span{H1 ∪ H2} is a Hilbert space containing H1 and
H2. It remains to prove that P3 is a projection map and that it is well defined. We
denote byQ0,Q1,Q2 the projections onto the spacesH0,H1 andH2 respectively.
Since H0 ⊂ H1, we can write H1 = H0 ⊕ (H1 ∩ H⊥0 ). Similarly H2 = H0 ⊕
(H2 ∩H⊥0 ). Finally, since H1 |⌣H0 H2,
H3 = H0 ⊕ (H1 ∩H⊥0 )⊕ (H2 ∩H⊥0 )
Let v3 ∈ H3. Let u0 = Q0(v3), u1 = PH⊥
∩H1(v3) = Q1(v3) − u0, u2 =
∩H2(v3) = Q2(v3) − u0. Then v3 = u0 + u1 + u2.
AsH1∩H2 = H0, we can write v3 in many different ways as a sum of elements
inH1 andH2. Given one such expression, v3 = v1+v2, with v1 ∈ H1 and v2 ∈ H2,
it is easy to see that P1(v1)+P2(v2) = P0(u0)+P1(u1)+P2(u2), and thus prove
that P3 is well defined.
Let TP be the theory of Hilbert spaces with a projection. It is axiomatized by
the theory of Hilbert spaces together with the axioms (1) and (2) that say that P
is a projection.
Consider first the finite dimensional models. Given an n-dimensional Hilbert
space Hn, there are only n + 1 many pairs (Hn, P), where P is a projection,
modulo isomorphism. They are classified by the dimension of P(H), which ranges
from 0 to n.
In order to characterize the existentially closedmodels of TP, note the following
facts:
(1) Let (H, P) be existentially closed, and (Hn, Pn) be ann-dimensional Hilbert
space with an orthonormal projection with the property that Pn(Hn) =
Hn. Then (H, P) ⊕ (Hn, Pn) is an extension of (H, P) with dim([P ⊕
Pn](H ⊕Hn)) ≥ n. Since n can be chosen as big as we want and (H, P)
is existentially closed, dim(P(H)) = ∞.
6 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
(2) Let (H, P) be existentially closed, and (Hn, P0) be ann-dimensionalHilbert
space with an orthonormal projection such that P0(Hn) = {0}. Then
(H, P)⊕ (Hn, P0) is an extension of (H,P) such that dim(([P⊕ P0](H⊕
⊥) ≥ n. Since n can be chosen as big as we want and (H, P) is
existentially closed, dim(P(H)⊥) = ∞.
The theory TPω extending T
P, stating that Pω is a projection and that there
are infinitely many pairwise orthonormal vectors v satisfying Pω(v) = v and
also infinitely many pairwise orthonormal vectors u satisfying Pω(u) = 0 gives
an axiomatization for the the model companion of TP, which corresponds to the
theory of beautiful pairs of Hilbert spaces. We will now study some properties of
Let (H, P) |= TPω and for any v ∈ H let dP(v) = ‖v − P(v)‖. Then dP(v)
measures the distance between v and the subspace P(H). The distance function
dP(x) is definable in (H, P). We will now prove the converse, that is, that we can
definably recover P from dP.
2.2. Lemma. Let (H, P) |= TPω. For any v ∈ Hω let dP(v) = ‖v − P(v)‖. Then
P(v) ∈ dcl(v) in the structure (H, dP).
Proof. Note that P(v) is the unique element x in P(H) satisfying ‖v−x‖ = dP(v).
Thus P(v) is the unique realization of the conditionϕ(x) = max{dP(x), |‖v−x‖−
dP(v)|} = 0. �
2.3. Proposition. Let (H, P) |= TPω. For any v ∈ Hω let dP(v) = ‖v − P(v)‖.
Then the projection function P(x) is definable in the structure (H, dP)
Proof. Let (H, P) |= TPω be κ-saturated for κ > ℵ0 and let dP(v) = ‖v − P(v)‖.
Since dP is definable in the structure (H, P), the new structure (H, dP) is still
κ-saturated. Let GP be the graph of the function P. Then by the previous lemma
GP is type-definable in (H, dP) and thus by [2, Proposition 9.24] P is definable in
the structure (H, dP). �
HILBERT SPACES WITH GENERIC PREDICATES 7
2.4. Notation. We write tp for L-types, tpP for LP-types and q�pP for quantifier
free LP-types. We write aclP for the algebraic closure in the language LP. We
follow a similar convention for dclP .
2.5. Lemma. TPω has quantifier elimination.
Proof. It suffices to show that quantifier free LP-types determine the LP-types.
Let (H, P) |= TPω and let ā = (a1, . . . , an), b̄ = (b1, . . . , bn) ∈ Hn. Assume that
q�pP(ā) = q�pP(b̄). Then
tp(P(a1), . . . , P(an)) = tp(P(b1), . . . , P(bn))
tp(a1 − P(a1), . . . , an − P(an)) = tp(b1 − P(b1), . . . , bn − P(bn)).
Let H0 = P(H) and let H1 = H
0 , both are then infinite dimensional Hilbert
spaces and H = H0 ⊕ H1. Let f0 ∈ Aut(H0) satisfy f0(P(a1), . . . , P(an)) =
(P(b1), . . . , P(bn)) and let f1 ∈ Aut(H1) be such that f1(a1 − P(a1), . . . , an −
P(an)) = (b1−P(b1), . . . , bn−P(bn)). Let f be the automorphism ofH induced
by by f0 and f1, that is, f = f0 ⊕ f1. Then f ∈ Aut(H, P) and f(a1, . . . , an) =
(b1, . . . , bn), so tpP(ā) = tpP(b̄). �
Characterization of types: By the previous lemma, the LP-type of an n-tuple
ā = (a1, . . . , an) inside a structure (H, P) |= T
ω is determined by the type of its
projections tp(P(a1), . . . , P(an), a1 − P(a1), . . . , an − P(an)). In particular, we
may regard (H, P) as the direct sum of the two independent pure Hilbert spaces
(P(H),+, 0, 〈, 〉) and (P(H)⊥,+, 0, 〈, 〉).
We may therefore characterize definable and algebraic closure, as follows.
2.6. Proposition. Let (H, P) |= TPω and let A ⊂ H. Then dclP(A) = aclP(A) =
dcl(A ∪ P(A)).
We leave the proof to the reader. Another consequence of the description of
types is:
8 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
2.7. Proposition. The theory TPω isω-stable.
Proof. Let (H, P) |= TPω be separable and let A ⊂ H be countable. Replacing
(H, P) for (H, P) ⊕ (H, P) if necessary, we may assume that
P(H)∩dclP(A)⊥ is infinite dimensional and that P(H)⊥∩dclP(A)⊥ is infinite
dimensional. Thus every Lp-type over A is realized in the structure (H, P) and
(S1(A), d) is separable. �
2.8. Proposition. Let (H, P) |= TPω be a κ-saturated domain and let A,B,C ⊂ H
be small. Then tpP(A/B ∪C) does not fork over C if and only if tp(A ∪ P(A)/B ∪
P(B) ∪ C ∪ P(C)) does not fork over C ∪ P(C).
Again the proof is straightforward.
3. Continuous random predicates
We now come to our main theory and to our first set of results. We study the
expansion of a Hilbert space with a distance function to a subset of H. Let dN
be a new unary predicate and let LN be the language of Hilbert spaces together
with dN. We denote the LN structures by (H, dN), where dN : H → [0, 1] and we
want to consider the structures where dN is a distance to a subset ofH. Instead of
measuring the actual distance to the subset, we truncate the distance at one. We
start by characterizing the functions dN corresponding to distances.
3.1. The basic theory T0. We denote by T0 the theory of Hilbert spaces together
with the next two axioms (compare with Theorem 9.11 in [2]):
(1) supxmin{1−
· dN(x), infymax{|dN(x) − ‖x− y‖|, dN(y)}} = 0
(2) supx supy[dN(y) − ‖x − y‖− dN(x)] ≤ 0
We say a point is black if dN(x) = 0 and white if dN(x) = 1. All other points
are gray, darker if d(x) is close to zero and whiter if dN(x) is close to one. This
terminology follows [10]. From the second axiom we get that dN is uniformly
continuous (with modulus of uniform continuity ∆(ǫ) = ǫ). Thus we can apply
the tools of continuous model theory to analyze these structures.
HILBERT SPACES WITH GENERIC PREDICATES 9
3.1. Lemma. Let (H, d) |= T0 be ℵ0-saturated and let N = {x ∈ H : dN(x) = 0}.
Then for any x ∈ H, dN(x) = dist(x,N).
Proof. Let v ∈ H and let w ∈ N. Then by the second axiom dN(v) ≤ ‖v − w‖
and thus dN(v) ≤ dist(v,N).
Now let v ∈ H. If dN(v) = 1 there is nothing to prove, so we may assume
that dN(v) < 1. Consider now the set of statements p(x) given by dN(x) = 0,
‖x− v‖ = dN(v).
Claim The type p(x) is approximately satisfiable.
Let ε > 0. We want to show that there is a realization of the statements
dN(x) ≤ ε, dN(v) ≤ ‖x − v‖ + ε. By the first axiom there is w such that
dN(w) ≤ ε and dN(v) ≤ ‖v −w‖+ ε.
Since (H, d) is ℵ0-saturated, there is w ∈ N such that ‖v − w‖ = dN(v) as
we wanted. �
There are several ages that need to be considered. We fix r ∈ [0, 1) and we
consider the class Kr of all models of T0 such that dN(0) = r. Note that in all
finite dimensional spaces in Kr we have at least a point v with dN(v) = 0.
3.2. Notation. If (Hi, d
N) |= T0 for i ∈ {0, 1}, we write (H0, d0N) ⊂ (H1, d1N) if
H0 ⊂ H1 and d0N = d1N ↾H0 (for each sort).
We will work in Kr. We start with constructing free amalgamations:
3.3. Lemma. Let (H0, d
N) ⊂ (Hi, diN) where i = 1, 2 and H1 |⌣H0 H2 be Hilbert
spaces with distance functions, all of them in Kr. Let H3 = span{H1, H2} and let
d3N(v) = min
(PH1(v))
2 + ‖PH2∩H⊥0 (v)‖
(PH2(v))
2 + ‖PH1∩H⊥0 (v)‖
Then (Hi, d
N) ⊂ (H3, d3N) for i = 1, 2, and (H3, d3N) ∈ Kr.
Proof. For arbitrary v ∈ H1,
(PH1(v))
2 + ‖PH2∩H⊥0 (v)‖
2 = d1N(v). Since
(H0, d
N) ⊂ (Hi, diN) we also have
10 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
(PH2(v))
2 + ‖PH1∩H⊥0 (v)‖
(PH0(v))
2 + ‖PH1∩H⊥0 (v)‖
2 ≥ d1N(v).
Similarly, for any v ∈ H2, d3N(v) = d2N(v).
Therefore (H3, d
N) ⊃ (Hi, diN) for i ∈ {1, 2}. Now we have to prove that the
function d3N that we defined is indeed a distance function.
Geometrically, d3N(v) takes the minimum of the distances of v to the selected
black subsets of H1 and H2. That is, the random subset of the amalgamation of
(H1, d
N) and (H2, d
N) is the union of the two random subsets. It is easy to check
that (H3, d
N) |= T0. Since each of (H1, d
N), (H2, d
N) belongs to Kr, we have
d1N(0) = r = d
N(0) and thus d
N(0) = r. �
The classK0 also has the JEP: let (H1, d
N), (H2, d
N) belong toK0 and assume
that H1 ⊥ H2. Let N1 = {v ∈ H1 : d1N(v) = 0} and let N2 = {v ∈ H2 :
d2N(v) = 0}. Let H3 = span(H1 ∪H2) and let N3 = N1 ∪N2 ⊂ H3 and finally,
let d3N(v) = dist(v,N3). Then (H3, d
N) is a witnesses of the JEP in K0.
3.4. Lemma. There is a model (H, dN) |= T0 such that H is a 2n-dimensional
Hilbert space and there are orthonormal vectors v1, . . . , vn ∈ H, u1, . . . , un ∈ H
such that dN((ui+ vj)/2) =
2/2 for i ≤ j, dN(0) = 0 and dN((ui+ vj)/2) = 0
for i > j.
Proof. LetH be a Hilbert space of dimension 2n, and fix some orthonormal basis
〈v1, . . . , vn, u1, . . . , un〉 for H. Let N = {(ui + vj)/2 : i > j} ∪ {0} and let
dN(x) = dist(x,N). Then dN(0) = 0 and dN((ui + vj)/2) = 0 for i > j. Since
‖(ui + vj)/2− (uk + vj)/2‖ =
2/2 for i 6= k and ‖(ui + vj)/2− 0‖ =
we get that dN(ui + vj) =
2/2 for i ≤ j �
A similar construction can be made in order to get the Lemma with dN(0) = r
for any r ∈ [0, 1]. In particular, if we fix an infinite cardinal κ and we amalga-
mate all possible pairs (H,d) in Kr for dim(H) ≤ κ, the theory of the resulting
structure will be unstable.
3.2. The model companion.
HILBERT SPACES WITH GENERIC PREDICATES 11
3.2.1. Basic notations. We now provide the axioms of the model companion of
T0 ∪ {dN(0) = 0}.
Call Td0 the theory of the structure built out of amalgamating all separable
Hilbert spaces together with a distance function belonging to the age K0. Infor-
mally speaking, Td0 = Th(lim−→(K0)). We show how to axiomatize Td0.
The idea for the axiomatization of this part (unlike our third example, in next
section) follows the lines of Theorem 2.4 of [7]. There are however important
differences in the behavior of algebraic closures and independence, due to the
metric character of our examples.
Let (M,dN) inK0 be an existentially closed structure and take some extension
(M1, dN) ⊃ (M,dN). Let x̄ = (x1, . . . , xn+k) be elements in M1 \M and let
z1, . . . , zn+k be their projections on M. Assume that for i ≤ n there are ȳ =
(y1, . . . , yn) inM1 \M that satisfy dN(xi) = ‖xi − yi‖ and dN(yi) = 0. Also
assume that for i > n, the witnesses for the distances to the black points belong
toM, that is, d2N(xi) = ‖xi − zi‖2 + d2N(zi) for i > n. Also, let us assume that
all points in x̄, ȳ live in a ball of radius L around the origin. Let ū = (u1, . . . , un)
be the projection of ȳ = (y1, . . . , yn) overM.
