UNIVERSAL PROPERTIES OF L(𝔽_{∞}) IN SUBFACTOR THEORY.

SORIN POPA AND DIMITRI SHLYAKHTENKO

Abstract. We construct a functor from the category of standard lattices, with embeddings as morphisms, into the category of inclusions N ⊂ M, with N ≅ M ≅ L(𝔽_{∞}), with commuting square embeddings as morphisms. This functor is a right inverse to the functor G assigning to an inclusion its standard lattice. We also prove that given a subfactor Q ⊂ P, with P an arbitrary II_{1} factor, there exists a subfactor Q̂ ⊂ P̂, with the same standard invariant as Q ⊂ P, Q̂ ≅ Q∗L(𝔽_{∞}), P̂ ≅ P ∗ L(𝔽_{∞}).

1. Introduction.

A standard lattice G is an abstraction of the system of higher relative commutants of a subfactor. That is to say, the relative commutants of an arbitrary finite-index inclusion of II_{1} factors satisfy the axioms of a standard lattice (see [Pop95a]). Conversely, as was proved in [Pop95a], given any standard lattice G, it can be realized as the system of higher relative commutants of some inclusion of II_{1} factors, constructed out of G. However, it remained an open problem, whether one can construct a “universal” II_{1} factor, that would contain subfactors with any given standard lattice as higher relative commutants.

The main result of this paper shows that one can realize any arbitrary standard lattice G as the system of higher relative commutants of a type II_{1} subfactor P_{−1} ⊂ P_{0}, where both P_{−1} and P_{0} are isomorphic to the free group factor L(𝔽_{∞}). In other words, L(𝔽_{∞}) is such a universal type II_{1} factor, whose subfactors realize all possible standard lattices.

Moreover, the association G ↦ F(G) = (P_{−1} ⊂ P_{0}) with P_{−1} ≅ P_{0} ≅ L(𝔽_{∞}) is functorial in the following sense. Let 𝔾 denote the category of standard lattices, whose morphisms are embeddings of standard λ-lattices (see Section 2 for the exact definition of this latter notion). For a II_{1} factor M, let 𝕊 denote the category of all subfactors P_{−1} ⊂ P_{0} with P_{−1} ≅ P_{0} ≅ L(𝔽_{∞}), having as morphisms commuting square embeddings: a morphism ı : (P_{−1}^{(1)} ⊂ P_{0}^{(1)}) → (P_{−1}^{(2)} ⊂ P_{0}^{(2)}) is an embedding of algebras which forms the following commuting square

\begin{aligned}P_{−1}^{(2)} & ⊂ & P_{0}^{(2)} \\ ∪ı & & ∪ı \\ P_{−1}^{(1)} & ⊂ & P_{0}^{(1)} \end{aligned} .

Then F : 𝔾 → 𝕊 is a functor, which is a right inverse to the functor G : 𝕊 → 𝔾, taking an inclusion P_{−1} ⊂ P_{0} to its standard lattice G_{P_{−1}⊂P_{0}}.

In fact, L(𝔽_{∞}) has a further universal property. The collection of standard lattices of all possible subfactors of a given factor M does not decrease when M is replaced by M ∗ L(𝔽_{∞}). More precisely, if N ⊂ M is an inclusion of II_{1} factors, then there exists an inclusion N̂ ⊂ M̂ (functorially associated to N ⊂ M), with N̂ ≅ N ∗ L(𝔽_{∞}), M̂ ≅ M ∗ L(𝔽_{∞}), and so that the standard lattices G_{N̂⊂M̂} and G_{N⊂M} are the same.

Date: Preliminary version; December 2, 2000.

We note that these results are generalization of earlier results about realization of finite-depth subfactors inside free group factors [Răd94], [Dyk95], irreducible subfactors in L(𝔽_{∞}) [US00] and finite-depth subfactors of M ∗ L(𝔽_{∞}), for M arbitrary [Shl99].

