SUBFACTORS OF FREE PRODUCTS OF RESCALINGS OF A II_{1}–FACTOR

KEN DYKEMA

Abstract. Let Q be any II_{1}–factor. It is shown that any standard lattice 𝒢 can be realized as the standard invariant of a free product of (several) rescalings of Q. In particular, if Q has fundamental group equal to the positive reals and if P is the free product of infinitely many copies of Q, then P has subfactors giving rise to all possible standard invariants. Similarly, given a II_{1}–subfactor N ⊂ M, it is shown there are subfactors N̂ ⊂ M̂ having the same standard invariant as N ⊂ M but where M̂, respectively N̂, is the free product of M, respectively N, with rescalings of Q.

1. Introduction

The systems of higher relative commutants of finite index subfactors of II_{1}–factors were classified by S. Popa [7], when he proved certain conditions on a lattice of finite dimensional commuting squares (together with certain other data such as Markov traces) to be equivalent to it arising as the higher relative commutants of a subfactor. Such an abstract lattice is called a standard lattice. The question of whether there is a single II_{1}–factor M whose subfactors give rise to all standard lattices was answered in the affirmative by Popa and Shlyakhtenko [8], where they proved this universal property for M = L(F_{∞}), the von Neumann algebra of the free group on infinitely many generators. Their proof has two main components. First, given a standard lattice 𝒢, they construct a non–degenerate commuting square

\begin{aligned}B & ⊂ & A \\ ∪ & & ∪ \\ D & ⊂ & C \end{aligned}  (1)

of type I von Neumann algebras having atomic centers, which encodes the standard lattice. Then, for any II_{1}–factor Q, they consider the resulting inclusion N ⊂ M of amalgamated free product von Neumann algebras

ℳ = (Q ⊗ B, τ_{Q} ⊗ id_{B}) ∗_{B} (A, E_{B}^{A})  (2)
𝒩 = (Q ⊗ D, τ_{Q} ⊗ id_{D}) ∗_{D} (C, E_{D}^{C}) ,

where τ_{Q} is the tracial state on Q and E_{B}^{A} and E_{D}^{C} are the conditional expectations from the commuting square (1). By relating it to a construction from [7], they show that 𝒩 ⊂ ℳ is an inculsion of II_{∞} factors which is an infinite amplification of an inclusion N ⊂ M of II_{1}–factors whose system of higher relative commutants is the original standard lattice 𝒢. The second main part of their proof is to show that if Q is taken to be L(F_{∞}), then both N and M are isomorphic to L(F_{∞}).

Date: 20 February 2002.

Supported in part by NSF grant DMS–0070558. The author would like to thank also the Mathematical Sciences Research Institute, where part of this work was done. Research at MSRI is supported in part by NSF grant DMS–9701755.

In this paper, we identify N and M for general Q as a free scaled product [4] of rescalings of Q. This allows us to generalize Popa and Shlyakhtenko’s results. For example, (Theorem 4.2), we show that for every standard lattice 𝒢 of finite depth and for every II_{1}–factor Q there is a subfactor N ⊆ M whose standard lattice is 𝒢 and where

M = Q_{s_{1}} ∗ ⋯ ∗ Q_{s_{m}}
N = Q_{t_{1}} ∗ ⋯ ∗ Q_{t_{n}} ∗ L(F_{b})

for some m, n ∈ N, positive real numbers s_{1}, …, s_{m} and t_{1}, …, t_{n} and b > 1 − n depending only on 𝒢. (Here we are using the notation of [6] for L(F_{a}) with a possibly negative). Taking Q to be the hyperfinite II_{1}–factor R and using the result [2] that the n–fold free product of R is L(F_{n}), this gives after rescaling if necessary that if 1 < t < ∞ and if 𝒢 is a standard lattice with finite depth, then L(F_{t}) has a subfactor whose standard invariant is 𝒢; This result was first proved by Rădulescu [9].

We also show, (Theorem 4.4), that for every standard lattice 𝒢 and for every II_{1}– factor Q, there is a subfactor N ⊂ M whose standard lattice is 𝒢 and such that

M = *_{i=1}^{∞} Q_{s_{i}}
N = *_{i=1}^{∞} Q_{t_{i}}

for some positive numbers s_{1}, s_{2}, … and t_{1}, …, t_{2} depending only on 𝒢. As a consequence of these two results, we obtain (Theorem 4.5) that if Q is a II_{1}–factor whose fundamental group is equal to the set of all positive reals then the free product Q ∗ Q ∗ ⋯ of infinitely many copies of Q has the universal property with respect to subfactors, namely, its subfactors give rise to all standard lattices as their standard invariants.

Popa and Shlyakhtenko [8] also proved that if N ⊂ M is any subfactor of a II_{1}– factor then there is a subfactor pair N̂ ⊆ M̂ having the same standard lattice as N ⊆ M and such that N̂ ≅ N ∗ L(F_{∞}) and M̂ ≅ M ∗ L(F_{∞}). Their proofs involved amalgamated free products of the form

(Q ⊗ B, τ_{Q} ⊗ id_{B}) ∗_{B} (M ⊗ B(H), E) ,  (3)

where E is a conditional expectation onto a copy of the type I algebra B embedded in the II_{∞}–factor M ⊗ B(H), and where they use Q = L(F_{∞}).

We will identify the free product (3) for an arbitrary II_{1}–factor Q in terms of free subproducts, and thereby prove generalizations of the above mentioned result where L(F_{∞}) in the formula for N̂ and M̂ is replaced by free products of rescalings of Q. (See Theorems 4.6, 4.7 and 4.8.)

The organization of the rest of the paper is as follows: in §2 and, respectively, §3, isomorphism results are proved for amalgamated free products as in (3) and, respectively (2). (However, the simplifying assumption is made that the algebra B is commutative — an easy trick reduces the general case to this. Moreeover, because it entails no extra difficulty and in fact makes the proof more transparent, we consider free products where the algebra Q ⊗ B of (3) and (2) is replaced by a direct sum ⨁_{i∈I} Q(i) of II_{1}–factors whose center is B.) In §4 we prove the existence of subfactors, including those described above.