Let ϕ(x̄, ȳ, z̄, ū) be a formula such that ϕ(x̄, ȳ, z̄, ū) = 0 describes the val-
ues of the inner products between all the elements of the tuples, that is, it de-
termines the (Hilbert space) geometric locus of the tuple (x̄, ȳ, z̄, ū). The state-
ment ϕ(x̄, ȳ, z̄, ū) = 0 expresses the position of the potentially new points x̄, ȳ
with respect to their projections into a model. Since dN(xi) = ‖xi − yi‖ and
dN(yi) = 0, we have ‖xi − yj‖ ≥ ‖xi − yi‖ for j ≤ n, i ≤ n. Also, for i > n,
d2N(xi) = ‖xi − zi‖2 + d2N(zi), and get ‖xi − yj‖2 ≥ ‖xi − zi‖2 + d2N(zi) for
j ≤ n.
Note that as (M1, dN) ⊃ (M,dN), for all z ∈ M, d2N(z) ≤ ‖z − yi‖2 =
‖z − ui‖2 + ‖yi − ui‖2 for i ≤ n. We may also assume that there is a positive
real ηϕ such that ‖xi − zi‖ ≥ ηϕ for i ≤ n + k and ‖yi − ui‖ ≥ ηϕ for i ≤ n.
3.2.2. An informal description of the axioms. We want to express that for any pa-
rameters z̄, ū in the structure
12 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
if we can find realizations x̄, ȳ of ϕ(x̄, ȳ, z̄, ū) = 0 such that for allw and i ≤ n,
d2N(w) ≤ ‖w− ui‖2 + ‖ui − yi‖2, ‖xi − yi‖2 ≤ ‖xi − zi‖2 + d2N(zi) for i ≤ n,
‖xi − yj‖2 ≥ ‖xi − zj‖2 + d2N(zj) for i > n and j ≤ n,
then there are tuples x̄ ′, ȳ ′ such that ϕ(x̄ ′, ȳ ′, z̄, ū) = 0, dN(y
i) = 0, dN(x
‖x ′i − y ′i‖ for i ≤ n and d2N(xj) = ‖xj − zj‖2 + d2N(zj) for j > n.
That is, for any z̄, ū in the structure, if we can find realizations x̄, ȳ of the
Hilbert space locus given byϕ, and we prescribe “distances” dN that do not clash
with the dN information we already had, in such a way that for i ≤ n, the yi’s
are black and are witnesses for the distance to the black set for the xi’s, and for
i > n the xi’s do not require new witnesses, then we can actually find arbitrarily
close realizations, with the prescribed distances.
The only problemwith this idea is that we do not have an implication in contin-
uous logic. We replace the expression “p → q” by a sequence of approximations
indexed by ε.
3.2.3. The axioms of TN.
3.5. Notation. Let z̄, ū be tuples inM and let x ∈M1. By Pspan(z̄ū)(x) we mean
the projection of x in the space spanned by (z̄, ū).
For fixed ε ∈ (0, 1), let f : [0, 1] → [0, 1] be a continuous function such that
whenever ϕ(t̄) < f(ε) and ϕ(t̄ ′) = 0, then
(a): ‖Pspan(z̄ū)(xi) − zi‖ < ε.
(b): ‖Pspan(z̄ū)(yi) − ui‖ < ε.
(c): |‖ti − tj‖− ‖t ′i − t ′j‖| < ε where t̄ is the concatenation of x̄, ȳ, z̄, ū.
Choosing ε small enough, we may assume that
(d): ‖xi − Pspan(z̄ū)(xi)‖ ≥ ηϕ/2 for i ≤ n + k.
(e): ‖yi − Pspan(z̄ū)(yi)‖ ≥ ηϕ/2 for i ≤ n.
Let δ = 2
ε(L+ 2) and consider the following axiom ψϕ,ε (which we write
as a positive bounded formula for clarity) where the quantifiers range over a ball
of radius L+ 1:
HILBERT SPACES WITH GENERIC PREDICATES 13
∀z̄∀ū
∀x̄∀ȳϕ(x̄, ȳ, z̄, ū) ≥ f(ε)∨∃w
i≤n(d
N(w) ≥ ‖w−ui‖2+‖yi−ui‖2+
i>n,j≤n(‖xi − yj‖2 ≤ ‖xi − zi‖2 + d2N(zi) + ε2)∨
i,j≤n,j6=i(‖xi−yj‖ ≤ ‖xi−yi‖−ε)∨
i≤n(‖xi−zi‖2+d2N(zi) ≤ ‖xi−yi‖2−ε2)
∨∃x̄∃ȳ
(ϕ(x̄, ȳ, z̄, ū) ≤ f(ε)∧
i≤n dN(yi) ≤ δ)∧
i≤n |dN(xi)−‖xi−yi‖| ≤
i>n |d
N(xi) − ‖xi − zi‖2 − d2N(zi)| ≤ 4δL
Let TN be the theory T0 together with this scheme of axiomsψϕ,ε indexed by all
Hilbert space geometric locus formulas ϕ(x̄, ȳ, z̄, ū) = 0 and ε ∈ (0, 1) ∩ Q. The
radius of the ball that contains all elements, L, as well as n and k are determined
from the configuration of points described by the formula ϕ(x̄, ȳ, z̄, ū) = 0.
3.2.4. Existentially closed models of T0.
3.6. Theorem. Assume that (M,dN) |= T0 is existentially closed. Then (M,dN) |=
Proof. Fix ε > 0 and ϕ as above. Let z̄ ∈Mn+k, ū ∈Mn and assume that there
are x̄, ȳwithϕ(x̄, ȳ, z̄, ū) < f(ε) and d2N(w) < ‖w−ui‖2+‖yi−ui‖2+ε2 for all
w ∈M, ‖xi−yi‖2 < ‖xi−zi‖2+d2N(zi)+ε2 for i ≤ n, ‖xi−yj‖ > ‖xi−yi‖−ε
for i, j ≤ n, i 6= j, ‖xi − yj‖2 > ‖xi − zi‖2 + d2N(zj) + ε2 for i > n, j ≤ n. Let
ε ′ < ε be such that ϕ(x̄, ȳ, z̄, ū) < f(ε ′) and
(f): d2N(w) < ‖w − ui‖2 + ‖yi − ui‖2 + ε ′2 for allw ∈M.
(g1): ‖xi − yi‖2 > ‖xi − zi‖2 + dN(zi) + ε ′2 for i ≤ n.
(g2): ‖xi − yj‖ > ‖xi − yi‖− ε ′ for i, j ≤ n, i 6= j
(h): ‖xi − yj‖2 ≥ ‖xi − zi‖2 + d2N(zi) + ε ′2 for i > n, j ≤ n.
We construct an extension (H,dN) ⊃ (M,dN) where the conclusion of the
axiom indexed by ε ′ holds. Since (M,dN) is existentially closed and the conclu-
sion of the axiom is true for (H,dN) replacing ε for ε
′ < ε, then the conclusion
of the axiom indexed by ε will hold for (M,dN).
So let H ⊃ M be such that dim(H ∩ M⊥) = ∞. Let a1, . . . , an+k and
c1, . . . , cn ∈ H be such that tp(ā, c̄/z̄ū) = tp(x̄, ȳ/z̄ū) and āc̄ |
⌣z̄ū
M. We
14 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
can write ai = a
i + z
i and ci = c
i + u
i for some z
i ∈ M and a ′i, c ′i ∈M⊥.
By (d) and (e) ‖a ′i‖ ≥ η/2 for i ≤ n + k and ‖c ′i‖ ≥ η/2 for i ≤ n. Now let
ĉi = c
i + u
i + δ
′c ′i/‖c ′i‖, where δ ′ =
2ε ′(L+ 2).
Let the black points in H be the ones fromM plus the points ĉ1, . . . , ĉn. Now
we check that the conclusion of the axiom ψϕ,ε ′ holds.
(1) ϕ(ā, c̄, z̄, ū) ≤ f(ε ′) since tp(ā, c̄/z̄ū) = tp(x̄, ȳ/z̄ū).
(2) Since ‖ci − ĉi‖ ≤ δ ′ and ĉi is black we have dN(ci) ≤ δ ′.
(3) We check that the distance from ai to the black set is as prescribed for
i ≤ n. dN(ai) ≤ ‖ai − ĉi‖ ≤ ‖ai − ci‖+ δ ′ for i ≤ n.
Also, for i 6= j, i, j ≤ n, using (g2)we prove ‖ai− ĉj‖ ≥ ‖ai−cj‖−δ ′ ≥
‖ai − ci‖ − ε ′ − δ ′ ≥ ‖ai − ci‖− 2δ ′. Finally by (a) ‖ai − PM(ai)‖2 +
d2N(PM(ai)) ≥ (‖ai − zi‖− ε ′)2+(dN(zi)− ε ′)2 ≥ ‖ai− zi‖2− 2Lε ′ +
ε ′2 + d2N(zi) − 2ε
′ + ε ′2 and by (g1), we get ‖ai − zi‖2 − 2Lε ′ + ε ′2 +
d2N(zi) − 2ε
′ + ε ′2 ≥ ‖ai − ci‖2 − 2Lε ′ − 2ε ′ ≥ ‖ai − ci‖2 − 4δ ′2.
(4) We check that dN(ai) is as desired for i > n. Clearly ‖aj − ĉi‖ ≥ ‖aj −
ci‖ − δ ′, so ‖aj − ĉi‖2 ≥ ‖aj − ci‖2 + δ ′2 − 2δ ′2L and by (h) we get
‖aj − ci‖2 + δ ′2 − 4δ ′L ≥ ‖aj − zj‖2 + d2N(zj) − 4δ ′L − ε ′2 + δ ′2 ≥
‖aj − zj‖2 + d2N(zj) − 4δ ′L.
It remains to show that (M,dN) ⊂ (H,dN), i.e., the function dN onH extends
the function dN on M . Since we added the black points in the ball of radius
L + 1, we only have to check that for any w ∈ M in the ball of radius L + 2,
d2N(w) ≤ ‖w − ĉi‖2 = ‖w − u ′i‖2 + ‖c ′i + δ ′(c ′i/‖c ′i‖)‖2.
But by (f) d2N(w) ≤ ‖w − ui‖2 + ‖ci − ui‖2 + ε ′2, so it suffices to show that
‖w− ui‖2 + ‖ci − ui‖2 + ε ′2 ≤ ‖w− u ′i‖2 + ‖c ′i‖2 + 2δ ′‖c ′i‖+ δ ′2
By (a) ‖w− u ′i‖2 ≥ (‖w − ui‖− ε ′)2 and is enough to prove that
‖w− ui‖2 + ‖ci − ui‖2 + ε ′2 ≤ (‖w − ui‖− ε ′)2 + ‖c ′i‖2 + 2δ ′‖c ′i‖+ δ ′2
But (‖w−ui‖−ε ′)2+‖c ′i‖2+2δ ′‖c ′i‖+δ ′2 = ‖w−ui‖2−2ε ′‖w−ui‖+ε ′2+
‖c ′i‖2+2δ ′‖c ′i‖+δ ′2 and ‖ci−ui‖2 ≤ ‖ci−u ′i‖2+2ε ′‖ci−u ′i‖+ε ′2 = ‖c ′i‖2+
HILBERT SPACES WITH GENERIC PREDICATES 15
2ε ′‖c ′i‖+ε ′2. Thus, after simplifying, we only need to check 2ε ′‖w−ui‖+ε ′2 ≤
δ ′2 which is true since 2ε ′‖w−ui‖+ ε ′2 ≤ 2ε ′(2L+ 2) + ε ′2 ≤ 4ε ′(L+ 2). �
3.7. Theorem. Assume that (M,dN) |= TN. Then (M,dN) is existentially closed.
Proof. Let (H,dN) ⊃ (M,dN) and assume that (H,dN) is ℵ0-saturated. Let
ψ(x̄, v̄) be a quantifier free LN-formula, where x̄ = (x1, . . . xn+k) and v̄ = (v1, . . . vl).
Suppose that there are a1, . . . , an+k ∈ H \ M and e1, . . . el ∈ M such that
(H,dN) |= ψ(ā, ē) = 0. After enlarging the formula ψ if necessary, we may
assume that ψ(x̄, v̄) = 0 describes the values of dN(xi) for i ≤ n+ k, the values
of dN(vj) for j ≤ l and the inner products between those elements. We may as-
sume that for i ≤ n there is ρ > 0 such that dN(ai)−d(ai, z) ≥ 2ρ for all z ∈M
with dN(z) ≤ ρ. Since (H,dN) isℵ0-saturated, there are c1, . . . cn ∈ H such that
dN(ai) = ‖ai− ci‖ and dN(ci) = 0. Then d(ci,M) ≥ ρ. Fix ε > 0, ε < ρ, 1. We
may also assume that for i > n, |d2N(ai)−‖ai−PM(ai)‖2−d2N(PM(ai))| ≤ ε/2.
Also, assume that all points mentioned so far live in a ball of radius L around the
origin. Let b1, . . . , bn+k ∈M be the projections of a1, . . . , an+k ontoM and let
d1, . . . , dn ∈ M be the projections of c1, . . . , cn ontoM. Let ϕ(x̄, ȳ, z̄, ū) = 0
be an L-statement that describes the inner products between the elements listed
and such that ϕ(ā, c̄, b̄, d̄) = 0. Using the axioms we can find ā ′, c̄ ′ inM such
that ϕ(ā ′, c̄ ′, b̄, d̄) ≤ f(ε), dN(c ′i) ≤ δ for i ≤ n, |dN(a ′i) − ‖a ′i − c ′i‖| ≤ δ for
i ≤ n and |d2N(ai) − ‖ai − bi‖2 −d2N(bi)| ≤ 4Lδ, where δ =
2ε(L + 2). Since
ε > 0 was arbitrary we get (M,dn) |= infx1 . . . infxn+k ψ(x̄, v̄) = 0. �
4. Model theoretic analysis of TN
We prove three theorems in this section about the theory TN:
• TN is not simple,
• TN is not even NTP2! (Of course, this implies the previous, but we will
provide the proof of non-simplicity as well.)