To prove the first result above, to a standard λ-lattice G = (A_{ij}) we associate in a functorial way a λ-Markov commuting square

\begin{aligned}B_{−1} & ⊂ & B_{0} \\ ∪ & & ∪ \\ A_{−1} & ⊂ & A_{0} \end{aligned} ,

with A_{i} semifinite type I with discrete center, and with the top row being the infinite amplification of the standard model inclusion R_{−1} ⊂ R_{0} associated to G. Thus B_{i} are hyperfinite type II_{∞} algebras (possibly with non-trivial center). Most importantly, the commuting square satisfies (B′_{i} ∩ A_{j}) ≃ (A_{ij}) in its Jones tower, thus encoding the standard lattice G = (A_{ij}).

Similarly, to prove the second result, given an arbitrary inclusion of II_{1} factors M_{−1} ⊂ M_{0}, we associate to it a canonical λ-Markov commuting square

M_{−1} ⊂ M_{0}
∪  ∪
A_{−1} ⊂ A_{0}

in which the bottom row is as before, but the top row is the infinite amplification of the inclusion M_{−1} ⊂ M_{0}. Also, we have again (M′_{i} ∩ A_{j}) ≃ (M′_{i} ∩ M_{j}), in the corresponding Jones tower.

To construct such canonical commuting squares out of a given standard lattice or a subfactor, we use inductive limits of non-unital embeddings naturally associated to the duality isomorphisms in the Jones tower.

Using the above commuting squares, we construct amalgamated free product inclusions (along the lines of [Pop93],[Pop95a])

B_{−1} ∗_{A_{−1}} (Q ⊗ A_{−1}) ⊂ B_{0} ∗_{A_{0}} (Q ⊗ A_{0}). (1.1)

and

M_{−1} ∗_{A_{−1}} (Q ⊗ A_{−1}) ⊂ M_{0} ∗_{A_{0}} (Q ⊗ A_{0}) (1.2)

The crucial point is that (1.1) has the standard lattice G as its relative commutants, while (1.2) has the same higher relative commutants as M_{−1} ⊂ M_{0}.

We then identify the algebras involved in the inclusions (1.1) and (1.2). In fact, we show that B_{i} ∗_{A_{i}} (Q ⊗ A_{i}) ≅ L(𝔽_{∞}) and M_{i} ∗_{A_{i}} (Q ⊗ A_{i}) ≅ (M_{i} ∗ L(𝔽_{∞})) ⊗ B(H), thereby proving the claimed universal properties of L(𝔽_{∞}).

The techniques needed for this identification come from free probability theory pioneered by Voiculescu ([VDN92]). The main observation is that the amalgamated free product algebra B_{i} ∗_{A_{i}} (Q ⊗ A_{i}) is generated by B_{i} and Q ≅ L(𝔽_{∞}); furthermore, Q has as generators an infinite semicircular system X_{1}, X_{2}, … [Voi90]. The position of this family relative to B_{i} is encoded in the statement that {X_{n}} form an operator-valued semicircular system over B_{i} in the sense of [Shl98], [Shl99]. The rest of the proof involves manipulations with this semicircular system — in ways that parallel earlier random-matrix techniques of Voiculescu [Voi90],[VDN92], further developed by Radulescu [Răd92], [Răd94] and Dykema, [Dyk94], [Dyk93], [DR00].

Acknowledgement.The second-named author would like to thank the Wiley W. Manuel Courthouse in Oakland, CA, where an early part of the work was carried out while on breaks from jury duty. The authors would also like to thank MSRI and the organizers of the stimulating program on Operator algebras. Research supported in part by the NSF grant no. DMS-9801324, for S.P., and by an NSF postdoctoral fellowship, for D.S.