Acknowledgements. The author thanks S. Popa and F. Rădulescu for helpful conversations.

2. Amalgamated free products, I

In this section, we consider amalgamated free products related to those of the form (3) when B is taken to be commutative, and we show the result can be expressed in terms of free scaled products of von Neumann algebras. See [4] for the definition of and results about free scaled products.

Take an index set I equal to {1, 2, …, m} for some integer m ≥ 2 or to the natural numbers (in which case we write m = ∞) and let Q = ⨁_{i∈I} Q(i), where each Q(i) is a II_{1}–factor. let B be the center of Q and let p_{i} denote the minimal projection of B that is the support of the ith summand Q(i) of Q. Let E_{B}^{Q} : Q → B be the conditional expectation that is the center–valued trace on Q. Let M be a type II_{1} or II_{∞} factor with fixed faithful (finite or semifinite) trace Tr. Let N denote the II_{1}–factor obtained as eMe if M is type II_{∞}, where e ∈ M is a projection of trace 1, and as M_{Tr(1)^{−1}} if M is type II_{1}; (thus if M is type II_{1} and Tr(1) ≥ 1 then N ≅ eMe where e is as above). Suppose B is embedded in M as a unital subalgebra in such a way that Tr(p_{i}) < ∞ for all i ∈ I and let E_{B}^{M} : M → B be the Tr–preserving conditional expectation. Finally, let β(i) = Tr(p_{i}).

In this and the next section, when X_{1} and X_{2} are subsets of an algebra, we will use the symbols Λ^{o}(X_{1}, X_{2}) to denote the set of all alternating words in X_{1} and X_{2}, i.e. all words of the form a_{1}a_{2} ⋯ a_{n}, where a_{j} ∈ X_{i(j)} and i(j) ≠ i(j + 1). In fact, we will frequently blur the distinction between the set of words per se and the set of elements in the algebra that are equal to such words.

Proposition 2.1. Consider the amalgamated free product of von Neumann algebras

(M, E) = (M, E_{B}^{M}) ∗_{B} (Q, E_{B}^{Q}) .  (4)

If ∑_{i=1}^{m} β(i) < ∞ then M is a II_{1}–factor, while if ∑_{i=1}^{m} β(i) = ∞ then M is a II_{∞}–factor. In either case, we have

p_{1}Mp_{1} ≅ N_{β(1)} *_{i∈I} [\frac{β(i)}{β(1)}, Q(i)] .  (5)

Proof. We denote also by Tr the semifinite trace Tr↾_{B} ∘ E on M. The method of [3] shows that E is faithful, hence Tr is faithful on M; we will show the isomorphism (5), from which it will follow that M is a factor with faithful trace Tr, which is finite or semifinite depending on ∑_{i=1}^{m} β(i).

Let p_{[1,n]} = p_{1} + p_{2} + ⋯ + p_{n} and let β[1, n] = Tr(p_{[1,n]}). For every n ∈ I, let

M(n) = W^{∗}(p_{[1,n]}Mp_{[1,n]} ∪ p_{[1,n]}Q) .

Then M(1) is generated by p_{1}Mp_{1} ≅ N_{β(1)} and p_{1}Q ≅ Q(1), which are free, so

M(1) ≅ N_{β(1)} ∗ Q(1) .

Now for n ∈ I, n ≥ 2, we will find M(n) in terms of M(n − 1). Let K ∈ N be such that β[1, n − 1] ≥ β(n)/K and let (e_{ij})_{1≤i,j≤K} be a system of matrix units in M such that ∑_{1}^{K} e_{ii} = p_{n}. Let A = span {e_{ij} | 1 ≤ i, j ≤ K} and let P = W^{∗}(A ∪ p_{n}Q). Clearly A and p_{n}Q are free with respect to β(n)^{−1}Tr↾_{p_{n}Mp_{n}}, so

P ≅ Q(n) ∗ M_{K}(C) ≅ Q(n) ∗ L(F_{1−K^{−2}}).

Let v ∈ M be such that v^{∗}v = e_{11} and q := vv^{∗} ≤ p_{[1,n−1]} . Then

M(n) = W^{∗}(M(n − 1) ∪ P ∪ {v}) , qM(n)q = W^{∗}(qM(n − 1)q ∪ vPv^{∗}) .

We claim that qM(n − 1)q and vPv^{∗} are free with respect to Kβ(n)^{−1}Tr↾_{qMq}. This is equivalent to freeness of v^{∗}M(n − 1)v and e_{11}Pe_{11} with respect to Kβ(n)^{−1}Tr↾_{e_{11}Me_{11}}, which is what we will show. Using the trace–preserving conditional expectation P → A along with the facts that e_{11} is a minimal projection in A and that {p_{n}} ∪ Λ^{o}(A ∩ ker Tr, p_{n}Q ∩ ker Tr) spans a dense subspace of P, we find that a dense subspace of e_{11}Pe_{11} ∩ ker Tr is spanned by

Φ̃ := ⋃_{1≤i,j≤K} e_{1i}Φe_{j1} ,

where Φ is the set of all words belonging to Λ^{o}(A ∩ ker Tr, p_{n}Q ∩ ker Tr) whose first and last letters come from p_{n}Q ∩ ker Tr. On the other hand, since

p_{[1,n−1]}B ∪ Λ^{o}(p_{[1,n−1]}Mp_{[1,n−1]} ∩ ker E, p_{[1,n−1]}Q ∩ ker E)

spans a dense subspace of M(n − 1), we see that v^{∗}M(n − 1)v ∩ ker Tr is contained in the closed linear span of

Ψ̃ := (e_{11}Me_{11} ∩ ker Tr) ∪ Ψ ,

where Ψ is the set of all words belonging to Λ^{o}(M ∩ker E, p_{[1,n−1]}Q∩ker E) with first letter from e_{11}Mp_{[1,n−1]} and with last letter from p_{[1,n−1]}Me_{11}. Now to show freeness of qM(n − 1)q and vPv^{∗}, it will suffice to show