16 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
• TN is NSOP1. Therefore, in spite of having a tree property, our theory is
still “close to being simple” in the precise sense of not having the SOP1
tree property.
These results place TN in a very interesting situation in the stability hierarchy
for continuous logic.
4.1. Notation. We write tp for types of elements in the language L and tpN for
types of elements in the language LN. Similarly we denote by aclN the algebraic
closure in the language LN and by acl the algebraic closure for pure Hilbert spaces.
Recall that for a set A, acl(A) = dcl(A), and this corresponds to the closure of
the space spanned by A (Fact 1.2).
4.2. Observation. The theory TN does not have elimination of quantifiers. We
use the characterization of quantifier elimination given in Theorem 8.4.1 from
[9]. Let H1 be a two dimensional Hilbert space, let {u1, u2} be an orthonormal
basis for H1 and let N1 = {0, u0 +
u1} and let d
N(x) = min{1, dist(x,N1)}.
Then (H1, d
N) |= T0. Let a = u0, b = u0 −
u1 and c = u0 +
u1. Note
that d1N(b) =
. Let (H ′1, d
N) ⊃ (H1, d1N) be existentially closed. Now let H2
be an infinite dimensional separable Hilbert space and let {vi : i ∈ ω} be an
orthonormal basis. LetN2 = {x ∈ H : ‖x− v1‖ = 14 , Pspan(v1)(x) = v1} ∪ {0} and
let d2N(x) = min{1, dist(x,N2)}. Let (H
N) ⊃ (H2, d2N) be existentially closed.
Then (span(a), d1N ↾span(a))
∼= (span(v1), d
N ↾span(v1)) and they can be identified
say by a function F. But (H ′1, d
N) and (H
N) cannot be amalgamated over this
common substructure: If they could, then we would have dist(F(b), v1 +
vi) =
dist(b, v1 +
vi) <
for some i > 1 and thus d1N(b) <
, a contradiction.
In this case, the main reason for this failure of amalgamation resides in the
fact that (span(a), d1N ↾span(a))
∼= (span(v1), d
N ↾span(v1)) is not a model of T0:
informally, the distance values around v1 are determinedby an “external attractor”
(the black point u0 +
u1 or the black ring orthogonal to v1 at distance
) that
the subspace (span(a), d1N ↾span(a)) simply cannot see. This violates Axiom (1) in
HILBERT SPACES WITH GENERIC PREDICATES 17
the description of T0. This “noise external to the substructure” accounts for the
failure of amalgamation, and ultimately for the lack of quantifier elimination.
In [7, Corollary 2.6], the authors show that the algebraic closure of the expan-
sion of a simple structure with a generic subset corresponds to the algebraic in
the original language. However, in our setting, the new algebraic closure aclN(X)
does not agree with the old algebraic closure acl(X):
4.3. Observation. The previous construction shows that aclN does not coincide
with acl. Indeed, c ∈ aclN(a) \ acl(a) - the set of solutions of the type tpN(c/a)
is {c}, but c /∈ dcl(a) as c /∈ span(a).
However, models of the basic theory T0 are LN-algebraically closed. The proof
is similar to [7, Proposition 2.6(3)]:
4.4. Lemma. Let (M,dN) |= TN and let A ⊂ M be such that A = dcl(A) and
(A,dN ↾A) |= T0. Let a ∈M. Then a ∈ aclN(A) if and only if a ∈ A.
Proof. Assume a /∈ A. We will show that a /∈ aclN(A). Let a ′ |= tp(a/A) be
such that a ′ |
M. Let (M ′, dN) be an isomorphic copy of (M,dN) over A
through f : M →A M ′ such that f(a) = a ′. We may assume thatM ′ |
Since (A,dN ↾A) is an amalgamation base, (N,dN) = (M ⊕A M ′, dN) |= T0.
Let (N ′, dN) ⊃ (N,dN) be an existentially closed structure. Then tpN(a/A) =
tpN(a
′/A) and therefore a /∈ aclN(A). �
As TN is model complete, the types in the extended language are determined by
the existential formulas within them, i.e. formulas of the form inf ȳϕ(ȳ, x̄) = 0
Another difference with the work of Chatzidakis and Pillay is that the analogue
to [7, Proposition 2.5] no longer holds. Let a, b, c be as in Observation 4.3; notice
that (span(a), dN ↾span(a))
∼= (span(v1), dN ↾span(v1)). However, (H
1, dN, a) 6≡
(H ′2, dN, v1). Instead, we can show the following weaker version of the Proposi-
tion.
18 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
4.5.Proposition. Let (M,dN) and (N,dN) bemodels of TN and letA be a common
subset ofM and N such that (span(A), dN ↾span(A)) |= T0. Then
(M,dN) ≡A (N,dN).
Proof. Assume thatM∩N = span(A). Since (span(A), dN ↾span(A)) |= T0, it is an
amalgamation base and therefore we may consider the free amalgam (M⊕span(A)
N,dN) of (M,dN) and (N,dN) over (span(A), dN ↾span(A)). Let now (E, dN) be
a model of TN extending (M⊕span(A) N,dN). By the model completeness of TN,
we have that (M,dN) ≺ (E, dN) and (N,dN) ≺ (E, dN) and thus (M,dN) ≡A
(N,dN). �
4.1. Generic independence. In this section we define an abstract notion of in-
dependence and study its properties.
Fix (U, dN) |= TN be a κ-universal domain.
4.6. Definition. Let A,B,C ⊂ U be small sets. We say that A is ∗-independent
from B over C and write A |∗
B if aclN(A ∪ C) is independent (in the sense of
Hilbert spaces) from aclN(C ∪ B) over aclN(C). That is, A |∗
B if for all a ∈
aclN(A ∪ C), PB∪C(a) = PC(a), where B ∪ C = aclN(C ∪ B) and C = aclN(C).
4.7. Proposition. The relation |∗
satisfies the following properties (here A, B, etc.,
are any small subsets of U):
(1) Invariance under automorphisms of U.
(2) Symmetry: A |∗
B⇐⇒ B |∗
(3) Transitivity: A |∗
BD if and only if A |∗
B and A |∗
(4) Finite Character: A |∗
B if and only ā |∗
B for all ā ∈ A finite.
(5) Local Character: If ā is any finite tuple, then there is countable B0 ⊆ B such
that ā |∗
(6) Extension property over models of T0. If (C,dN ↾C) |= T0, then we can find
A ′ such that tpN(A/C) = tpN(A
′/C) and A ′ |∗
(7) Existence over models: ā |∗
M for any ā.
(8) Monotonicity: āā ′ |∗
b̄b̄ ′ implies ā |
HILBERT SPACES WITH GENERIC PREDICATES 19
Proof. (1) Is clear.
(2) It follows from the fact that independence in Hilbert spaces satisfies Sym-
metry (see Proposition 1.3).
(3) It follows from the fact that independence in Hilbert spaces satisfies Tran-
sitivity (see Proposition 1.3).
(4) Clearly A |∗
B implies that ā |∗
B for all ā ∈ A finite. On the other
hand if ā |∗
B for all ā ∈ A finite, then for a dense subset A0 of A,
B and thus A |∗
(5) Local Character: let ā be a finite tuple. Since independence in Hilbert
spaces satisfies local character, there is B1 ⊆ aclN(B) countable such that
ā |∗
B. Now let B0 ⊆ B be countable such that aclN(B0) ⊃ B1. Then
ā |∗
(6) LetC be such that (C,dN ↾C) |= T0. LetD ⊃ A∪C be such that (D,dN ↾D
) |= T0 and let E ⊃ B ∪ C be such that (E, dN ↾E) |= T0. Changing D for
another set D ′ with tpN(D
′/C) = tpN(D/C), we may assume that the
space generated by D ′ ∪ E is the free amalgamation of D ′ and E over C.
By lemma 4.4 D ′, E are algebraically closed andD ′ |∗
(7) It follows from the definition of ∗-independence.
(8) It follows from the definition of ∗-independence and transitivity.
Therefore we have a natural independence notion that satisfies many good
properties, but not enough to guarantee the simplicity of TN.
We will show below that the theory TN has both TP2 and NSOP1. This places
it in an interesting area of the stability hierarchy for continuous model theory:
while having the tree property TP2 and therefore lacking the good properties of
NTP2 theories, it still has a quite well-behaved independence notion |
, good
enough to guarantee that it does not have the SOP1 tree property. Therefore,
although the theory is not simple, it is reasonably close to this family of theories.
4.2. The failure of simplicity.
20 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
4.8. Theorem. The theory TN is not simple.
The proof’s idea uses a characterization of simplicity in terms of the number of
partial types due to Shelah (see [11]; see also Casanovas [6] for further analysis): T
is simple iff for all κ, λ such thatNT(κ, λ) < 2κ+λω, whereNT(κ, λ) counts the
supremum of the cardinalities of families of pairwise incompatible partial types
of size ≤ κ over a set of cardinality≤ λ. This holds for continuous logic as well.
We show that TN fails this criterion.
Proof. Fix κ an infinite cardinal and λ ≥ κ. We will find a complete submodel
Mf of the monster model, of density character λ, and λ
κ many types over sub-
sets of Mf of power κ in such a way that we guarantee that they are pairwise
incompatible in a uniform way.
Now also fix some orthonormal basis ofMf, listed as
{bi|i < κ} ∪ {aj|j < λ} ∪ {cX|X ∈ Pκ(λ)}.
Also fix, for every X ∈ Pκ(λ), a bijection
fX : {bi|i < κ} → {aj|j ∈ X}.
Let the “black points” ofMf consist of the set
N = {cX + bi + (1/2)fX(bi) | i < κ,X ∈ Pκ(λ)} ∪ {0}
and as usual define dN(x) as the distance from x to N. This is a submodel of the
monster.
Let AX := {bi|i < κ} ∪ {aj|j ∈ X} for each X ∈ Pκ(λ).
The crux of the proof is to notice that if X 6= Y then the types tp(cX/AX) and
tp(cY/AY) are incompatible, thereby witnessing that there are λ
κ many incom-
patible types:
Suppose there is some c such that tp(c/AX) = tp(cX/AX) and tp(c/AY) =
tp(cY/AY). Take (wlog) j ∈ Y \ X. Pick ℓ < κ such that fY(bℓ) = aj. Let k ∈ X
be such that fX(bℓ) = ak.
HILBERT SPACES WITH GENERIC PREDICATES 21
InMf, the distance to black of cX+bℓ−
ak is 1: by definition, cX+bℓ+
cX + bℓ +
fX(bℓ) ∈ N and the only difference between cX + bℓ − 12ak and
cX + bℓ +
ak is the sign in front of an element of an orthonormal basis.
Therefore the distance to black of d = c+ bℓ −
ak is also 1 (in the monster).
However, e = c + bℓ +
aj must be a black point, since e
′ = cY + bℓ +
aj is
black (by definition of N and since aj = fY(bℓ) and tp(c/AY) = tp(cY/AY)).
On the other hand, the distance from e to d is
< 1. This contradicts that
the color of d is 1. �
This stands in sharp contrast with respect to the result by Chatzidakis and
Pillay in the (discrete) first order case. The existence of these incompatible types
is rendered possible here by the presence of “euclidean” interactions between the
elements of the basis chosen.
So far we have two kinds of expansions of Hilbert spaces by predicates: either
they remain stable (as in the case of the distance to a Hilbert subspace as in the
previous section) or they are not even simple.
4.3. TN has the tree property TP2.
4.9. Theorem. The theory TN has the tree property TP2.
Proof. We will construct a complete submodel M |= T0 of the monster model,
of density character 2ℵ0 , and a quantifier free formula ϕ(x;y, z) that witnesses
TP2 inside M. Since this model can be embedded in the monster model of TN
preserving the distance to black points, this will show that TN has TP2.
We fix some orthonormal basis ofM, listed as
{bi|i < ω} ∪ {cn,i|n, i < ω} ∪ {af|f : ω→ ω}.
Also let the “black points” ofM consist of the set
N = {af + bn + (1/2)cn,f(n) | n < ω, f : ω→ ω} ∪ {0}
and as usual define dN(x) as the distance from x to N. This is a model of T0 and
thus a submodel of the monster.
22 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
Let ϕ(x, y, z) = max{1− dN(x+ y− (1/2)z), dN(x+ y − (1/2)z)}.
Claim1 For each i, the conditions {ϕ(x, bi, ci,j) = 0 : j ∈ ω} are 2-inconsistent.
Assume otherwise, so we can find a (in an extension ofM) such that dN(a +
bi + (1/2)ci,j) = 0 and dN(a + bi − (1/2)ci,l) = 1 for some j < l. But then
d(a+bi+(1/2)ci,j, a+bi−(1/2)ci,l) = d((1/2)ci,j,−(1/2)ci,l) =
2/2 < 1.
Sincea+bi+(1/2)ci,j is a black point, we get thatdN(a+bi−(1/2)ci,l) ≤
a contradiction.
Claim 2 For each f the conditions {ϕ(x, bi, ci,f(i)) = 0 : i ∈ ω} are consistent.
Indeed fix f and consider af, then by construction dN(af+bn+(1/2)cn,f(n)) =
0 and d(af + bn − (1/2)cn,f(n), af + bn + (1/2)cn,f(n)) = 1, so dN(af + bn −
(1/2)cn,f(n)) ≤ 1.
Now we check the distance to the other points in N. It is easy to see that
d(af + bn − (1/2)cn,f(n), af + bm + (1/2)cm,f(m)) > 1 form 6= n, d(af + bn −
(1/2)cn,f(n), ag + bk + (1/2)ck,g(k)) > 1 for g 6= f and all indexes k. Finally,
d(af + bn − (1/2)cn,f(n), 0) > 1. This shows that af is a witness for the claim.
4.4. TN and the property NSOP1. Chernikov and Ramsey have proved that
whenever a first order discrete theory satisfies the following properties (for ar-
bitrary models and tuples), then the theory satisfies theNSOP1 property (see [8,
Prop. 5.3]).
• Strong finite character: whenever ā depends on b̄ overM, there is a for-
mula ϕ(x, b̄, m̄) ∈ tp(ā/b̄M) such that every ā ′ |= ϕ(x̄, b̄, m̄) depends
on b̄ overM.