2. Some canonical commuting squares associated to a subfactor.

Let M_{−1} ⊂ M_{0} be an inclusion of Type II_{1} factors with finite Jones index. In this section we will associate to it a system of λ-Markov commuting squares of semifinite von Neumann algebras with trace preserving expectations

M_{−1} ⊂ M_{0}
∪  ∪
C^{0} = B_{−1} ⊂ B_{0}
∪  ∪
A_{−1} ⊂ A_{0}

in which the upper commuting square is the ∞-amplification of

\begin{aligned}M_{−1} & ⊂ & M_{0} \\ ∪ & & ∪ \\ M_{−1}^{st} & ⊂ & M_{0}^{st} \end{aligned} ,

M_{−1}^{st} ⊂ M_{0}^{st} being the standard model associated with M_{−1} ⊂ M_{0}, and in which A_{−1} ⊂ A_{0} is an inclusion of type I von Neumann algebras with atomic centers and inclusion matrix given by the graph of M_{−1} ⊂ M_{0}. The construction of the lower commuting square

\begin{aligned}B_{−1} & ⊂ & B_{0} \\ ∪ & & ∪ \\ A_{−1} & ⊂ & A_{0} \end{aligned}

will in fact only depend on the standard invariant G = G_{M_{−1},M_{0}} of M_{−1} ⊂ M_{0} and will be functorial in G. This commuting square will completely encode G, as it will satisfy B′_{i} ∩ A_{j} = M′_{i} ∩ A_{j} ≃ M′_{i} ∩ M_{j} in the Jones tower for C^{0}, respectively for M_{−1} ⊂ M_{0}.

The commuting square C^{0} will be constructed as an inductive limit of non-unital Tracepreserving embeddings of the commuting squares

M_{2n−1} ⊂ M_{2n}
∪  ∪
A_{−∞,2n−1} ⊂ A_{−∞,2n}
∪  ∪
A_{−1,2n−1} ⊂ A_{−1,2n}

where A_{ij} = M′_{i} ∩ M_{j}, j, i ∈ ℤ are the higher relative commutants in some tunnel-tower … ⊂

M_{−2} ⊂^{e_{−1}} M_{−1} ⊂^{e_{0}} M_{0} ⊂^{e_{1}} M_{1} ⊂^{e_{2}} M_{2} ⊂ … for M_{−1} ⊂ M_{0} and A_{−∞,j} = \overline{∪_{k≤j}A_{kj}}.

Lemma 2.1. For each k ≥ 0, n ≥ 0, let α_{n}^{k} be the map from M_{2n+k} into M_{2n+k+2} given by α_{n}^{k}(x) = λ^{−k−1}e_{2n+1}e_{2n+2} ⋯ e_{2n+k+1}e_{2n+k+2}xe_{2n+k+2} ⋯ e_{2n+1}, x ∈ M_{2n+k}. Then α_{n}^{k} are nonunital ∗-isomorphisms and they satisfy:

1^{∘}. α_{n}^{k}(M_{2n+j−1}) = e_{2n+1}M_{2n+j+1}e_{2n+1}, j = 0, 1, …, k + 1, with α_{n}^{k+1}_{|M_{2n+j−1}} = α_{n}^{k} if j ≤ k + 1.

3^{∘}. α_{n}^{k}(x) = σ′(x)e_{2n+1}, x ∈ M′_{2n−1} ∩ M_{2n+k}, where σ′ is the duality isomorphism on ∪_{i,j∈ℤ}A_{ij} (see e.g. [Pop99]).

2^{∘}. α_{n}^{k}(A_{i,2n+j−1}) = e_{2n+1}A_{i,2n+j+1}e_{2n+1}, j = 0, 1, …, k + 1, −∞ ≤ i ≤ −1.

4^{∘}. If Tr_{n} is the rescaled trace on ∪_{k}M_{2n+k} given by Tr_{n} = λ^{−n}τ then Tr_{n+1}(α_{n}^{k}(x)) = Tr_{n}(x), ∀x ∈ M_{2n+k}, ∀k ≥ 0.