Λ^{o}(Φ̃, Ψ̃) ⊆ ker Tr .  (6)

Let x ∈ Λ^{o}(Φ̃, Ψ̃). If the first letter in x comes from e_{11}Φe_{j1} ⊆ Φ̃ then substitute e_{11} = (e_{11} − K^{−1}p_{n}) + K^{−1}p_{n} for this first e_{11}, while if the last letter in x comes from e_{1i}Φe_{11} ⊆ Φ̃ then substitute this value for this last e_{11} and then distribute. Using e_{j1}e_{11}Mp_{[1,n−1]} ⊆ ker E and e_{j1}(e_{11}Me_{11} ∩ ker Tr)e_{i1} ⊆ ker E for all i, j ∈ {1, …, K} and regrouping, we see that x is equal to a linear combination of at most four words from Λ^{o}(M ∩ ker E, Q ∩ ker E), and hence Tr(x) = 0 by freeness. We have shown (6) and thereby freeness of qM(n − 1)q and vPv^{∗}.

What we have shown above implies there is an isomorphism

p_{[1,n−1]}M(n)p_{[1,n−1]} ⟶^{∼} M(n − 1) ∗ [\frac{β(n)}{Kβ[1,n−1]}, vPv^{∗}]  (7)

intertwining the inclusion M(n − 1) ↪ p_{[1,n−1]}M(n)p_{[1,n−1]} and the canonical embedding of M(n − 1) in the RHS of (7). By results of [6],

vPv^{∗} ≅ P_{1/K} ≅ Q(n)_{1/K} ∗ L(F_{K^{2}−1}) ,

so using the technology of free scaled products [4], we get isomorphisms

M(n − 1) ∗ [\frac{β(n)}{Kβ[1,n−1]}, Q(n)_{1/K} ∗ L(F_{K^{2}−1})] ⟶^{∼}

⟶^{∼} M(n − 1) ∗ [\frac{β(n)}{Kβ[1,n−1]}, Q(n)_{1/K}] ∗ [\frac{β(n)}{Kβ[1,n−1]}, L(F_{K^{2}−1})]

⟶^{∼} M(n − 1) ∗ L(F_{(β(n)/β[1,n−1])^{2}(1−K^{−2})}) ∗ [\frac{β(n)}{Kβ[1,n−1]}, Q(n)_{1/K}]

⟶^{∼} M(n − 1) ∗ [\frac{β(n)}{β[1,n−1]}, Q(n)] ,

all intertwining the canonical embeddings of M(n − 1) into the free scaled products; (the last isomorphism above is obtained from Theorem 5.5 of [4], and the others follow routinely from the definition of free scaled product). Now compressing by p_{1} and appealing to Theorem 4.9 of [4], we get an isomorphism

p_{1}M(n)p_{1} ⟶^{∼} p_{1}M(n − 1)p_{1} ∗ [\frac{β(n)}{β(1)}, Q(n)]  (8)

intertwining the inclusion p_{1}M(n − 1)p_{1} ↪ p_{1}M(n)p_{1} and the canonical embedding of p_{1}M(n − 1)p_{1} in the RHS of (8).

Taking the inductive limit as n increases gives the desired isomorphism (5). □

The following corollary is a direct consequence of Proposition 2.1 and results from [6] and [4].

Corollary 2.2. Let

(M, E) = (M, E_{B}^{M}) ∗_{B} (Q, E_{B}^{Q})

with accompanying notation be as in Proposition 2.1.

(A) If I = {1, …, m} is finite then

M ≅ M ∗ Q(1)_{\frac{1}{β(1)}} ∗ Q(2)_{\frac{1}{β(2)}} ∗ ⋯ ∗ Q(m)_{\frac{1}{β(m)}} ∗ L(F_{t})

where

t = −m + ∑_{i=1}^{m} β(i)^{2} .

(B) If I = N is infinite and if one or more of the following conditions hold:

(i) Q(i) ≅ Q(i) ∗ L(F_{∞}) for some i ∈ N, (ii) N ≅ N ∗ L(F_{∞}),

(iii) ∑_{i=1}^{∞} β(i)^{2} = ∞,

then

p_{1}Mp_{1} ≅ N_{β(1)} ∗ ( *_{i=1}^{∞} Q(i)_{\frac{β(1)}{β(i)}} ) .

3. Amalgamated free products, II

In this section, we will describe amalgamated free product von Neumann algebra related to those in (2), but with B commutative, in terms of free scaled products.

We begin by letting A be a separable, type I von Neumann algebra with atomic center Z(A) and let B be a unital commutative subalgebra of A with the property that B ∩Z(A) = C1. We assume that there is a normal, faithful, semifinite trace Tr_{A} on A such that Tr_{A}(p) < ∞ for every minimal projection p in B. Let E_{B}^{A} : A → B be the Tr_{A}–preserving conditional expectation onto B. Let (p_{i})_{i∈I} be the minimal projections in B, where we take either I = {1, …, n}, some n ∈ N, or I = N, in which case we set n = ∞. Let Q(i) be a II_{1}–factor (i ∈ I) and let 𝒬 = ⨁_{i∈I} Q(i), with the center of 𝒬 identified with B in such a way that p_{i} is the support projection of the ith summand Q(i). Let E_{B}^{𝒬} : 𝒬 → B be the conditional expectation that is the center valued trace on 𝒬. Our goal in this section is to write the amalgamated free product von Neumann algebra

(𝒫, E) = (A, E_{B}^{A}) ∗_{B} (𝒬, E_{B}^{𝒬})

as a free scaled product of II_{1}–factors.

We have implicitly choosen an ordering of the minimal projections in B. Let (q_{j})_{j∈J} be the minimal projections in Z(A). Define J_{m} recursively (m ∈ I) by

J_{1} = {j ∈ J | q_{j}p_{1} ≠ 0}
J_{m} = {j ∈ J | q_{j}p_{m} ≠ 0}\(J_{1} ∪ ⋯ ∪ J_{m−1}) (m ∈ I\{1}) .