• Existence over models: ā |
M for any ā.
• Monotonicity: āā ′ |
b̄b̄ ′ implies ā |
• Symmetry: ā |
b̄ ⇐⇒ b̄ |
• Independent amalgamation: c̄0 |
c̄1, b̄0 |
c̄0, b̄1 |
c̄1, b̄0 ≡M b̄1
implies there exists b̄ with b̄ ≡c̄0M b̄0, b̄ ≡c̄1M b̄1.
HILBERT SPACES WITH GENERIC PREDICATES 23
We prove next that in TN, |
∗ satisfies analogues of these five properties -
we may thereby conclude that TN can be regarded (following the analogy) as a
NSOP1 continuous theory.
In what remains of the paper, we prove that TN satisfies these properties.
We focus our efforts in strong finite character and independent amalgamation,
the other properties were proved in Proposition 4.7.
We need the following setting:
Let M be the monster model of TN andA ⊂M. FixAwithA ⊂ A ⊂ M be such
that A |= T0 and let ā = (a0, . . . , an) ∈ M. We say that (ā, A,B) is minimal if
tp(B/A) = tp(A/A) and for all b̄ ∈ M, if tp(b̄/A) = tp(ā/A) then
‖ prB(b0)‖+ · · · + ‖ prB(bn)‖ ≥ ‖ prB(a0)‖+ · · · + ‖ prB(an)‖.
By compactness, for all p ∈ S(A) there is a minimal (ā, A,B) such that ā |= p.
Now let cl0(A) be the set of all x such that for some minimal (ā, A,B), x =
prB(a0) (the first coordinate of ā).
4.10. Lemma. If tp(B/A) = tp(A/A) and x ∈ cl0(A) then x ∈ B.
Proof. Suppose not. Let B witness this and let C and ā be such that (ā, A,C) is
minimal and x = prC(a0). Since C |= T0, wlog B |⌣C a (independence in the
sense of Hilbert spaces). But then prB(ai) = prB(prC(ai)) for each i and thus
‖ prB(a0)‖+ · · ·+‖ prB(an)‖ < ‖ prC(a0)‖+ · · ·+‖ prC(an)‖. This contradicts
minimality. �
A direct consequence of the previous lemma is that cl0(A) ⊂ bclN(A) =
∩A⊂B|=TNB, as cl0(A) belongs to every model of the theory TN.
We now define the essential closure ecl. Let clα+1(A) = cl0(clα(A)) for all
ordinals α, clδ(A) =
α<δ(clα(A)), and ecl(A) =
α∈On clα(A).
4.11. Lemma. For all ā, B,A, if ecl(A) = A then there is b̄ such that tp(b̄/A) =
tp(ā/A) and b̄ |
Proof. Choose A |= T0 such that A ⊂ A and c̄ such that tp(c̄/A) = tp(ā/A)
and (c̄, A,A) is minimal. Since cl0(A) = A, prA(ci) ∈ A for all i ≤ n (c̄ =
24 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
(c0, . . . , cn)), i.e. c̄ |
A. Now choose b̄ such that tp(b̄/A) = tp(c̄/A) and
B. Then b̄ is as needed. �
4.12. Corollary. ecl(A) = aclN(A).
Proof. Clearly aclN(A) ⊂ bclN(A). On the other hand, assume that x /∈ aclN(A).
Let B be a model of TN such that A ⊂ B. By Lemma 4.11, we may assume that
B. Then x /∈ B, so x /∈ bcl(A), so x /∈ ecl(A). �
4.13. Theorem. Suppose ecl(A) = A, A ⊂ B,C, B |∗
C (i.e. ecl(B) |
ecl(C)),
ā |∗
B, b̄ |∗
C and tp(ā/A) = tp(b̄/A). Then there is c̄ such that tp(c̄/B) =
tp(ā/B), tp(c̄/C) = tp(b̄/C) and c̄ |∗
Proof. Wlog ecl(B) = B and ecl(C) = C. By Lemma 4.11 we can find modelsA0,
A1, B
∗ and C∗ of T0 such that Aā ⊂ A0, Ab̄ ⊂ A1, B ⊂ B∗ and C ⊂ C∗, such
that B∗ |∗
C∗,A0 |
B∗ andA1 |
C∗. We can also find models of T0,A
and D∗ such that A0B
∗ ⊂ A∗0, A1C∗ ⊂ A∗1 and B∗C∗ ⊂ D∗ and wlog we may
assume that ā and b̄ are chosen so thatA∗0 |⌣B∗ D
∗,A∗1 |⌣C∗ D
∗, and that there is
an automorphism F of the monster model fixingA pointwise such that F(ā) = b̄,
F(A0) = A1 and F(A
0) |⌣A1
A∗1 . Notice that now
D∗ and A1 |
We can now find Hilbert spaces A∗, A∗∗0 , A
1 and E such that
(i) E is generated byD∗A∗∗0 A
(ii) A ⊂ A∗ ⊂ A∗∗0 ∩A∗∗1 , B∗ ⊂ A∗∗0 , C∗ ⊂ A∗∗1 ,
(iii) There are Hilbert space isomorphisms G : A∗∗0 → A
0 and H : A
1 → A
such that
a) F ◦G ↾ A∗ = H ↾ A∗,
b) G ↾ B∗ = idB∗ , H ↾ C
∗ = idC∗ ,
c) G ∪ idD∗ generate an isomorphism
〈A∗∗0 D∗〉 → 〈A∗0D∗〉
HILBERT SPACES WITH GENERIC PREDICATES 25
d) H ∪ idD∗ generate an isomorphism
〈A∗∗1 D∗〉 → 〈A∗1D∗〉
e) F ∪G ∪H generate an isomorphism
〈A∗∗0 A∗∗1 〉 → 〈F(A∗0)A∗1〉.
We can find these because non-dividing independence in Hilbert spaces has 3-
existence (the independence theorem holds for types over sets).
Now we choose the “black points” of our model: a ∈ E is black if one of the
following holds:
(i) a ∈ A∗∗0 and G(a) is black,
(ii) a ∈ A∗∗0 and H(a) is black,
(iii) a ∈ D∗ and is black.
Then in E we define the “distance to black” function simply as the real distance.
Then in D∗ there is no change and G and H remain isomorphisms after adding
this structure;D∗, A∗0, A
1 and F(A
0) witness this.
So we can assume that E is a submodel of the monster, and letting c̄ = G−1(a),
G witnesses that tp(c̄/B) = tp(ā/B) and H witnesses that tp(c̄/C) = tp(b̄/C).
We have already seen that A∗ |
D∗ and thus c̄ |∗
BC. �
4.14. Proposition. Suppose b̄ 6 |∗
C, A ⊂ B ∩ C and (wlog) C = bcl(C). Then
there exists a formula χ ∈ tp(b̄/C) such that for all ā |= χ, ā 6 |∗
Proof. By compactness, we can find ε > 0 such that (letting b̄ = (b0, . . . , bn),
(ā = (a0, . . . , an)),
∀ā |= tp(b̄/B), ‖ prC(a0)‖+· · ·+‖ prC(an)‖ ≥ ε+‖ prbcl(A)(a0)‖+· · ·+‖ prbcl(A)(an)‖.
Again by compactness we can find χ ∈ tp(b̄/B) such that (1) holds when we
replace tp(b̄/B) by χ and ε by ε/2, that is:
26 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
∀ā |= χ, ‖ prC(a0)‖+· · ·+‖ prC(an)‖ ≥ ε/2+‖ prbcl(A)(a0)‖+· · ·+‖ prbcl(A)(an)‖.
and in particular ā 6 |∗
C, as we wanted �
References
[1] Itaï Ben Yaacov, Lovely pairs of models: the non first order case, Journal of Symbolic Logic, volume
69 (2004), 641-662.
[2] Itaï Ben Yaacov, Alexander Berenstein, Ward C. Henson and Alexander Usvyatsov, Model The-
ory for metric structures in Model Theory with Applications to Algebra and Analysis, Volume
2, Cambridge University Press, 2008.
[3] Itaï Ben Yaacov, Alexander Usvyatsov and Moshe Zadka, Generic automorphism of a Hilbert
space, preprint avaliable at http://ptmat.fc.ul.pt/∼alexus/papers.html.
[4] Alexander Berenstein, Hilbert Spaces with Generic Groups of Automorphisms, Arch. Math. Logic,
vol 46 (2007) no. 3, 289–299.
[5] Alexander Berenstein and Steven Buechler, Simple stable homogeneous expansions of Hilbert
spaces. Ann. Pure Appl. Logic 128 (2004), no. 1-3, 75–101.
[6] Enrique Casanovas, The number of types in simple theories. Ann. Pure Appl. Logic 98 (1999),
69–86.
[7] Zoé Chatzidakis and Anand Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic
95 (1998), no. 1-3, 71–92.
[8] Artem Chernikov and Nicholas Ramsey, On model-theoretic tree properties, Journal of Mathe-
matical Logic, 16 (2), 2016.
[9] Wilfrid Hodges,Model Theory, Cambridge University Press 1993.
[10] Bruno Poizat, Le carré de l’égalité, Journal of Symbolic Logic 64 (1999), 1339–1355.
[11] Saharon Shelah, Simple Unstable Theories, Ann. Math Logic 19 (1980), 177–203.
http://ptmat.fc.ul.pt/~alexus/papers.html
HILBERT SPACES WITH GENERIC PREDICATES 27
Alexander Berenstein, Universidad de los Andes, Departamento de Matemáticas, Cra
1 # 18A-10, Bogotá, Colombia.
URL: www.matematicas.uniandes.edu.co/~aberenst
Tapani Hyttinen, University of Helsinki, Department of Mathematics and Statistics,
Gustaf Hällströminkatu 2b. Helsinki 00014, Finland.
E-mail address: tapani.hyttinen@helsinki.fi
Andrés Villaveces, Universidad Nacional de Colombia, Departamento de Matemáticas,
Av. Cra 30 # 45-03, Bogotá 111321, Colombia.
E-mail address: avillavecesn@unal.edu.co
	1. Introduction
	1.1. Model theory of Hilbert spaces (quick review)
	2. Random subspaces and beautiful pairs
	3. Continuous random predicates
	3.1. The basic theory T0
	3.2. The model companion
	4. Model theoretic analysis of TN
	4.1. Generic independence
	4.2. The failure of simplicity
	4.3. TN has the tree property TP2
	4.4. TN and the property NSOP1
	References

  We study the model theory of expansions of Hilbert spaces by generic
predicates. We first prove the existence of model companions for generic
expansions of Hilbert spaces in the form first of a distance function to a
random substructure, then a distance to a random subset. The theory obtained
with the random substructure is {\omega}-stable, while the one obtained with
the distance to a random subset is $TP_2$ and $NSOP_1$. That example is the
first continuous structure in that class.

Introduction
This paper deals with Hilbert spaces expanded with random predicates in the
framework of continuous logic as developed in [2]. The model theory of Hilbert
spaces is very well understood, see [2, Chapter 15] or [5]. However, we briefly
review some of its properties at the end of this section.
In this paper we build several new expansions, by various kinds of random
predicates (random substructure and the distance to a random subset) of Hilbert
spaces, and study them within the framework of continuous logic. While our
constructions are not exactlymetric Fraïssé (failing the hereditaryproperty), some
of them are indeed amalgamation classes and we study the model theory of their
limits.
Several papers deal with generic expansions of Hilbert spaces. Ben Yaacov,
Usvyatsov and Zadka [3] studied the expansion of a Hilbert space with a generic
This workwas partially supported by Colciencias grantMétodos de Estabilidad en Clases No Esta-
bles. The second and third author were also sponsored by Catalonia’s Centre de Recerca Matemàtica
(Intensive Research Program in Strong Logics) and by the University of Helsinki in 2016 for part
of this work. The third author was also partially sponsored by Colciencias (Proy. 1101-05-13605
CT-210-2003).
http://arxiv.org/abs/0704.1633v2
2 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
automorphism. The models of this theory are expansions of Hilbert spaces with a
unitary map whose spectrum is S1. A model of this theory can be constructed by
amalgamating together the collection ofn-dimensional Hilbert spaces with a uni-
tary map whose eigenvalues are the n-th roots of unity as n varies in the positive
integers. More work on generic automorphisms can be found in [4], where the
first author of this paper studied Hilbert spaces expanded with a random group
of automorphisms G.
There are also several papers about expansions of Hilbert spaces with random
subspaces. In [5] the first author and Buechler identified the saturated models of
the theory of beautiful pairs of a Hilbert space. An analysis of lovely pairs (the
generalization of beautiful pairs (belles paires) to simple theories) in the setting of
compact abstract theories is carried out in [1]. In the (very short) second section of
this paper we build the beautiful pairs of Hilbert spaces as the model companion
of the theory of Hilbert spaces with an orthonormal projection. We provide an
axiomatization for this class and we show that the projection operator into the
subspace is interdefinable with a predicate for the distance to the subspace. We
also prove that the theory of beautiful pairs of Hilbert spaces is ω-stable. Many
of the properties of beautiful pairs of Hilbert spaces are known from the literature
or folklore, so this section is mostly a compilation of results.
In the third section we add a predicate for the distance to a random subset. This
construction was inspired by the idea of finding an analogue to the first order
generic predicates studied by Chatzidakis and Pillay in [7]. The axiomatization
we found for the model companion was inspired in the ideas of [7] together with
the following observation: in Hilbert spaces there is a definable function that
measures the distance between a point and a model. We prove that the theory
of Hilbert spaces with a generic predicate is unstable. We also study a natural
notion of independence in a monster model of this theory and prove some of its
properties. Several natural independence notions have various good properties,
but the theory fails to be simple and even fails to be NTP2.
1.1. Model theory of Hilbert spaces (quick review).
HILBERT SPACES WITH GENERIC PREDICATES 3
1.1.1. Hilbert spaces. We follow [2] in our study of the model theory of a real
Hilbert space and its expansions. We assume the reader is familiar with the basic
concepts of continuous logic as presented in [2]. A Hilbert spaceH can be seen as
amulti-sorted structure (Bn(H), 0,+, 〈, 〉, {λr : r ∈ R})0<n<ω, whereBn(H) is the
ball of radiusn,+ stands for addition of vectors (defined fromBn(H)×Bn(H) into
B2n(H)), 〈, 〉 : Bn(H) × Bn(H) → [−n2, n2] is the inner product, 0 is a constant
for the zero vector and λr : Bn(H) → Bn(⌈|r|⌉)H is the multiplication by r ∈ R.