Proof. Since [x, e_{2n+k+2}] = 0 for x ∈ M_{2n+k} and λ^{−k−1/2}e_{2n+1} ⋯ e_{2n+k+2} is a partial isometry, α_{n}^{k} follows a *-isomorphism. For the properties 1^{∘} − 4^{∘} we have:

1^{∘}. Since e_{2n+j+1}M_{2n+j−1}e_{2n+j+1} = M_{2n+j−1}e_{2n+j+1} = e_{2n+j+1}M_{2n+j+1}e_{2n+j+1} it follows that e_{2n+1} … e_{2n+k+2}M_{2n+j−1}e_{2n+k+2} … e_{2n+1} = e_{2n+1} … e_{2n+j+1}M_{2n+j−1}e_{2n+j+1} … e_{2n+1} = e_{2n+1} … e_{2n+j+1}M_{2n+j+1}e_{2n+j+1} … e_{2n+1} = e_{2n+1}M_{2n+j+1}e_{2n+1}.

2^{∘}. Since e_{2n+1}, e_{2n+2}, …, e_{2n+k+2} ∈ A_{−1,2n+k+2} and A_{i,2n+j−1}e_{2n+j+1} = e_{2n+j+1}A_{i,2n+j+1}e_{2n+j+1} for each j = 0, 1, …, k + 1 and −∞ ≤ i ≤ −1, this part follows by 1^{∘}.

3^{∘}. This is trivial by the definition of σ′.

4^{∘}. Since τ(xe_{2n+k+2}) = λτ(x) for x ∈ M_{2n+k}, one gets τ(α_{n}^{k}(x)) = λτ(x) so that Tr_{n+1}(α_{n}^{k}(x)) = Tr_{n}(x).

Notation 2.2. To simplify the notations we will denote by C_{n} the system of commuting squares

M_{2n−1} ⊂ M_{2n} ⊂ … ⊂ M_{2n+k} ⊂ …
∪ ∪ ∪
A_{−∞,2n−1} ⊂ A_{−∞,2n} ⊂ … ⊂ A_{−∞,2n+k} ⊂ …
∪ ∪ ∪
A_{−1,2n−1} ⊂ A_{−1,2n} ⊂ … ⊂ A_{−1,2n+k} ⊂ …

with C_{n}^{k} denoting its truncation up to k, k = 0, 1, . . .. Thus, with these notations Lemma 2.1 states that α_{n}^{k} identifies the commuting square (C_{n}^{k}, Tr_{n}) with the “corner” e_{2n+1}C_{n+1}^{k}e_{2n+1} of the commuting square (C_{n+1}^{k}, Tr_{n+1}) endowed with the restriction of the trace Tr_{n+1}.

Moreover, since by 2.1.1^{∘} we have α_{n}^{k+1}_{|M_{2n+j−1}} = α_{n}^{k}, for 0 ≤ j ≤ k + 1, with the sequence {α_{n}^{k}(x)}_{k} being constant from a certain point on, for each x ∈ M_{2n+j}, ∀j, we immediately get the following:

Corollary 2.3. For each n ≥ 0 and x ∈ ∪_{j≥0}M_{2n+j} let α_{n}(x) =^{def} lim_{k→∞}α_{n}^{k}(x). Then we have

1^{∘}. α_{n}(C_{n}) = e_{2n+1}C_{n+1}e_{2n+1}.

2^{∘}. α_{n}(x) = σ′(x)e_{2n+1}, x ∈ ∪_{j}A_{2n−1,2n+j} = ∪_{j}(M′_{2n−1} ∩ M_{2n+j}), where σ′ is the duality endomomorphism on ∪_{j}A_{0j} that sends A_{ij} onto A_{i+2,j+2}, ∀j ≥ i ≥ 0 (as defined in [Pop99]).

3^{∘}. Tr_{n+1} ∘α_{n} = Tr_{n} and α_{n} takes the Tr_{n}-preserving expectations (= τ-preserving expectations) in C_{n} into the restrictions to e_{2n+1}C_{n+1}e_{2n+1} of the Tr_{n+1}-preserving expectations in C_{n+1}.