Let β(i) = Tr_{A}(p_{i}), (i ∈ I) and if f_{j} is a minimal projection of A that lies under q_{j}, then let α(j) = Tr_{A}(f_{j}) (j ∈ J). For every m ∈ I\{1}, let e_{m} be a minimal projection in A such that e_{m} ≤ p_{1} + ⋯ + p_{m−1} and e_{m} is equivalent to a subprojection of p_{m}. (Such a projection must exist by the condition Z(A) ∩ B = C.) Let j(m) ∈ J be such that e_{m} ≤ q_{j(m)} and let γ(m) = α(j(m)) = Tr_{A}(e_{m}).

Proposition 3.1. Let

(𝒫, E) = (A, E_{B}^{A}) ∗_{B} (𝒬, E_{B}^{𝒬})

be the free product of von Neumann algebras with amalgamation over B. If ∑_{i=1}^{n} β(i) < ∞ then 𝒫 is a II_{1}–factor, while if ∑_{i=1}^{n} β(i) = ∞ then 𝒫 is a II_{∞}– factor. In either case,

p_{1}𝒫p_{1} ≅ (Q(1) ∗ L(F_{r})) *_{i=2}^{n} [\frac{γ(i)}{β(1)}, Q(i)_{\frac{γ(i)}{β(i)}} ] ,  (9)

where

r = \frac{1}{β(1)^{2}} (β(1)^{2} − ∑_{j∈J_{1}} α(j)^{2}) + \frac{1}{β(1)^{2}} ∑_{m=2}^{n} (β(m)^{2} − γ(m)^{2} − ∑_{j∈J_{m}} α(j)^{2}) .

Proof. The condition Z(A) ∩ B = C ensures that 𝒫 is a factor. We have the normal, faithful, semifinite trace Tr = Tr_{A} ∘ E on 𝒫, which is finite if and only if ∑_{i=1}^{n} β(i) < ∞.

It remains to find the isomorphism class of p_{1}𝒫p_{1}. For m ∈ I, let

p_{[1,m]} = p_{1} + ⋯ + p_{m}
β[1, m] = β(1) + ⋯ + β(m) = Tr(p_{[1,m]})

and let

𝒩(m) = W^{∗}(p_{m}𝒬 ∪ p_{m}Ap_{m})
𝒫(m) = W^{∗}(p_{[1,m]}𝒬 ∪ p_{[1,m]}Ap_{[1,m]}) .

Then

𝒩(m) ≅ Q(m) ∗ L(F_{s(m)})

where s(m) is the free dimension^{1} of p_{m}Ap_{m} with respect to β(m)^{−1}Tr↾_{p_{m}Ap_{m}}. We have 𝒫(1) = 𝒩(1) and

s(1) = \frac{1}{β(1)^{2}} (β(1)^{2} − ∑_{j∈J_{1}} α(j)^{2}) .

Note that 𝒫 is the increasing union of 𝒫(1) ⊆ 𝒫(2) ⊆ ⋯ and thus p_{1}𝒫p_{1} is the increasing union of p_{1}𝒫(1)p_{1} ⊆ p_{1}𝒫(2)p_{1} ⊆ · · ·. We will show that each p_{1}𝒫(m)p_{1} is isomorphic to a free scaled product of II_{1}–factors in such a way that the inclusions p_{1}𝒫(m)p_{1} ⊆ p_{1}𝒫(m + 1)p_{1} become canonical embeddings. Taking the inductive limit will allow us to express p_{1}𝒫p_{1} as a free scaled product of II_{1}–factors.

Fix m such that m, m + 1 ∈ I. Let

K_{m} = ⋃_{i=1}^{m} J_{i},
K_{m}^{(1)} = {k ∈ K_{m} | q_{k}p_{m+1} ≠ 0}
K_{m}^{(0)} = K_{m}\K_{m}^{(1)} .

The hypothesis Z(A)∩B = C ensures that K_{m}^{(1)} is nonempty; in fact, j(m+1) ∈ K_{m}^{(1)}. Make an ordering K_{m}^{(1)} = {k(i) | i ∈ L} where L = {0, 1, …, ℓ } or L = {0} ∪ N, such that k(0) = j(m + 1). For every k ∈ K_{m}^{(1)}, let v_{k} ∈ A be such that v_{k}^{∗}v_{k} is a minimal projection in A, v_{k}^{∗}v_{k} ≤ p_{[1,m]} and v_{k}v_{k}^{∗} ≤ p_{m+1}. Note that Tr(v_{k}^{∗}v_{k}) = α(k); we assume without loss of generality that v_{k(0)}^{∗}v_{k(0)} = e_{m+1}.

Let

ℛ(0) = W^{∗}(𝒫(m) ∪ 𝒩(m + 1) ∪ {v_{k(0)}}) .

Then

p_{[1,m]}ℛ(0)p_{[1,m]} = W^{∗}(𝒫(m) ∪ v_{k(0)}^{∗}𝒩(m + 1)v_{k(0)}) .

We will show that the pair

e_{m+1}𝒫(m)e_{m+1},  v_{k(0)}^{∗}𝒩(m + 1)v_{k(0)}  (10)

1We are using the “free dimension” that was introduced in [1] — see [4] for further discusion.

Since each p_{m}Ap_{m} is finite dimensional, its free dimension is well defined, and we may use the term “free dimension” without inhibition.

is free with respect to γ(m + 1)^{−1}Tr↾_{e_{m}𝒫e_{m}}. Using C^{o} to denote C ∩ ker E for any subalgebra C of 𝒫, since e_{m} is a minimal projection in A, 𝒫(m)^{o} is densely spanned by

Θ := e_{m}(Λ^{o}((𝒬p_{[1,m]})^{o}, (p_{[1,m]}Ap_{[1,m]})^{o})\(p_{[1,m]}Ap_{[1,m]})^{o})e_{m}

and (v_{k(0)}^{∗}𝒩(m + 1)v_{k(0)})^{o} is densely spanned by

Ω := v_{k(0)}^{∗}(Λ^{o}((𝒬p_{m+1})^{o}, (p_{m+1}Ap_{m+1})^{o})\(p_{m+1}Ap_{m+1})^{o})v_{k(0)} .