We denote by L the language of Hilbert spaces and by T the theory of Hilbert
spaces.
By a universal domainH of T we mean a Hilbert spaceH which is κ-saturated
and κ-strongly homogeneous with respect to types in the language L, where κ is
a cardinal larger than 2ℵ0 . Constructing such a structure is straightforward —just
consider a Hilbert space with an orthonormal basis of cardinality at least κ.
We will assume that the reader is familiar with the metric versions of definable
closure and non-dividing. The reader can check [2, 5] for the definitions.
1.1. Notation. Let dcl stand for the definable closure and acl stand for the alge-
braic closure in the language L.
1.2. Fact. Let A ⊂ H be small. Then dcl(A) = acl(A) = the smallest Hilbert
subspace ofH containing A.
Proof. See Lemma 3 in [5, p. 80] �
Recall a characterization of non-dividing in pure Hilbert spaces (that will be
useful in the more sophisticated constructions in forthcoming sections):
1.3. Proposition. Let B,C ⊂ H be small, let (a1, . . . , an) ∈ Hn and assume that
C = dcl(C), soC is a Hilbert subspace ofH. Denote by PC the projection onC. Then
tp(a1, . . . , an/C ∪ B) does not divide over C if and only if for all i ≤ n and all
b ∈ B, ai − PC(ai) ⊥ b− PC(b).
Proof. Proved as Corollary 2 and Lemma 8 of [5, pp. 81–82]. �
4 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
For A,B,C ⊂ H small, we say that A is independent from B over C if for all
n ≥ 1 and ā ∈ An, tp(ā/C ∪ B) does not divide over C.
Under non-dividing independence, types over sets are stationary. In particular,
the independence theorem holds over sets, and we may refer to this property as
3-existence. It is also important to point out that non-dividing is trivial, that is,
for all sets B,C and tuples (a1, . . . , an) from H, tp(a1, . . . , an/C ∪ B) does not
divide over C if and only if tp(ai/B ∪ C) does not divide over C for i ≤ n.
2. Random subspaces and beautiful pairs
We now deal with the easiest situation: a Hilbert space with an orthonormal
projection operator onto a subspace. Let Lp = L ∪ {P} where P is a new unary
function andwe consider structures of the form (H, P), whereP : H→ H is a pro-
jection into a subspace. Note that P : Bn(H) → Bn(H) and that P is determined
by its action onB1(H). Recall that projections are bounded linear operators, char-
acterized by two properties:
(1) P2 = P
(2) P∗ = P
The second condition means that for any u, v ∈ H, 〈P(u), v〉 = 〈u, P(v)〉.
A projection also satisfies, for any u, v ∈ H, ‖P(u) − P(v)‖ ≤ ‖u − v‖. In
particular, it is a uniformly continuous map and its modulus of uniform continuity
is ∆P(ǫ) = ǫ.
We start by showing that the class of Hilbert spaces with projections has the
free amalgamation property:
2.1. Lemma. Let (H0, P0) ⊂ (Hi, Pi) where i = 1, 2 and H1 |
H2 be (possibly
finite dimensional) Hilbert spaces with projections. Then H3 = span{H1, H2} is a
Hilbert space and P3(v3) = P1(v1) + P2(v2) is a well defined projection, where
v3 = v1 + v2 and v1 ∈ H1, v2 ∈ H2.
HILBERT SPACES WITH GENERIC PREDICATES 5
Proof. It is clear that H3 = span{H1 ∪ H2} is a Hilbert space containing H1 and
H2. It remains to prove that P3 is a projection map and that it is well defined. We
denote byQ0,Q1,Q2 the projections onto the spacesH0,H1 andH2 respectively.
Since H0 ⊂ H1, we can write H1 = H0 ⊕ (H1 ∩ H⊥0 ). Similarly H2 = H0 ⊕
(H2 ∩H⊥0 ). Finally, since H1 |⌣H0 H2,
H3 = H0 ⊕ (H1 ∩H⊥0 )⊕ (H2 ∩H⊥0 )
Let v3 ∈ H3. Let u0 = Q0(v3), u1 = PH⊥
∩H1(v3) = Q1(v3) − u0, u2 =
∩H2(v3) = Q2(v3) − u0. Then v3 = u0 + u1 + u2.
AsH1∩H2 = H0, we can write v3 in many different ways as a sum of elements
inH1 andH2. Given one such expression, v3 = v1+v2, with v1 ∈ H1 and v2 ∈ H2,
it is easy to see that P1(v1)+P2(v2) = P0(u0)+P1(u1)+P2(u2), and thus prove
that P3 is well defined.
Let TP be the theory of Hilbert spaces with a projection. It is axiomatized by
the theory of Hilbert spaces together with the axioms (1) and (2) that say that P
is a projection.
Consider first the finite dimensional models. Given an n-dimensional Hilbert
space Hn, there are only n + 1 many pairs (Hn, P), where P is a projection,
modulo isomorphism. They are classified by the dimension of P(H), which ranges
from 0 to n.
In order to characterize the existentially closedmodels of TP, note the following
facts:
(1) Let (H, P) be existentially closed, and (Hn, Pn) be ann-dimensional Hilbert
space with an orthonormal projection with the property that Pn(Hn) =
Hn. Then (H, P) ⊕ (Hn, Pn) is an extension of (H, P) with dim([P ⊕
Pn](H ⊕Hn)) ≥ n. Since n can be chosen as big as we want and (H, P)
is existentially closed, dim(P(H)) = ∞.
6 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
(2) Let (H, P) be existentially closed, and (Hn, P0) be ann-dimensionalHilbert
space with an orthonormal projection such that P0(Hn) = {0}. Then
(H, P)⊕ (Hn, P0) is an extension of (H,P) such that dim(([P⊕ P0](H⊕
⊥) ≥ n. Since n can be chosen as big as we want and (H, P) is
existentially closed, dim(P(H)⊥) = ∞.
The theory TPω extending T
P, stating that Pω is a projection and that there
are infinitely many pairwise orthonormal vectors v satisfying Pω(v) = v and
also infinitely many pairwise orthonormal vectors u satisfying Pω(u) = 0 gives
an axiomatization for the the model companion of TP, which corresponds to the
theory of beautiful pairs of Hilbert spaces. We will now study some properties of
Let (H, P) |= TPω and for any v ∈ H let dP(v) = ‖v − P(v)‖. Then dP(v)
measures the distance between v and the subspace P(H). The distance function
dP(x) is definable in (H, P). We will now prove the converse, that is, that we can
definably recover P from dP.
2.2. Lemma. Let (H, P) |= TPω. For any v ∈ Hω let dP(v) = ‖v − P(v)‖. Then
P(v) ∈ dcl(v) in the structure (H, dP).
Proof. Note that P(v) is the unique element x in P(H) satisfying ‖v−x‖ = dP(v).
Thus P(v) is the unique realization of the conditionϕ(x) = max{dP(x), |‖v−x‖−
dP(v)|} = 0. �
2.3. Proposition. Let (H, P) |= TPω. For any v ∈ Hω let dP(v) = ‖v − P(v)‖.
Then the projection function P(x) is definable in the structure (H, dP)
Proof. Let (H, P) |= TPω be κ-saturated for κ > ℵ0 and let dP(v) = ‖v − P(v)‖.
Since dP is definable in the structure (H, P), the new structure (H, dP) is still
κ-saturated. Let GP be the graph of the function P. Then by the previous lemma
GP is type-definable in (H, dP) and thus by [2, Proposition 9.24] P is definable in
the structure (H, dP). �
HILBERT SPACES WITH GENERIC PREDICATES 7
2.4. Notation. We write tp for L-types, tpP for LP-types and q�pP for quantifier
free LP-types. We write aclP for the algebraic closure in the language LP. We
follow a similar convention for dclP .
2.5. Lemma. TPω has quantifier elimination.
Proof. It suffices to show that quantifier free LP-types determine the LP-types.
Let (H, P) |= TPω and let ā = (a1, . . . , an), b̄ = (b1, . . . , bn) ∈ Hn. Assume that
q�pP(ā) = q�pP(b̄). Then
tp(P(a1), . . . , P(an)) = tp(P(b1), . . . , P(bn))
tp(a1 − P(a1), . . . , an − P(an)) = tp(b1 − P(b1), . . . , bn − P(bn)).
Let H0 = P(H) and let H1 = H
0 , both are then infinite dimensional Hilbert
spaces and H = H0 ⊕ H1. Let f0 ∈ Aut(H0) satisfy f0(P(a1), . . . , P(an)) =
(P(b1), . . . , P(bn)) and let f1 ∈ Aut(H1) be such that f1(a1 − P(a1), . . . , an −
P(an)) = (b1−P(b1), . . . , bn−P(bn)). Let f be the automorphism ofH induced
by by f0 and f1, that is, f = f0 ⊕ f1. Then f ∈ Aut(H, P) and f(a1, . . . , an) =
(b1, . . . , bn), so tpP(ā) = tpP(b̄). �
Characterization of types: By the previous lemma, the LP-type of an n-tuple
ā = (a1, . . . , an) inside a structure (H, P) |= T
ω is determined by the type of its
projections tp(P(a1), . . . , P(an), a1 − P(a1), . . . , an − P(an)). In particular, we
may regard (H, P) as the direct sum of the two independent pure Hilbert spaces
(P(H),+, 0, 〈, 〉) and (P(H)⊥,+, 0, 〈, 〉).
We may therefore characterize definable and algebraic closure, as follows.
2.6. Proposition. Let (H, P) |= TPω and let A ⊂ H. Then dclP(A) = aclP(A) =
dcl(A ∪ P(A)).
We leave the proof to the reader. Another consequence of the description of
types is:
8 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
2.7. Proposition. The theory TPω isω-stable.
Proof. Let (H, P) |= TPω be separable and let A ⊂ H be countable. Replacing
(H, P) for (H, P) ⊕ (H, P) if necessary, we may assume that
P(H)∩dclP(A)⊥ is infinite dimensional and that P(H)⊥∩dclP(A)⊥ is infinite
dimensional. Thus every Lp-type over A is realized in the structure (H, P) and
(S1(A), d) is separable. �
2.8. Proposition. Let (H, P) |= TPω be a κ-saturated domain and let A,B,C ⊂ H
be small. Then tpP(A/B ∪C) does not fork over C if and only if tp(A ∪ P(A)/B ∪
P(B) ∪ C ∪ P(C)) does not fork over C ∪ P(C).
Again the proof is straightforward.
3. Continuous random predicates
We now come to our main theory and to our first set of results. We study the
expansion of a Hilbert space with a distance function to a subset of H. Let dN
be a new unary predicate and let LN be the language of Hilbert spaces together
with dN. We denote the LN structures by (H, dN), where dN : H → [0, 1] and we
want to consider the structures where dN is a distance to a subset ofH. Instead of
measuring the actual distance to the subset, we truncate the distance at one. We
start by characterizing the functions dN corresponding to distances.
3.1. The basic theory T0. We denote by T0 the theory of Hilbert spaces together
with the next two axioms (compare with Theorem 9.11 in [2]):
(1) supxmin{1−
· dN(x), infymax{|dN(x) − ‖x− y‖|, dN(y)}} = 0
(2) supx supy[dN(y) − ‖x − y‖− dN(x)] ≤ 0
We say a point is black if dN(x) = 0 and white if dN(x) = 1. All other points
are gray, darker if d(x) is close to zero and whiter if dN(x) is close to one. This
terminology follows [10]. From the second axiom we get that dN is uniformly
continuous (with modulus of uniform continuity ∆(ǫ) = ǫ). Thus we can apply
the tools of continuous model theory to analyze these structures.
HILBERT SPACES WITH GENERIC PREDICATES 9
3.1. Lemma. Let (H, d) |= T0 be ℵ0-saturated and let N = {x ∈ H : dN(x) = 0}.
Then for any x ∈ H, dN(x) = dist(x,N).
Proof. Let v ∈ H and let w ∈ N. Then by the second axiom dN(v) ≤ ‖v − w‖
and thus dN(v) ≤ dist(v,N).
Now let v ∈ H. If dN(v) = 1 there is nothing to prove, so we may assume
that dN(v) < 1. Consider now the set of statements p(x) given by dN(x) = 0,
‖x− v‖ = dN(v).
Claim The type p(x) is approximately satisfiable.
Let ε > 0. We want to show that there is a realization of the statements
dN(x) ≤ ε, dN(v) ≤ ‖x − v‖ + ε. By the first axiom there is w such that
dN(w) ≤ ε and dN(v) ≤ ‖v −w‖+ ε.
Since (H, d) is ℵ0-saturated, there is w ∈ N such that ‖v − w‖ = dN(v) as
we wanted. �
There are several ages that need to be considered. We fix r ∈ [0, 1) and we
consider the class Kr of all models of T0 such that dN(0) = r. Note that in all
finite dimensional spaces in Kr we have at least a point v with dN(v) = 0.
3.2. Notation. If (Hi, d
N) |= T0 for i ∈ {0, 1}, we write (H0, d0N) ⊂ (H1, d1N) if
H0 ⊂ H1 and d0N = d1N ↾H0 (for each sort).
We will work in Kr. We start with constructing free amalgamations:
3.3. Lemma. Let (H0, d
N) ⊂ (Hi, diN) where i = 1, 2 and H1 |⌣H0 H2 be Hilbert
spaces with distance functions, all of them in Kr. Let H3 = span{H1, H2} and let
d3N(v) = min
(PH1(v))
2 + ‖PH2∩H⊥0 (v)‖
(PH2(v))
2 + ‖PH1∩H⊥0 (v)‖
Then (Hi, d
N) ⊂ (H3, d3N) for i = 1, 2, and (H3, d3N) ∈ Kr.