4^{∘}. The top row of commuting squares in C_{n} is a sequence of basic constructions of the initial homogeneous λ-Markov commuting square of inclusions

C_{n}^{0} = \begin{aligned}M_{2n−1} & ⊂ & M_{2n} \\ ∪ & & ∪ \\ A_{−∞,2n−1} & ⊂ & A_{−∞,2n}. \end{aligned}

Moreover, α_{n} (1_{C_{n}}) = α_{n} (1_{M_{2n}}) = e_{2n+1} ∈ M_{2n+2} (the latter regarded as an algebra in C_{n+1}) has scalar central trace in A_{−∞,2n+1} (as an algebra in C_{n+1}), so that C_{n+1} is the λ^{−1}-amplification of C_{n}.

Proof. α_{n} is well defined because for each x and k large enough one has α_{n}(x) = α_{n}^{k}(x) (by 2.1.1^{∘}). Then properties 1^{∘}-3^{∘} are just reformulations of 2.1.1^{∘}–2.1.4^{∘}. The last property 4^{∘} is well known (see e.g. [Pop95b]).

Definition 2.4. We define C to be the system of inclusions of von Neumann algebras

M_{−1} ⊂ M_{0} ⊂ … ⊂ M_{k} ⊂ …
∪ ∪ ∪
B_{−1} ⊂ B_{0} ⊂ … ⊂ B_{k} ⊂ …
∪ ∪ ∪
A_{−1} ⊂ A_{0} ⊂ … ⊂ A_{k} ⊂ …

obtained as the inductive limit of the sequence of non-unital trace preserving embeddings of commuting squares

(C_{0}; Tr_{0}) ↪^{α_{0}} (C_{1}; Tr_{1}) ↪^{α_{1}} (C_{2}; Tr_{2}) ↪ .

By this we mean the following:

(2.4.1). We first take the (non-unital!) algebraic inductive limit M_{i}^{0} of M_{i} ↪^{α_{0}} M_{i+2} ↪^{α_{1}} M_{i+4} ↪ . . .. We note that M_{−1}^{0} ⊂ M_{0}^{0} ⊂ ...., in a natural way.

(2.4.2). For each n ≥ 0, j ≥ −1 and x ∈ M_{i+2n} we denote by α̃_{n}(x) = … α_{n+1} ∘ α_{n}(x) the image of x in M_{i}^{0}. With this notation, one clearly has M_{i}^{0} = ∪_{n}α̃_{n}(M_{i+2n}).

(2.4.3). On M_{i}^{0} one takes the C^{∗}-norm defined by ∥x∥ = ∥α̃_{n}(x)∥_{M_{i+2n}}, if x ∈ M_{i+2n}.

(2.4.4). One defines a positive tracial functional Tr on M_{i}^{0} by Tr(α̃_{n}(x)) = Tr_{n} x, if x ∈ M_{i+2n}. (2.4.5). One defines M_{i} to be the completion of M_{i}^{0} in the topology of convergence in the norm

∥x∥_{2,Tr} = Tr(x^{∗}x)^{1/2} on bounded sets (in C^{∗}-norm) (note that M_{i} can also be defined through the GNS construction for (M_{i}^{0}, Tr)).

(2.4.6). Note that Tr extends to a normal semifinite faithful trace on M_{i}, still denoted Tr. Moreover, M_{i} this way defined clearly satisfy M_{i} ⊂ M_{i+1} with Tr_{M_{i+1}|M_{i}} = Tr_{M_{i}} (the notation being self-explanatory).

(2.4.7). Define B_{i}, respectively A_{i}, i ≥ −1, as the closure in the same topology of ∥ ∥_{2,Tr} - convergence on bounded sets of the *-subalgebras \overline{_{⋃n}α̃_{n}(A_{−∞})}, respectively \overline{_{⋃n}α̃_{n}(A_{−1,i+2n})}, of M_{i}^{0}.

(2.4.8). We note that the trace Tr on M_{i} restricts to semifinite traces on both B_{i} and A_{i}, for each i ≥ −1.