To show freeness of the pair (10), it will suffice to show Λ^{o}(Θ, Ω) ⊆ ker E. However, since

(p_{[1,m]}Ap_{[1,m]})v_{k(0)}^{∗}(p_{m+1}Ap_{m+1}) ⊆ ker E_{B}^{A} ,

given x ∈ Λ^{o}(Θ, Ω), by regrouping we see that x is equal to some x′ ∈ Λ^{o}(𝒬^{o}, A^{o}), which is contained in ker E by freeness. This shows that the pair (10) is free. Therefore, there is an isomorphism

p_{[1,m]}ℛ(0)p_{[1,m]} ⟶^{∼} 𝒫(m) ∗ [\frac{γ(m+1)}{β[1,m]} , 𝒩(m + 1)_{\frac{γ(m+1)}{β(m+1)}} ]  (11)

intertwining the inclusion 𝒫(m) ↪ p_{[1,m]}ℛ(0)p_{[1,m]} and the canonical embedding of 𝒫(m) in the RHS of (11). By [4], ℛ(0) is a factor.

Define ℛ(i) recursively for every i ∈ L\{0} by

ℛ(i) = W^{∗}(ℛ(i − 1) ∪ {v_{k(i)}}).

We will show that every ℛ(i) is a factor that can be written as a free subproduct involving ℛ(i − 1). Fixing i, suppose that ℛ(i − 1) is a factor. Let w_{i} ∈ ℛ(i − 1) be such that w_{i}^{∗}w_{i} = v_{k(i)}^{∗}v_{k(i)} and w_{i}w_{i}^{∗} = v_{k(i)}v_{k(i)}^{∗}. Let f = w_{i}^{∗}w_{i} and g = w_{i}w_{i}^{∗}; note that f ≤ p_{[1,m]}q_{k(i)} and g ≤ p_{m+1}q_{k(i)}. We will show that in f𝒫f and with respect to α(k(i))^{−1}Tr↾_{f𝒫f} , w_{i}^{∗}v_{k(i)} is a Haar unitary and is ∗–free from fℛ(i − 1)f. Let

C = W^{∗}(p_{[1,m]}Ap_{[1,m]} ∪ p_{m+1}Ap_{m+1} ∪ {v_{k(0)}, …, v_{k(i−1)}}) .

Then fCg = {0} and ℛ(i − 1) is densely spanned by Λ^{o}(C^{o}, (p_{[1,m+1]}𝒬)^{o}). In order to show that w_{i}^{∗}v_{k(i)} is a Haar unitary and is ∗–free from fℛ(i − 1)f, it will suffice to show

Λ^{o}((fℛ(i − 1)f)^{o} ∪ fℛ(i − 1)g ∪ gℛ(i − 1)f ∪ (gℛ(i − 1)g)^{o}, {v_{k(i)}, v_{k(i)}^{∗}}) ⊆ ker E .

Because f and g are minimal projections in C and fCg = {0},

(fℛ(i − 1)f)^{o} ∪ fℛ(i − 1)g ∪ gℛ(i − 1)f∪(gℛ(i − 1)g)^{o} ⊆
⊆ \overline{span} (Λ^{o}(C^{o}, (p_{[1,m+1]}𝒬)^{o})\C^{o}) .

But since Cv_{k(i)}C ⊆ ker E, we have

Λ^{o}(Λ^{o}(C^{o}, (p_{[1,m+1]}𝒬)^{o})\C^{o}, {v_{k(i)}, v_{k(i)}^{∗}}) ⊆ ker E .

Thus w_{i}^{∗}v_{k(i)} is a Haar unitary and is ∗–free from fℛ(i − 1)f. This gives a ∗– isomorphism

p_{[1,m]}ℛ(i)p_{[1,m]} ⟶^{∼} p_{[1,m]}ℛ(i − 1)p_{[1,m]} ∗ [\frac{α(k(i))}{β[1,m]} , L(Z)]  (12)

intertwining the inclusion p_{[1,m]}ℛ(i − 1)p_{[1,m]} ↪ p_{[1,m]}ℛ(i)p_{[1,m]} and the canonical embedding of p_{[1,m]}ℛ(i − 1)p_{[1,m]} into the RHS of (12).

Since 𝒫(m + 1) = \overline{⋃_{i∈L} ℛ(i)}, by composing the isomorphisms (11) and (12), we get an isomorphism

\begin{aligned}p_{[1,m]}𝒫(m + 1)p_{[1,m]} & ⟶^{∼} \\ & (𝒫(m) ∗ [\frac{γ(m+1)}{β[1,m]} , 𝒩(m + 1)_{\frac{γ(m+1)}{β(m+1)}} ]) *_{i∈L} [\frac{α(k(i))}{β[1,m]} , L(Z)] \end{aligned} (13)

intertwining the inclusion 𝒫(m) ↪ p_{[1,m]}𝒫(m + 1)p_{[1,m]} and the canonical embedding of 𝒫(m) in the RHS of (13). Using that 𝒩(m + 1) ≅ Q(m + 1) ∗ L(F_{s(m+1)}) with

s(m + 1) = 1 − ∑_{j∈K_{m}^{(1)}∪J_{m}} (\frac{α(j)}{β(m+1)})^{2} ,

by results from [4] there is an isomorphism

p_{1}𝒫(m + 1)p_{1} ⟶^{∼} (p_{1}𝒫(m)p_{1} ∗ L(F_{r(m+1)})) ∗ [\frac{γ(m+1)}{β(1)} , Q(m + 1)_{\frac{γ(m+1)}{β(m+1)}} ] , (14)

where

r(m + 1) = \frac{1}{β(1)^{2}} (β(m + 1)^{2} − γ(m + 1)^{2} − ∑_{j∈J_{m}} α(j)^{2}) ,

intertwining the inclusion p_{1}𝒫(m)p_{1} ↪ p_{1}𝒫(m + 1)p_{1} and the canonical embedding of p_{1}𝒫(m)p_{1} in the RHS of (14). Taking the inductive limit as m → ∞, we obtain the isomorphism (9). □

The next corollary follows directly from Proposition 3.1 and results of [6] and [4].