Proof. For arbitrary v ∈ H1,
(PH1(v))
2 + ‖PH2∩H⊥0 (v)‖
2 = d1N(v). Since
(H0, d
N) ⊂ (Hi, diN) we also have
10 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
(PH2(v))
2 + ‖PH1∩H⊥0 (v)‖
(PH0(v))
2 + ‖PH1∩H⊥0 (v)‖
2 ≥ d1N(v).
Similarly, for any v ∈ H2, d3N(v) = d2N(v).
Therefore (H3, d
N) ⊃ (Hi, diN) for i ∈ {1, 2}. Now we have to prove that the
function d3N that we defined is indeed a distance function.
Geometrically, d3N(v) takes the minimum of the distances of v to the selected
black subsets of H1 and H2. That is, the random subset of the amalgamation of
(H1, d
N) and (H2, d
N) is the union of the two random subsets. It is easy to check
that (H3, d
N) |= T0. Since each of (H1, d
N), (H2, d
N) belongs to Kr, we have
d1N(0) = r = d
N(0) and thus d
N(0) = r. �
The classK0 also has the JEP: let (H1, d
N), (H2, d
N) belong toK0 and assume
that H1 ⊥ H2. Let N1 = {v ∈ H1 : d1N(v) = 0} and let N2 = {v ∈ H2 :
d2N(v) = 0}. Let H3 = span(H1 ∪H2) and let N3 = N1 ∪N2 ⊂ H3 and finally,
let d3N(v) = dist(v,N3). Then (H3, d
N) is a witnesses of the JEP in K0.
3.4. Lemma. There is a model (H, dN) |= T0 such that H is a 2n-dimensional
Hilbert space and there are orthonormal vectors v1, . . . , vn ∈ H, u1, . . . , un ∈ H
such that dN((ui+ vj)/2) =
2/2 for i ≤ j, dN(0) = 0 and dN((ui+ vj)/2) = 0
for i > j.
Proof. LetH be a Hilbert space of dimension 2n, and fix some orthonormal basis
〈v1, . . . , vn, u1, . . . , un〉 for H. Let N = {(ui + vj)/2 : i > j} ∪ {0} and let
dN(x) = dist(x,N). Then dN(0) = 0 and dN((ui + vj)/2) = 0 for i > j. Since
‖(ui + vj)/2− (uk + vj)/2‖ =
2/2 for i 6= k and ‖(ui + vj)/2− 0‖ =
we get that dN(ui + vj) =
2/2 for i ≤ j �
A similar construction can be made in order to get the Lemma with dN(0) = r
for any r ∈ [0, 1]. In particular, if we fix an infinite cardinal κ and we amalga-
mate all possible pairs (H,d) in Kr for dim(H) ≤ κ, the theory of the resulting
structure will be unstable.
3.2. The model companion.
HILBERT SPACES WITH GENERIC PREDICATES 11
3.2.1. Basic notations. We now provide the axioms of the model companion of
T0 ∪ {dN(0) = 0}.
Call Td0 the theory of the structure built out of amalgamating all separable
Hilbert spaces together with a distance function belonging to the age K0. Infor-
mally speaking, Td0 = Th(lim−→(K0)). We show how to axiomatize Td0.
The idea for the axiomatization of this part (unlike our third example, in next
section) follows the lines of Theorem 2.4 of [7]. There are however important
differences in the behavior of algebraic closures and independence, due to the
metric character of our examples.
Let (M,dN) inK0 be an existentially closed structure and take some extension
(M1, dN) ⊃ (M,dN). Let x̄ = (x1, . . . , xn+k) be elements in M1 \M and let
z1, . . . , zn+k be their projections on M. Assume that for i ≤ n there are ȳ =
(y1, . . . , yn) inM1 \M that satisfy dN(xi) = ‖xi − yi‖ and dN(yi) = 0. Also
assume that for i > n, the witnesses for the distances to the black points belong
toM, that is, d2N(xi) = ‖xi − zi‖2 + d2N(zi) for i > n. Also, let us assume that
all points in x̄, ȳ live in a ball of radius L around the origin. Let ū = (u1, . . . , un)
be the projection of ȳ = (y1, . . . , yn) overM.
Let ϕ(x̄, ȳ, z̄, ū) be a formula such that ϕ(x̄, ȳ, z̄, ū) = 0 describes the val-
ues of the inner products between all the elements of the tuples, that is, it de-
termines the (Hilbert space) geometric locus of the tuple (x̄, ȳ, z̄, ū). The state-
ment ϕ(x̄, ȳ, z̄, ū) = 0 expresses the position of the potentially new points x̄, ȳ
with respect to their projections into a model. Since dN(xi) = ‖xi − yi‖ and
dN(yi) = 0, we have ‖xi − yj‖ ≥ ‖xi − yi‖ for j ≤ n, i ≤ n. Also, for i > n,
d2N(xi) = ‖xi − zi‖2 + d2N(zi), and get ‖xi − yj‖2 ≥ ‖xi − zi‖2 + d2N(zi) for
j ≤ n.
Note that as (M1, dN) ⊃ (M,dN), for all z ∈ M, d2N(z) ≤ ‖z − yi‖2 =
‖z − ui‖2 + ‖yi − ui‖2 for i ≤ n. We may also assume that there is a positive
real ηϕ such that ‖xi − zi‖ ≥ ηϕ for i ≤ n + k and ‖yi − ui‖ ≥ ηϕ for i ≤ n.
3.2.2. An informal description of the axioms. We want to express that for any pa-
rameters z̄, ū in the structure
12 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
if we can find realizations x̄, ȳ of ϕ(x̄, ȳ, z̄, ū) = 0 such that for allw and i ≤ n,
d2N(w) ≤ ‖w− ui‖2 + ‖ui − yi‖2, ‖xi − yi‖2 ≤ ‖xi − zi‖2 + d2N(zi) for i ≤ n,
‖xi − yj‖2 ≥ ‖xi − zj‖2 + d2N(zj) for i > n and j ≤ n,
then there are tuples x̄ ′, ȳ ′ such that ϕ(x̄ ′, ȳ ′, z̄, ū) = 0, dN(y
i) = 0, dN(x
‖x ′i − y ′i‖ for i ≤ n and d2N(xj) = ‖xj − zj‖2 + d2N(zj) for j > n.
That is, for any z̄, ū in the structure, if we can find realizations x̄, ȳ of the
Hilbert space locus given byϕ, and we prescribe “distances” dN that do not clash
with the dN information we already had, in such a way that for i ≤ n, the yi’s
are black and are witnesses for the distance to the black set for the xi’s, and for
i > n the xi’s do not require new witnesses, then we can actually find arbitrarily
close realizations, with the prescribed distances.
The only problemwith this idea is that we do not have an implication in contin-
uous logic. We replace the expression “p → q” by a sequence of approximations
indexed by ε.
3.2.3. The axioms of TN.
3.5. Notation. Let z̄, ū be tuples inM and let x ∈M1. By Pspan(z̄ū)(x) we mean
the projection of x in the space spanned by (z̄, ū).
For fixed ε ∈ (0, 1), let f : [0, 1] → [0, 1] be a continuous function such that
whenever ϕ(t̄) < f(ε) and ϕ(t̄ ′) = 0, then
(a): ‖Pspan(z̄ū)(xi) − zi‖ < ε.
(b): ‖Pspan(z̄ū)(yi) − ui‖ < ε.
(c): |‖ti − tj‖− ‖t ′i − t ′j‖| < ε where t̄ is the concatenation of x̄, ȳ, z̄, ū.
Choosing ε small enough, we may assume that
(d): ‖xi − Pspan(z̄ū)(xi)‖ ≥ ηϕ/2 for i ≤ n + k.
(e): ‖yi − Pspan(z̄ū)(yi)‖ ≥ ηϕ/2 for i ≤ n.
Let δ = 2
ε(L+ 2) and consider the following axiom ψϕ,ε (which we write
as a positive bounded formula for clarity) where the quantifiers range over a ball
of radius L+ 1:
HILBERT SPACES WITH GENERIC PREDICATES 13
∀z̄∀ū
∀x̄∀ȳϕ(x̄, ȳ, z̄, ū) ≥ f(ε)∨∃w
i≤n(d
N(w) ≥ ‖w−ui‖2+‖yi−ui‖2+
i>n,j≤n(‖xi − yj‖2 ≤ ‖xi − zi‖2 + d2N(zi) + ε2)∨
i,j≤n,j6=i(‖xi−yj‖ ≤ ‖xi−yi‖−ε)∨
i≤n(‖xi−zi‖2+d2N(zi) ≤ ‖xi−yi‖2−ε2)
∨∃x̄∃ȳ
(ϕ(x̄, ȳ, z̄, ū) ≤ f(ε)∧
i≤n dN(yi) ≤ δ)∧
i≤n |dN(xi)−‖xi−yi‖| ≤
i>n |d
N(xi) − ‖xi − zi‖2 − d2N(zi)| ≤ 4δL
Let TN be the theory T0 together with this scheme of axiomsψϕ,ε indexed by all
Hilbert space geometric locus formulas ϕ(x̄, ȳ, z̄, ū) = 0 and ε ∈ (0, 1) ∩ Q. The
radius of the ball that contains all elements, L, as well as n and k are determined
from the configuration of points described by the formula ϕ(x̄, ȳ, z̄, ū) = 0.
3.2.4. Existentially closed models of T0.
3.6. Theorem. Assume that (M,dN) |= T0 is existentially closed. Then (M,dN) |=
Proof. Fix ε > 0 and ϕ as above. Let z̄ ∈Mn+k, ū ∈Mn and assume that there
are x̄, ȳwithϕ(x̄, ȳ, z̄, ū) < f(ε) and d2N(w) < ‖w−ui‖2+‖yi−ui‖2+ε2 for all
w ∈M, ‖xi−yi‖2 < ‖xi−zi‖2+d2N(zi)+ε2 for i ≤ n, ‖xi−yj‖ > ‖xi−yi‖−ε
for i, j ≤ n, i 6= j, ‖xi − yj‖2 > ‖xi − zi‖2 + d2N(zj) + ε2 for i > n, j ≤ n. Let
ε ′ < ε be such that ϕ(x̄, ȳ, z̄, ū) < f(ε ′) and
(f): d2N(w) < ‖w − ui‖2 + ‖yi − ui‖2 + ε ′2 for allw ∈M.
(g1): ‖xi − yi‖2 > ‖xi − zi‖2 + dN(zi) + ε ′2 for i ≤ n.
(g2): ‖xi − yj‖ > ‖xi − yi‖− ε ′ for i, j ≤ n, i 6= j
(h): ‖xi − yj‖2 ≥ ‖xi − zi‖2 + d2N(zi) + ε ′2 for i > n, j ≤ n.
We construct an extension (H,dN) ⊃ (M,dN) where the conclusion of the
axiom indexed by ε ′ holds. Since (M,dN) is existentially closed and the conclu-
sion of the axiom is true for (H,dN) replacing ε for ε
′ < ε, then the conclusion
of the axiom indexed by ε will hold for (M,dN).
So let H ⊃ M be such that dim(H ∩ M⊥) = ∞. Let a1, . . . , an+k and
c1, . . . , cn ∈ H be such that tp(ā, c̄/z̄ū) = tp(x̄, ȳ/z̄ū) and āc̄ |
⌣z̄ū
M. We
14 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
can write ai = a
i + z
i and ci = c
i + u
i for some z
i ∈ M and a ′i, c ′i ∈M⊥.
By (d) and (e) ‖a ′i‖ ≥ η/2 for i ≤ n + k and ‖c ′i‖ ≥ η/2 for i ≤ n. Now let
ĉi = c
i + u
i + δ
′c ′i/‖c ′i‖, where δ ′ =
2ε ′(L+ 2).
Let the black points in H be the ones fromM plus the points ĉ1, . . . , ĉn. Now
we check that the conclusion of the axiom ψϕ,ε ′ holds.
(1) ϕ(ā, c̄, z̄, ū) ≤ f(ε ′) since tp(ā, c̄/z̄ū) = tp(x̄, ȳ/z̄ū).
(2) Since ‖ci − ĉi‖ ≤ δ ′ and ĉi is black we have dN(ci) ≤ δ ′.
(3) We check that the distance from ai to the black set is as prescribed for
i ≤ n. dN(ai) ≤ ‖ai − ĉi‖ ≤ ‖ai − ci‖+ δ ′ for i ≤ n.
Also, for i 6= j, i, j ≤ n, using (g2)we prove ‖ai− ĉj‖ ≥ ‖ai−cj‖−δ ′ ≥
‖ai − ci‖ − ε ′ − δ ′ ≥ ‖ai − ci‖− 2δ ′. Finally by (a) ‖ai − PM(ai)‖2 +
d2N(PM(ai)) ≥ (‖ai − zi‖− ε ′)2+(dN(zi)− ε ′)2 ≥ ‖ai− zi‖2− 2Lε ′ +
ε ′2 + d2N(zi) − 2ε
′ + ε ′2 and by (g1), we get ‖ai − zi‖2 − 2Lε ′ + ε ′2 +
d2N(zi) − 2ε
′ + ε ′2 ≥ ‖ai − ci‖2 − 2Lε ′ − 2ε ′ ≥ ‖ai − ci‖2 − 4δ ′2.
(4) We check that dN(ai) is as desired for i > n. Clearly ‖aj − ĉi‖ ≥ ‖aj −
ci‖ − δ ′, so ‖aj − ĉi‖2 ≥ ‖aj − ci‖2 + δ ′2 − 2δ ′2L and by (h) we get
‖aj − ci‖2 + δ ′2 − 4δ ′L ≥ ‖aj − zj‖2 + d2N(zj) − 4δ ′L − ε ′2 + δ ′2 ≥
‖aj − zj‖2 + d2N(zj) − 4δ ′L.
It remains to show that (M,dN) ⊂ (H,dN), i.e., the function dN onH extends
the function dN on M . Since we added the black points in the ball of radius
L + 1, we only have to check that for any w ∈ M in the ball of radius L + 2,
d2N(w) ≤ ‖w − ĉi‖2 = ‖w − u ′i‖2 + ‖c ′i + δ ′(c ′i/‖c ′i‖)‖2.