(2.4.9). If for each n we choose an inclusion Q_{n} ⊂ P_{n} between two of the algebras in the commuting square C_{n}, but so that for each n the algebras are chosen at the same “spot”, and if we denote by E_{Q}^{P} the unique Tr-preserving expectation of the inductive limit P =^{def} \overline{_{⋃n}α̃_{n}(P_{n})} onto the inductive limit Q =^{def} \overline{_{⋃n}α̃_{n}(Q_{n})} then by Corollary 2.3 we have E_{Q}^{P}(α̃_{n}(x)) = α̃_{n}(E_{Q_{n}}^{P_{n}}(x)), for x ∈ P_{n}.

In particular, by 2.3.3^{∘}, the properties (2.4.8) and (2.4.9) above show that the system of inclusions C, endowed with the corresponding Tr-preserving expectations between its algebras, is a system of commuting squares.

We now examine more closely the main properties of C.

Lemma 2.5. If for each n ≥ 0 we let 1_{n} be the identity in C_{n}, i.e. 1_{n} = 1_{M_{2n−1}} = 1_{A_{−1,2n−1}} = 1_{M_{2n+k}} = 1_{A_{−1,2n+k}}, ∀k ≥ 0, and define p_{n} = α̃_{n}(1_{n}) then we have:

1^{∘}. p_{n} belong to A_{−1}, Tr p_{n} = λ^{−n}, ∀n, and p_{0} ≤ p_{1} ≤ p_{2} ≤ … with p_{n} ↗ 1_{A_{−1}}(= 1_{C}).

2^{∘}. For each n, p_{n}Cp_{n} is naturally isomorphic to C_{n}, via α̃_{n} (as commuting squares of trace preserving expectations).

3^{∘}. p_{n} has scalar central trace in p_{n+1}B_{−1}p_{n+1}, ∀n ≥ 0.

4^{∘}. For each j ≥ i ≥ −1 and x ∈ A_{ij} there exists a unique element α(x) in ∪_{k}M_{k} such that [α(x), p_{n}] = 0, ∀n, α(x)p_{n} = α̃_{n}(σ^{′n}(x)), where σ′ is the duality isomorphism as in 2.3.3^{∘}. Moreover, α is a ∗-isomorphism and α(A_{ij}) = M′_{i} ∩ M_{j} = M′_{i} ∩ A_{j}, ∀j ≥ i ≥ −1.

5^{∘}. α(e_{j}) belongs to A_{j}, ∀j ≥ 1 and α(e_{n+1}) implements the Tr-preserving conditional expectation of M_{n} onto M_{n−1}, ∀n ≥ 0.

Proof. 1^{∘} is clear by the definitions, and so is the equality p_{n}Cp_{n} = α̃_{n}(C_{n}) of condition 2^{∘}. Then p_{n} have scalar central trace in p_{n+1}B_{−1}p_{n+1} because e_{2n+1} has scalar central trace in A_{−∞,2n+1} (see e.g. [Pop95b]).

The first part in 4^{∘} is clear by property 2^{∘} in Corollary 2.3. Then the equality α(A_{ij}) = M′_{i}∩M_{j} is immediate by the definitions of α, M_{i}, M_{j}. Also, by the way it is defined, α(A_{ij}) is clearly contained in A_{j}, so α(A_{ij}) = M′_{i} ∩ A_{j}. (Note that in 5.2.1^{∘} we prove a stronger statement about these higher relative commutants, with the above argument being given in details).

Since e_{j} lies in A_{−1,j} for j ≥ 1, it follows by 4^{∘} that α(e_{j}) lies in A_{j}. Since e_{2k+1} implements the expectation of M_{2k} onto M_{2k−1}, it follows that α̃_{k}(e_{2k+1}) implements the conditional expectation of p_{k}M_{0}p_{k} onto p_{k}M_{−1}p_{k}. Since p_{k} ↗ 1 and α(e_{1})p_{k} = α̃_{k}(e_{2k+1}), we get the last part of 5^{∘} as well.