Corollary 3.2. Let A, B, 𝒬 = ⨁_{i∈I} Q(i) and

(𝒫, E) = (𝒬, E_{B}^{𝒬}) ∗_{B} (A, E_{B}^{A})

with accompanying notation be as in Proposition 3.1.

(A) If I = {1, …, n} is finite then

𝒫 ≅ Q(1)_{\frac{1}{β(1)}} ∗ Q(2)_{\frac{1}{β(2)}} ∗ ⋯ ∗ Q(n)_{\frac{1}{β(n)}} ∗ L(F_{t})

where

t = −n + 1 + ∑_{i=1}^{n} β(i)^{2} − ∑_{j∈J} α(j)^{2}
= −n + 1 + fdim (A) − fdim (B) .

(B) If I = N is infinite and if one or more of the following conditions hold:

(i) Q(i) ≅ Q(i) ∗ L(F_{∞}) for some i ∈ N,

(ii) ∑_{i=1}^{∞} γ(i)^{2} = ∞,

(iii) r = ∞, then

p_{1}𝒫p_{1} ≅ *_{i=1}^{∞} Q(i)_{\frac{β(1)}{β(i)}} .

4. Subfactors

In this section, we will recall constructions and results of Popa and Shlyakhtenko [8] and we will use them together with the results from §2 and §3 to prove the existence of subfactors in free subproducts and free products, including those mentioned in the introduction.

Let Q be any II_{1}–factor. Let M_{−1} ⊂ M_{0} be a II_{1}–subfactor of finite index λ, let 𝒢 = 𝒢_{M_{−1},M_{0}} be the associated standard λ–lattice and let Γ be the standard graph and Γ′ the second standard graph of M_{−1} ⊂ M_{0}. Popa and Shlyakhtenko [8] construct semifinite von Neumann algebras

\begin{aligned}ℳ_{−1} & ⊂ & ℳ_{0} \\ ∪ & & ∪ \\ 𝒜_{−1}^{0} & ⊂ & 𝒜_{0}^{0} \\ ∪ & & ∪ \\ 𝒜_{−1}^{−1} & ⊂ & 𝒜_{0}^{−1} \end{aligned}  (15)

with a specified faithful semifinite trace Tr on ℳ_{0} and they prove the following:

I. The trace preserving conditional expectations make (15) a diagram of com-

muting squares. II. ℳ_{i} ≅ M_{i} ⊗ B(ℋ), (i = −1, 0). III. Each algebra 𝒜_{i}^{k} is a type I von Neumann algebra with atomic center, and Tr

is finite on all minimal projections in 𝒜_{i}^{k}. IV. The graphs of the inclusions 𝒜_{−1}^{−1} ⊂ 𝒜_{0}^{−1} and 𝒜_{−1}^{0} ⊂ 𝒜_{0}^{0} are Γ and Γ′, respec-

tively. V. The commuting square

\begin{aligned}𝒜_{−1}^{0} & ⊂ & 𝒜_{0}^{0} \\ ∪ & & ∪ \\ 𝒜_{−1}^{−1} & ⊂ & 𝒜_{0}^{−1} \end{aligned}  (16)

depends only on the standard λ–lattice 𝒢, and the commuting square (16) is functorial in 𝒢, (see [8, Thm. 2.9] for details).

VI. Consider the amalgamated free products of von Neumann algebras

𝒫_{0} = 𝒜_{0}^{0} ∗_{𝒜_{0}^{−1}} (Q ⊗ 𝒜_{0}^{−1})  (17)
𝒫_{−1} = 𝒜_{−1}^{0} ∗_{𝒜_{−1}^{−1}} (Q ⊗ 𝒜_{−1}^{−1}) ,  (18)

where the amalgamation is with respect to the trace preserving conditional expectations. Then 𝒫_{−1} ⊂ 𝒫_{0} is an inclusion of type II_{∞} factors whose standard invariant (i.e. the standard invariant of p𝒫_{−1}p ⊂ p𝒫_{0}p where p ∈ 𝒫_{−1} is any finite projection) is 𝒢.

VII. Consider the amalgamated free products of von Neumann algebras

ℳ̂_{0} = ℳ_{0} ∗_{𝒜_{0}^{−1}} (Q ⊗ 𝒜_{0}^{−1})  (19)
ℳ̂_{−1} = ℳ_{−1} ∗_{𝒜_{−1}^{−1}} (Q ⊗ 𝒜_{−1}^{−1}) ,  (20)

where the amalgamation is with respect to the trace preserving conditional expectations. Then ℳ̂_{−1} ⊂ ℳ̂_{0} is an inclusion of type II_{∞} factors whose standard invariant (i.e. the standard invariant of pℳ̂_{−1}p ⊂ pℳ̂_{0}p where p ∈ ℳ̂_{−1} is any finite projection) is 𝒢.