But by (f) d2N(w) ≤ ‖w − ui‖2 + ‖ci − ui‖2 + ε ′2, so it suffices to show that
‖w− ui‖2 + ‖ci − ui‖2 + ε ′2 ≤ ‖w− u ′i‖2 + ‖c ′i‖2 + 2δ ′‖c ′i‖+ δ ′2
By (a) ‖w− u ′i‖2 ≥ (‖w − ui‖− ε ′)2 and is enough to prove that
‖w− ui‖2 + ‖ci − ui‖2 + ε ′2 ≤ (‖w − ui‖− ε ′)2 + ‖c ′i‖2 + 2δ ′‖c ′i‖+ δ ′2
But (‖w−ui‖−ε ′)2+‖c ′i‖2+2δ ′‖c ′i‖+δ ′2 = ‖w−ui‖2−2ε ′‖w−ui‖+ε ′2+
‖c ′i‖2+2δ ′‖c ′i‖+δ ′2 and ‖ci−ui‖2 ≤ ‖ci−u ′i‖2+2ε ′‖ci−u ′i‖+ε ′2 = ‖c ′i‖2+
HILBERT SPACES WITH GENERIC PREDICATES 15
2ε ′‖c ′i‖+ε ′2. Thus, after simplifying, we only need to check 2ε ′‖w−ui‖+ε ′2 ≤
δ ′2 which is true since 2ε ′‖w−ui‖+ ε ′2 ≤ 2ε ′(2L+ 2) + ε ′2 ≤ 4ε ′(L+ 2). �
3.7. Theorem. Assume that (M,dN) |= TN. Then (M,dN) is existentially closed.
Proof. Let (H,dN) ⊃ (M,dN) and assume that (H,dN) is ℵ0-saturated. Let
ψ(x̄, v̄) be a quantifier free LN-formula, where x̄ = (x1, . . . xn+k) and v̄ = (v1, . . . vl).
Suppose that there are a1, . . . , an+k ∈ H \ M and e1, . . . el ∈ M such that
(H,dN) |= ψ(ā, ē) = 0. After enlarging the formula ψ if necessary, we may
assume that ψ(x̄, v̄) = 0 describes the values of dN(xi) for i ≤ n+ k, the values
of dN(vj) for j ≤ l and the inner products between those elements. We may as-
sume that for i ≤ n there is ρ > 0 such that dN(ai)−d(ai, z) ≥ 2ρ for all z ∈M
with dN(z) ≤ ρ. Since (H,dN) isℵ0-saturated, there are c1, . . . cn ∈ H such that
dN(ai) = ‖ai− ci‖ and dN(ci) = 0. Then d(ci,M) ≥ ρ. Fix ε > 0, ε < ρ, 1. We
may also assume that for i > n, |d2N(ai)−‖ai−PM(ai)‖2−d2N(PM(ai))| ≤ ε/2.
Also, assume that all points mentioned so far live in a ball of radius L around the
origin. Let b1, . . . , bn+k ∈M be the projections of a1, . . . , an+k ontoM and let
d1, . . . , dn ∈ M be the projections of c1, . . . , cn ontoM. Let ϕ(x̄, ȳ, z̄, ū) = 0
be an L-statement that describes the inner products between the elements listed
and such that ϕ(ā, c̄, b̄, d̄) = 0. Using the axioms we can find ā ′, c̄ ′ inM such
that ϕ(ā ′, c̄ ′, b̄, d̄) ≤ f(ε), dN(c ′i) ≤ δ for i ≤ n, |dN(a ′i) − ‖a ′i − c ′i‖| ≤ δ for
i ≤ n and |d2N(ai) − ‖ai − bi‖2 −d2N(bi)| ≤ 4Lδ, where δ =
2ε(L + 2). Since
ε > 0 was arbitrary we get (M,dn) |= infx1 . . . infxn+k ψ(x̄, v̄) = 0. �
4. Model theoretic analysis of TN
We prove three theorems in this section about the theory TN:
• TN is not simple,
• TN is not even NTP2! (Of course, this implies the previous, but we will
provide the proof of non-simplicity as well.)
16 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
• TN is NSOP1. Therefore, in spite of having a tree property, our theory is
still “close to being simple” in the precise sense of not having the SOP1
tree property.
These results place TN in a very interesting situation in the stability hierarchy
for continuous logic.
4.1. Notation. We write tp for types of elements in the language L and tpN for
types of elements in the language LN. Similarly we denote by aclN the algebraic
closure in the language LN and by acl the algebraic closure for pure Hilbert spaces.
Recall that for a set A, acl(A) = dcl(A), and this corresponds to the closure of
the space spanned by A (Fact 1.2).
4.2. Observation. The theory TN does not have elimination of quantifiers. We
use the characterization of quantifier elimination given in Theorem 8.4.1 from
[9]. Let H1 be a two dimensional Hilbert space, let {u1, u2} be an orthonormal
basis for H1 and let N1 = {0, u0 +
u1} and let d
N(x) = min{1, dist(x,N1)}.
Then (H1, d
N) |= T0. Let a = u0, b = u0 −
u1 and c = u0 +
u1. Note
that d1N(b) =
. Let (H ′1, d
N) ⊃ (H1, d1N) be existentially closed. Now let H2
be an infinite dimensional separable Hilbert space and let {vi : i ∈ ω} be an
orthonormal basis. LetN2 = {x ∈ H : ‖x− v1‖ = 14 , Pspan(v1)(x) = v1} ∪ {0} and
let d2N(x) = min{1, dist(x,N2)}. Let (H
N) ⊃ (H2, d2N) be existentially closed.
Then (span(a), d1N ↾span(a))
∼= (span(v1), d
N ↾span(v1)) and they can be identified
say by a function F. But (H ′1, d
N) and (H
N) cannot be amalgamated over this
common substructure: If they could, then we would have dist(F(b), v1 +
vi) =
dist(b, v1 +
vi) <
for some i > 1 and thus d1N(b) <
, a contradiction.
In this case, the main reason for this failure of amalgamation resides in the
fact that (span(a), d1N ↾span(a))
∼= (span(v1), d
N ↾span(v1)) is not a model of T0:
informally, the distance values around v1 are determinedby an “external attractor”
(the black point u0 +
u1 or the black ring orthogonal to v1 at distance
) that
the subspace (span(a), d1N ↾span(a)) simply cannot see. This violates Axiom (1) in
HILBERT SPACES WITH GENERIC PREDICATES 17
the description of T0. This “noise external to the substructure” accounts for the
failure of amalgamation, and ultimately for the lack of quantifier elimination.
In [7, Corollary 2.6], the authors show that the algebraic closure of the expan-
sion of a simple structure with a generic subset corresponds to the algebraic in
the original language. However, in our setting, the new algebraic closure aclN(X)
does not agree with the old algebraic closure acl(X):
4.3. Observation. The previous construction shows that aclN does not coincide
with acl. Indeed, c ∈ aclN(a) \ acl(a) - the set of solutions of the type tpN(c/a)
is {c}, but c /∈ dcl(a) as c /∈ span(a).
However, models of the basic theory T0 are LN-algebraically closed. The proof
is similar to [7, Proposition 2.6(3)]:
4.4. Lemma. Let (M,dN) |= TN and let A ⊂ M be such that A = dcl(A) and
(A,dN ↾A) |= T0. Let a ∈M. Then a ∈ aclN(A) if and only if a ∈ A.
Proof. Assume a /∈ A. We will show that a /∈ aclN(A). Let a ′ |= tp(a/A) be
such that a ′ |
M. Let (M ′, dN) be an isomorphic copy of (M,dN) over A
through f : M →A M ′ such that f(a) = a ′. We may assume thatM ′ |
Since (A,dN ↾A) is an amalgamation base, (N,dN) = (M ⊕A M ′, dN) |= T0.
Let (N ′, dN) ⊃ (N,dN) be an existentially closed structure. Then tpN(a/A) =
tpN(a
′/A) and therefore a /∈ aclN(A). �
As TN is model complete, the types in the extended language are determined by
the existential formulas within them, i.e. formulas of the form inf ȳϕ(ȳ, x̄) = 0
Another difference with the work of Chatzidakis and Pillay is that the analogue
to [7, Proposition 2.5] no longer holds. Let a, b, c be as in Observation 4.3; notice
that (span(a), dN ↾span(a))
∼= (span(v1), dN ↾span(v1)). However, (H
1, dN, a) 6≡
(H ′2, dN, v1). Instead, we can show the following weaker version of the Proposi-
tion.
18 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
4.5.Proposition. Let (M,dN) and (N,dN) bemodels of TN and letA be a common
subset ofM and N such that (span(A), dN ↾span(A)) |= T0. Then
(M,dN) ≡A (N,dN).
Proof. Assume thatM∩N = span(A). Since (span(A), dN ↾span(A)) |= T0, it is an
amalgamation base and therefore we may consider the free amalgam (M⊕span(A)
N,dN) of (M,dN) and (N,dN) over (span(A), dN ↾span(A)). Let now (E, dN) be
a model of TN extending (M⊕span(A) N,dN). By the model completeness of TN,
we have that (M,dN) ≺ (E, dN) and (N,dN) ≺ (E, dN) and thus (M,dN) ≡A
(N,dN). �
4.1. Generic independence. In this section we define an abstract notion of in-
dependence and study its properties.
Fix (U, dN) |= TN be a κ-universal domain.
4.6. Definition. Let A,B,C ⊂ U be small sets. We say that A is ∗-independent
from B over C and write A |∗
B if aclN(A ∪ C) is independent (in the sense of
Hilbert spaces) from aclN(C ∪ B) over aclN(C). That is, A |∗
B if for all a ∈
aclN(A ∪ C), PB∪C(a) = PC(a), where B ∪ C = aclN(C ∪ B) and C = aclN(C).
4.7. Proposition. The relation |∗
satisfies the following properties (here A, B, etc.,
are any small subsets of U):
(1) Invariance under automorphisms of U.
(2) Symmetry: A |∗
B⇐⇒ B |∗
(3) Transitivity: A |∗
BD if and only if A |∗
B and A |∗
(4) Finite Character: A |∗
B if and only ā |∗
B for all ā ∈ A finite.
(5) Local Character: If ā is any finite tuple, then there is countable B0 ⊆ B such
that ā |∗
(6) Extension property over models of T0. If (C,dN ↾C) |= T0, then we can find
A ′ such that tpN(A/C) = tpN(A
′/C) and A ′ |∗
(7) Existence over models: ā |∗
M for any ā.
(8) Monotonicity: āā ′ |∗
b̄b̄ ′ implies ā |
HILBERT SPACES WITH GENERIC PREDICATES 19
Proof. (1) Is clear.
(2) It follows from the fact that independence in Hilbert spaces satisfies Sym-
metry (see Proposition 1.3).
(3) It follows from the fact that independence in Hilbert spaces satisfies Tran-
sitivity (see Proposition 1.3).
(4) Clearly A |∗
B implies that ā |∗
B for all ā ∈ A finite. On the other
hand if ā |∗
B for all ā ∈ A finite, then for a dense subset A0 of A,
B and thus A |∗
(5) Local Character: let ā be a finite tuple. Since independence in Hilbert
spaces satisfies local character, there is B1 ⊆ aclN(B) countable such that
ā |∗
B. Now let B0 ⊆ B be countable such that aclN(B0) ⊃ B1. Then
ā |∗
(6) LetC be such that (C,dN ↾C) |= T0. LetD ⊃ A∪C be such that (D,dN ↾D
) |= T0 and let E ⊃ B ∪ C be such that (E, dN ↾E) |= T0. Changing D for
another set D ′ with tpN(D
′/C) = tpN(D/C), we may assume that the
space generated by D ′ ∪ E is the free amalgamation of D ′ and E over C.
By lemma 4.4 D ′, E are algebraically closed andD ′ |∗
(7) It follows from the definition of ∗-independence.
(8) It follows from the definition of ∗-independence and transitivity.
Therefore we have a natural independence notion that satisfies many good
properties, but not enough to guarantee the simplicity of TN.
We will show below that the theory TN has both TP2 and NSOP1. This places
it in an interesting area of the stability hierarchy for continuous model theory:
while having the tree property TP2 and therefore lacking the good properties of
NTP2 theories, it still has a quite well-behaved independence notion |
, good
enough to guarantee that it does not have the SOP1 tree property. Therefore,
although the theory is not simple, it is reasonably close to this family of theories.
4.2. The failure of simplicity.
20 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
4.8. Theorem. The theory TN is not simple.
The proof’s idea uses a characterization of simplicity in terms of the number of
partial types due to Shelah (see [11]; see also Casanovas [6] for further analysis): T
is simple iff for all κ, λ such thatNT(κ, λ) < 2κ+λω, whereNT(κ, λ) counts the
supremum of the cardinalities of families of pairwise incompatible partial types
of size ≤ κ over a set of cardinality≤ λ. This holds for continuous logic as well.
We show that TN fails this criterion.
Proof. Fix κ an infinite cardinal and λ ≥ κ. We will find a complete submodel
Mf of the monster model, of density character λ, and λ
κ many types over sub-
sets of Mf of power κ in such a way that we guarantee that they are pairwise
incompatible in a uniform way.
Now also fix some orthonormal basis ofMf, listed as
{bi|i < κ} ∪ {aj|j < λ} ∪ {cX|X ∈ Pκ(λ)}.
Also fix, for every X ∈ Pκ(λ), a bijection
fX : {bi|i < κ} → {aj|j ∈ X}.
Let the “black points” ofMf consist of the set
N = {cX + bi + (1/2)fX(bi) | i < κ,X ∈ Pκ(λ)} ∪ {0}
and as usual define dN(x) as the distance from x to N. This is a submodel of the
monster.
Let AX := {bi|i < κ} ∪ {aj|j ∈ X} for each X ∈ Pκ(λ).
The crux of the proof is to notice that if X 6= Y then the types tp(cX/AX) and
tp(cY/AY) are incompatible, thereby witnessing that there are λ
κ many incom-
patible types:
Suppose there is some c such that tp(c/AX) = tp(cX/AX) and tp(c/AY) =
tp(cY/AY). Take (wlog) j ∈ Y \ X. Pick ℓ < κ such that fY(bℓ) = aj. Let k ∈ X
be such that fX(bℓ) = ak.