4.1. Consider the commuting square (16) determined by 𝒢. Let 𝒫_{0} be as in (17) and let q(0) ∈ A_{0}^{−1} be the sum of a maximal family of mutually orthogonal and nonequivalent minimal projections of A_{0}^{−1}. Letting B = q(0)A_{0}^{−1}q(0), we have that B is commutative and

q(0)𝒫_{0}q(0) = q(0)A_{0}^{0}q(0) ∗_{B} (Q ⊗ B) ,

where the amalgamation is with respect to the trace preserving conditional expectations. Fix any minimal projection q_{1}(0) in B. Let m(0) ∈ N ∪ {∞} be the number of minimal projections in B and let (β(0, i))_{i=1}^{m(0)} be the values of Tr on the minimal projections in B, with β(0, 1) = Tr(q_{1}(0)). Proposition 3.1 shows that for cretain r(0) ∈ [0, ∞] and γ(0, i) ≤ β(0, i), we have

q_{1}(0)𝒫_{0}q_{1}(0) ≅ \begin{cases}Q ∗ L(F_{r(0)}) & if m(0) = 1 \\ (Q ∗ L(F_{r(0)})) *_{i=2}^{m(0)} [\frac{γ(0, i)}{β(0, 1)}, Q_{\frac{γ(0,i)}{β(0,i)}} ] & if m(0) > 1 . \end{cases} (21)

Analogously, if 𝒫_{−1} is as in (18) and if q_{1}(−1) is any minimal projection in A_{−1}^{−1}, if m(−1) is the number of minimal projections in the center of A_{−1}^{−1}, letting (β(−1, i))_{i=1}^{m(−1)} be Tr applied to a maximal family of mutually inequivalent minimal projections in A_{−1}^{−1}, with β(−1, i) = Tr(q_{1}(−1)), we have

q_{1}(−1)𝒫_{−1}q_{1}(−1) ≅ \begin{cases}Q ∗ L(F_{r(−1)}) & if m(−1) = 1 \\ (Q ∗ L(F_{r(−1)})) *_{i=2}^{m(−1)} [\frac{γ(−1, i)}{β(−1, 1)}, Q_{\frac{γ(−1,i)}{β(−1,i)}} ] & if m(−1) > 1 \end{cases}

(22)

for certain r(−1) ∈ [0, ∞] and γ(−1, i) ≤ β(−1, i).

Theorem 4.2. Let 𝒢 be a standard lattice of finite depth. Then there are m, n ∈ N, positive real numbers s_{1}, …, s_{m} and t_{1}, …, t_{n} and there are a > 1 − m and b > 1 − n such that for any II_{1}–factor Q, there is an inclusion P_{−1} ⊂ P_{0} of II_{1}–factors whose standard invariant is 𝒢 and such that

P_{0} = Q_{s_{1}} ∗ ⋯ ∗ Q_{s_{m}} ∗ L(F_{a})
P_{−1} = Q_{t_{1}} ∗ ⋯ ∗ Q_{t_{n}} ∗ L(F_{b}) .

Proof. Since 𝒢 is of finite depth, it follows from [8] (see point IV above) that the centers of A_{0}^{−1} and A_{−1}^{−1} are finite dimensional. Thus m(0) and m(1) in (21) and (22) are finite. By results in [4], from (21) and (22) we get

q_{1}(0)𝒫_{0}q_{1}(0) ≅ Q_{s′_{1}} ∗ ⋯ ∗ Q_{s′_{m(0)}} ∗ L(F_{a′})
q_{1}(−1)𝒫_{−1}q_{1}(−1) ≅ Q_{t′_{1}} ∗ ⋯ ∗ Q_{t′_{m(1)}} ∗ L(F_{b′})

for some a′ > 1 − m(0) and b′ > 1 − m(−1), where s′_{i} = β(0, i)/β(0, 1) and t′_{i} = β(−1, i)/β(−1, 1). By results of [8] (see point VI above), the II_{1}–subfactor q𝒫_{−1}q ⊂ q𝒫_{0}q has standard invariant 𝒢, where q is any finite projection in 𝒫_{−1}. But q𝒫_{0}q is a rescaling of q_{1}(0)𝒫_{0}q_{1}(0) by λ := Tr(q)/Tr(q_{1}(0)), and thus by results of [6] we have

q𝒫_{0}q ≅ Q_{λs′_{1}} ∗ ⋯ ∗ Q_{λs′_{m(0)}} ∗ L(F_{a})

where a = λ^{−2}a′ + (m(0) − 1)(λ^{−2} − 1). Rescaling q_{1}(−1)𝒫_{−1}q_{1}(−1) yields

q𝒫_{−1}q ≅ Q_{t_{1}} ∗ ⋯ ∗ Q_{t_{m(1)}} ∗ L(F_{b})

for appropriate values of t_{i} and b. □

Remark 4.3. In the proof of Theorem 4.2, λ can be chosen to arrange either a = 0, provided m(0) > 1, or b = 0, provided m(1) > 1.

Theorem 4.4. Given a standard lattice 𝒢, there are some positive real numbers s_{1}, s_{2}, … and t_{1}, t_{2}, … such that for any II_{1}–factor Q, there is an inclusion P_{−1} ⊂ P_{0} of II_{1}–factors whose standard invariant is 𝒢 and such that

\begin{aligned}P_{0} & ≅ *_{i=1}^{∞} Q_{s_{i}} \\ P_{−1} & ≅ *_{i=1}^{∞} Q_{t_{i}} \end{aligned}  (23)

Proof. If 𝒢 is of finite depth, let Q̃ = Q ∗ Q ∗ ⋯ be a the free products of infinitely many copies of Q. By [5], a free product of infinitely many II_{1}–factors absorbs a free product with L(F_{∞}), and the rescaling by λ of a free product of infinitely many factors is isomorphic to the free product of the same factors each rescaled by λ. Applying these facts and Theorem 4.2 to Q̃, we find a subfactor P_{−1} ⊂ P_{0} whose standard invariant is 𝒢 and where

P_{0} ≅ (Q_{s_{1}} ∗ Q_{s_{1}} ∗ ⋯) ∗ ⋯ ∗ (Q_{s_{n}} ∗ Q_{s_{n}} ∗ ⋯)
P_{−1} ≅ (Q_{t_{1}} ∗ Q_{t_{1}} ∗ ⋯) ∗ ⋯ ∗ (Q_{t_{m}} ∗ Q_{t_{m}} ∗ ⋯) .

If 𝒢 has infinite depth, let Q̃ = Q ∗ L(F_{∞}). Let 𝒫_{0} and 𝒫_{−1} be as in (17) and (18), but with Q replaced by Q̃. Then we have the isomorphisms as in (21) and (22) but with Q replaced by Q̃. Since 𝒢 has infinite depth, m(0) and m(−1) are infinite. Since Q̃ absorbs a free product with L(F_{∞}), using [4, Theorem 3.11] and the above mentioned results about rescalings and absorbtion of L(F_{∞}), we have the following isomorphisms:

\begin{aligned}q_{1}(0)𝒫_{0}q_{1}(0) & ≅ & Q̃ ∗ ( *_{i=2}^{∞} Q̃_{\frac{β(0,i)}{β(0,1)}} ) & ≅ & *_{i=1}^{∞} Q_{\frac{β(0,i)}{β(0,1)}} \\ q_{1}(−1)𝒫_{−1}q_{1}(−1) & ≅ & Q̃ ∗ ( *_{i=2}^{∞} Q̃_{\frac{β(−1,i)}{β(−1,1)}} ) & ≅ & *_{i=1}^{∞} Q_{\frac{β(−1,i)}{β(−1,1)}} \end{aligned}

As in the proof of Theorem 4.2, we get a subfactor P_{−1} = q𝒫_{−1}q ⊂ P_{0} = q𝒫_{0}q with isomorphisms (23) as desired. □

The following theorem is a special case of the previous one, and gives that II_{1}– factors other than L(F_{∞}) have the universal property for subfactors.

Theorem 4.5. Let Q be any II_{1}–factor whose fundamental group is equal to the positive real numbers, and let P = Q ∗ Q ∗ ⋯ be the free product of infinitely many copies of Q. Then for any standard lattice 𝒢 there is a subfactor N = N_{𝒢} of P whose standard invariant is 𝒢; moreover N is isomorphic to P.

Consider the commuting square (15) determined by the subfactor M_{−1} ⊂ M_{0} and let ℳ̂_{0} be as in (19). With notation as in 4.1, we have

q(0)ℳ̂_{0}q(0) = q(0)ℳ_{0}q(0) ∗_{B} (Q ⊗ B),
q(−1)ℳ̂_{−1}q(−1) = q(−1)ℳ_{−1}q(−1) ∗_{B} (Q ⊗ B) ,

where the amalgamated free products are with respect to Tr–preserving conditional expectations. From Proposition 2.1, we have

q_{1}(0)ℳ̂_{0}q_{1}(0) ≅ \begin{cases}(M_{0})_{β(0,1)} & if m(0) = 1 \\ (M_{0})_{β(0,1)} *_{i=2}^{m(0)} [\frac{β(0,i)}{β(0,1)}, Q] & if m(0) > 1 \end{cases}

and similarly

q_{1}(−1)ℳ̂_{0}q_{1}(−1) ≅ \begin{cases}(M_{−1})_{β(−1,1)} & if m(−1) = 1 \\ (M_{−1})_{β(−1,1)} *_{i=2}^{m(−1)} [\frac{β(−1,i)}{β(−1,1)}, Q] & if m(−1) > 1 . \end{cases}

From these, and in a manner analogous to the proofs of Theorems 4.2, 4.4 and 4.5, one proves the following results.

Theorem 4.6. Let Q be any II_{1}–factor. Then for any finite depth subfactor M_{−1} ⊂ M_{0} of II_{1}–factors, there are m, n ∈ N, positive real numbers s_{1}, …, s_{m} and t_{1}, …, t_{n} and there are a > 1 − m and b > 1 − n, all depending only on the standard invariant of M_{−1} ⊂ M_{0}, and there is a II_{1}–subfactor M̂_{−1} ⊂ M̂_{0} having the same standard invariant as M_{−1} ⊂ M_{0} and where

M̂_{0} ≅ M_{0} ∗ Q_{s_{1}} ∗ ⋯ ∗ Q_{s_{m}} ∗ L(F_{a})
M̂_{−1} = M_{−1} ∗ Q_{t_{1}} ∗ ⋯ ∗ Q_{t_{n}} ∗ L(F_{b}) .

Theorem 4.7. Let Q be any II_{1}–factor. Then for any finite depth subfactor M_{−1} ⊂ M_{0} of II_{1}–factors, there are positive real numbers s_{1}, …, s_{m} and t_{1}, …, t_{n} depending only on the standard invariant of M_{−1} ⊂ M_{0} and there is a II_{1}–subfactor M̂_{−1} ⊂ M̂_{0} having the same standard invariant as M_{−1} ⊂ M_{0} and where

M̂_{0} ≅ M_{0} ∗ ( *_{i=1}^{∞} Q_{s_{i}})

M̂_{−1} = M_{−1} ∗ ( *_{i=1}^{∞} Q_{t_{i}}) .

Theorem 4.8. Let Q be any II_{1}–factor having fundamental group equal to R_{+}^{∗}. Then for any subfactor N ⊂ M of a II_{1}–factor, there is a subfactor N̂ ⊂ M̂ whose standard lattice is equal to that of N ⊂ M and such that

N̂ ≅ N ∗ ( *_{i=1}^{∞} Q),  M̂ ≅ M ∗ ( *_{i=1}^{∞} Q) .

References

[1] K. Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1993), 97-119.

[2] , Interpolated free group factors, Pacific J. Math. 163 (1994), 123-135.

[3] , Faithfulness of free product states, J. Funct. Anal. 154 (1998), 223-229.

[4] , Free subproducts and free scaled products of II_{1}–factors, J. Funct. Anal. (to appear).

[5] K. Dykema, F. Rădulescu, Compressions of free products of von Neumann algebras, Math. Ann. 316 (2000), 61-82.

[6] , Rescalings of free products of II_{1}–factors, Proc. Amer. Math. Soc. (to appear).

[7] S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), 427-445.

[8] S. Popa, D. Shlyakhtenko, Universal properties of L(F_{∞}) in subfactor theory, preprint (2000).

[9] F. Rădulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994), 347-389.

Department of Mathematics, Texas A&M University, College Station TX 77843– 3368, USA

  E-mail address: Ken.Dykema@math.tamu.edu