HILBERT SPACES WITH GENERIC PREDICATES 21
InMf, the distance to black of cX+bℓ−
ak is 1: by definition, cX+bℓ+
cX + bℓ +
fX(bℓ) ∈ N and the only difference between cX + bℓ − 12ak and
cX + bℓ +
ak is the sign in front of an element of an orthonormal basis.
Therefore the distance to black of d = c+ bℓ −
ak is also 1 (in the monster).
However, e = c + bℓ +
aj must be a black point, since e
′ = cY + bℓ +
aj is
black (by definition of N and since aj = fY(bℓ) and tp(c/AY) = tp(cY/AY)).
On the other hand, the distance from e to d is
< 1. This contradicts that
the color of d is 1. �
This stands in sharp contrast with respect to the result by Chatzidakis and
Pillay in the (discrete) first order case. The existence of these incompatible types
is rendered possible here by the presence of “euclidean” interactions between the
elements of the basis chosen.
So far we have two kinds of expansions of Hilbert spaces by predicates: either
they remain stable (as in the case of the distance to a Hilbert subspace as in the
previous section) or they are not even simple.
4.3. TN has the tree property TP2.
4.9. Theorem. The theory TN has the tree property TP2.
Proof. We will construct a complete submodel M |= T0 of the monster model,
of density character 2ℵ0 , and a quantifier free formula ϕ(x;y, z) that witnesses
TP2 inside M. Since this model can be embedded in the monster model of TN
preserving the distance to black points, this will show that TN has TP2.
We fix some orthonormal basis ofM, listed as
{bi|i < ω} ∪ {cn,i|n, i < ω} ∪ {af|f : ω→ ω}.
Also let the “black points” ofM consist of the set
N = {af + bn + (1/2)cn,f(n) | n < ω, f : ω→ ω} ∪ {0}
and as usual define dN(x) as the distance from x to N. This is a model of T0 and
thus a submodel of the monster.
22 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
Let ϕ(x, y, z) = max{1− dN(x+ y− (1/2)z), dN(x+ y − (1/2)z)}.
Claim1 For each i, the conditions {ϕ(x, bi, ci,j) = 0 : j ∈ ω} are 2-inconsistent.
Assume otherwise, so we can find a (in an extension ofM) such that dN(a +
bi + (1/2)ci,j) = 0 and dN(a + bi − (1/2)ci,l) = 1 for some j < l. But then
d(a+bi+(1/2)ci,j, a+bi−(1/2)ci,l) = d((1/2)ci,j,−(1/2)ci,l) =
2/2 < 1.
Sincea+bi+(1/2)ci,j is a black point, we get thatdN(a+bi−(1/2)ci,l) ≤
a contradiction.
Claim 2 For each f the conditions {ϕ(x, bi, ci,f(i)) = 0 : i ∈ ω} are consistent.
Indeed fix f and consider af, then by construction dN(af+bn+(1/2)cn,f(n)) =
0 and d(af + bn − (1/2)cn,f(n), af + bn + (1/2)cn,f(n)) = 1, so dN(af + bn −
(1/2)cn,f(n)) ≤ 1.
Now we check the distance to the other points in N. It is easy to see that
d(af + bn − (1/2)cn,f(n), af + bm + (1/2)cm,f(m)) > 1 form 6= n, d(af + bn −
(1/2)cn,f(n), ag + bk + (1/2)ck,g(k)) > 1 for g 6= f and all indexes k. Finally,
d(af + bn − (1/2)cn,f(n), 0) > 1. This shows that af is a witness for the claim.
4.4. TN and the property NSOP1. Chernikov and Ramsey have proved that
whenever a first order discrete theory satisfies the following properties (for ar-
bitrary models and tuples), then the theory satisfies theNSOP1 property (see [8,
Prop. 5.3]).
• Strong finite character: whenever ā depends on b̄ overM, there is a for-
mula ϕ(x, b̄, m̄) ∈ tp(ā/b̄M) such that every ā ′ |= ϕ(x̄, b̄, m̄) depends
on b̄ overM.
• Existence over models: ā |
M for any ā.
• Monotonicity: āā ′ |
b̄b̄ ′ implies ā |
• Symmetry: ā |
b̄ ⇐⇒ b̄ |
• Independent amalgamation: c̄0 |
c̄1, b̄0 |
c̄0, b̄1 |
c̄1, b̄0 ≡M b̄1
implies there exists b̄ with b̄ ≡c̄0M b̄0, b̄ ≡c̄1M b̄1.
HILBERT SPACES WITH GENERIC PREDICATES 23
We prove next that in TN, |
∗ satisfies analogues of these five properties -
we may thereby conclude that TN can be regarded (following the analogy) as a
NSOP1 continuous theory.
In what remains of the paper, we prove that TN satisfies these properties.
We focus our efforts in strong finite character and independent amalgamation,
the other properties were proved in Proposition 4.7.
We need the following setting:
Let M be the monster model of TN andA ⊂M. FixAwithA ⊂ A ⊂ M be such
that A |= T0 and let ā = (a0, . . . , an) ∈ M. We say that (ā, A,B) is minimal if
tp(B/A) = tp(A/A) and for all b̄ ∈ M, if tp(b̄/A) = tp(ā/A) then
‖ prB(b0)‖+ · · · + ‖ prB(bn)‖ ≥ ‖ prB(a0)‖+ · · · + ‖ prB(an)‖.
By compactness, for all p ∈ S(A) there is a minimal (ā, A,B) such that ā |= p.
Now let cl0(A) be the set of all x such that for some minimal (ā, A,B), x =
prB(a0) (the first coordinate of ā).
4.10. Lemma. If tp(B/A) = tp(A/A) and x ∈ cl0(A) then x ∈ B.
Proof. Suppose not. Let B witness this and let C and ā be such that (ā, A,C) is
minimal and x = prC(a0). Since C |= T0, wlog B |⌣C a (independence in the
sense of Hilbert spaces). But then prB(ai) = prB(prC(ai)) for each i and thus
‖ prB(a0)‖+ · · ·+‖ prB(an)‖ < ‖ prC(a0)‖+ · · ·+‖ prC(an)‖. This contradicts
minimality. �
A direct consequence of the previous lemma is that cl0(A) ⊂ bclN(A) =
∩A⊂B|=TNB, as cl0(A) belongs to every model of the theory TN.
We now define the essential closure ecl. Let clα+1(A) = cl0(clα(A)) for all
ordinals α, clδ(A) =
α<δ(clα(A)), and ecl(A) =
α∈On clα(A).
4.11. Lemma. For all ā, B,A, if ecl(A) = A then there is b̄ such that tp(b̄/A) =
tp(ā/A) and b̄ |
Proof. Choose A |= T0 such that A ⊂ A and c̄ such that tp(c̄/A) = tp(ā/A)
and (c̄, A,A) is minimal. Since cl0(A) = A, prA(ci) ∈ A for all i ≤ n (c̄ =
24 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
(c0, . . . , cn)), i.e. c̄ |
A. Now choose b̄ such that tp(b̄/A) = tp(c̄/A) and
B. Then b̄ is as needed. �
4.12. Corollary. ecl(A) = aclN(A).
Proof. Clearly aclN(A) ⊂ bclN(A). On the other hand, assume that x /∈ aclN(A).
Let B be a model of TN such that A ⊂ B. By Lemma 4.11, we may assume that
B. Then x /∈ B, so x /∈ bcl(A), so x /∈ ecl(A). �
4.13. Theorem. Suppose ecl(A) = A, A ⊂ B,C, B |∗
C (i.e. ecl(B) |
ecl(C)),
ā |∗
B, b̄ |∗
C and tp(ā/A) = tp(b̄/A). Then there is c̄ such that tp(c̄/B) =
tp(ā/B), tp(c̄/C) = tp(b̄/C) and c̄ |∗
Proof. Wlog ecl(B) = B and ecl(C) = C. By Lemma 4.11 we can find modelsA0,
A1, B
∗ and C∗ of T0 such that Aā ⊂ A0, Ab̄ ⊂ A1, B ⊂ B∗ and C ⊂ C∗, such
that B∗ |∗
C∗,A0 |
B∗ andA1 |
C∗. We can also find models of T0,A
and D∗ such that A0B
∗ ⊂ A∗0, A1C∗ ⊂ A∗1 and B∗C∗ ⊂ D∗ and wlog we may
assume that ā and b̄ are chosen so thatA∗0 |⌣B∗ D
∗,A∗1 |⌣C∗ D
∗, and that there is
an automorphism F of the monster model fixingA pointwise such that F(ā) = b̄,
F(A0) = A1 and F(A
0) |⌣A1
A∗1 . Notice that now
D∗ and A1 |
We can now find Hilbert spaces A∗, A∗∗0 , A
1 and E such that
(i) E is generated byD∗A∗∗0 A
(ii) A ⊂ A∗ ⊂ A∗∗0 ∩A∗∗1 , B∗ ⊂ A∗∗0 , C∗ ⊂ A∗∗1 ,
(iii) There are Hilbert space isomorphisms G : A∗∗0 → A
0 and H : A
1 → A
such that
a) F ◦G ↾ A∗ = H ↾ A∗,
b) G ↾ B∗ = idB∗ , H ↾ C
∗ = idC∗ ,
c) G ∪ idD∗ generate an isomorphism
〈A∗∗0 D∗〉 → 〈A∗0D∗〉
HILBERT SPACES WITH GENERIC PREDICATES 25
d) H ∪ idD∗ generate an isomorphism
〈A∗∗1 D∗〉 → 〈A∗1D∗〉
e) F ∪G ∪H generate an isomorphism
〈A∗∗0 A∗∗1 〉 → 〈F(A∗0)A∗1〉.
We can find these because non-dividing independence in Hilbert spaces has 3-
existence (the independence theorem holds for types over sets).
Now we choose the “black points” of our model: a ∈ E is black if one of the
following holds:
(i) a ∈ A∗∗0 and G(a) is black,
(ii) a ∈ A∗∗0 and H(a) is black,
(iii) a ∈ D∗ and is black.
Then in E we define the “distance to black” function simply as the real distance.
Then in D∗ there is no change and G and H remain isomorphisms after adding
this structure;D∗, A∗0, A
1 and F(A
0) witness this.
So we can assume that E is a submodel of the monster, and letting c̄ = G−1(a),
G witnesses that tp(c̄/B) = tp(ā/B) and H witnesses that tp(c̄/C) = tp(b̄/C).
We have already seen that A∗ |
D∗ and thus c̄ |∗
BC. �
4.14. Proposition. Suppose b̄ 6 |∗
C, A ⊂ B ∩ C and (wlog) C = bcl(C). Then
there exists a formula χ ∈ tp(b̄/C) such that for all ā |= χ, ā 6 |∗
Proof. By compactness, we can find ε > 0 such that (letting b̄ = (b0, . . . , bn),
(ā = (a0, . . . , an)),
∀ā |= tp(b̄/B), ‖ prC(a0)‖+· · ·+‖ prC(an)‖ ≥ ε+‖ prbcl(A)(a0)‖+· · ·+‖ prbcl(A)(an)‖.
Again by compactness we can find χ ∈ tp(b̄/B) such that (1) holds when we
replace tp(b̄/B) by χ and ε by ε/2, that is:
26 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES
∀ā |= χ, ‖ prC(a0)‖+· · ·+‖ prC(an)‖ ≥ ε/2+‖ prbcl(A)(a0)‖+· · ·+‖ prbcl(A)(an)‖.
and in particular ā 6 |∗
C, as we wanted �
References
[1] Itaï Ben Yaacov, Lovely pairs of models: the non first order case, Journal of Symbolic Logic, volume
69 (2004), 641-662.
[2] Itaï Ben Yaacov, Alexander Berenstein, Ward C. Henson and Alexander Usvyatsov, Model The-
ory for metric structures in Model Theory with Applications to Algebra and Analysis, Volume
2, Cambridge University Press, 2008.
[3] Itaï Ben Yaacov, Alexander Usvyatsov and Moshe Zadka, Generic automorphism of a Hilbert
space, preprint avaliable at http://ptmat.fc.ul.pt/∼alexus/papers.html.
[4] Alexander Berenstein, Hilbert Spaces with Generic Groups of Automorphisms, Arch. Math. Logic,
vol 46 (2007) no. 3, 289–299.
[5] Alexander Berenstein and Steven Buechler, Simple stable homogeneous expansions of Hilbert
spaces. Ann. Pure Appl. Logic 128 (2004), no. 1-3, 75–101.
[6] Enrique Casanovas, The number of types in simple theories. Ann. Pure Appl. Logic 98 (1999),
69–86.
[7] Zoé Chatzidakis and Anand Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic
95 (1998), no. 1-3, 71–92.
[8] Artem Chernikov and Nicholas Ramsey, On model-theoretic tree properties, Journal of Mathe-
matical Logic, 16 (2), 2016.
[9] Wilfrid Hodges,Model Theory, Cambridge University Press 1993.
[10] Bruno Poizat, Le carré de l’égalité, Journal of Symbolic Logic 64 (1999), 1339–1355.
[11] Saharon Shelah, Simple Unstable Theories, Ann. Math Logic 19 (1980), 177–203.
http://ptmat.fc.ul.pt/~alexus/papers.html
HILBERT SPACES WITH GENERIC PREDICATES 27
Alexander Berenstein, Universidad de los Andes, Departamento de Matemáticas, Cra
1 # 18A-10, Bogotá, Colombia.
URL: www.matematicas.uniandes.edu.co/~aberenst
Tapani Hyttinen, University of Helsinki, Department of Mathematics and Statistics,
Gustaf Hällströminkatu 2b. Helsinki 00014, Finland.
E-mail address: tapani.hyttinen@helsinki.fi
Andrés Villaveces, Universidad Nacional de Colombia, Departamento de Matemáticas,
Av. Cra 30 # 45-03, Bogotá 111321, Colombia.
E-mail address: avillavecesn@unal.edu.co
	1. Introduction
	1.1. Model theory of Hilbert spaces (quick review)
	2. Random subspaces and beautiful pairs
	3. Continuous random predicates
	3.1. The basic theory T0
	3.2. The model companion
	4. Model theoretic analysis of TN
	4.1. Generic independence
	4.2. The failure of simplicity
	4.3. TN has the tree property TP2
	4.4. TN and the property NSOP1
	References