The next lemma clarifies the structure of the inclusions A_{−1} ⊂ A_{0} ⊂ . . .. To state it, let us denote by Γ = Γ_{M_{−1},M_{0}} = (a_{kl})_{k∈K,l∈L} the standard graph of M_{−1} ⊂ M_{0} (or, equivalently, of G = G_{M_{−1},M_{0}}), which describes the sequence of inclusions A_{−1,−1} ⊂ A_{−1,0} ⊂ A_{−1,1} ⊂ . . .. Thus, if ∗ ∈ K denotes the initial vertex of Γ and K_{n} = (ΓΓ^{t})^{n}({∗}), L_{n} = (ΓΓ^{t})^{n}Γ({∗}), then K = ∪_{n}K_{n}, L_{n} = ∪_{n}L_{n} with the sets K_{n}, L_{n} having the following significance:

The set of simple summands of Z(A_{−1,2n−1}) (respectively Z(A_{−1,2n})) naturally identifies with the set K_{n} (respectively L_{n}), with the inclusion K_{n} ⊂ K_{n+1} (respectively L_{n} ⊂ L_{n+1}) corresponding to the embedding of Z(A_{−1,2n−1}) into Z(A_{−1,2n+1}) (respectively of Z(A_{−1,2n}) into Z(A_{−1,2n+2})) given by the applications

Z(A_{−1,j}) ∋ z ↦ z′ ∈ Z(A_{−1,j+2}),

with z′ the unique element in Z(A_{−1,j+2}) such that ze_{j+2} = z′e_{j+2}.

Moreover, the inclusion graphs of A_{−1,2n−1} ⊂ A_{−1,2n} (respectively A_{−1,2n} ⊂ A_{−1,2n+1}) are given by_{K_{n}}Γ (respectively_{L_{n}}Γ^{t}).

Also, there exists a unique vector s⃗ = (s_{k})_{k∈K} such that s_{∗} = 1, ΓΓ^{t}s⃗ = λ^{−1}s⃗ and such that if t⃗ = (t_{l})_{l∈L} = λΓ^{t}s⃗ then (λ^{n}s_{k})_{k∈K_{n}} resp. (λ^{n}t_{l})_{l∈L_{n}} give the traces of the minimal projections in A_{−1,2n−1} resp. A_{−1,2n}.

With these notations at hand we have:

Lemma 2.6. A_{−1} ⊂ A_{0} ⊂ … are inclusions of atomic von Neumann algebras. More precisely, for each n ≥ 0 the reduced sequence of inclusions p_{n}(A_{−1} ⊂ A_{0} ⊂ …)p_{n} is isomorphic via α̃_{n} to the sequence of inclusions (A_{−1,2n−1} ⊂ A_{−1,2n} ⊂ …), with the trace Tr on the former corresponding to the trace Tr_{n} on the latter. Moreover, if one identifies the set of factor summands of A_{−1} which contain non-zero parts of the projection p_{n} with the set of factor summands of A_{−1,2n−1}, i.e., with K_{n}, via the identification of p_{n}A_{−1}p_{n} with A_{−1,2n−1}, then the inclusion matrix for A_{−1} ⊂ A_{0} is given by Γ, while the trace Tr is given on the minimal projections of A_{−1} by the eigenvector s⃗ = (s_{k})_{k∈K} and on the minimal projections of A_{0} by the vector t⃗. Similarly, the inclusion graph for A_{i} ⊂ A_{i+1} is given by Γ if i is odd and by Γ^{t} if i is even, with the trace vector for the minimal projections of A_{2k−1} (resp. A_{2k}) being given by λ^{k}s⃗ (resp. λ^{k}t⃗).

Proof. We have already noted in 2.5 that the non-unital isomorphism α̃_{n} takes the sequence of inclusions (A_{−1,−1} ⊂ A_{−1,0} ⊂ A_{−1,1} ⊂ …) onto the sequence of inclusions p_{n}(A_{−1} ⊂ A_{0} ⊂ …)p_{n}, with Tr ∘α̃_{n} = Tr_{n}. Since A_{ij} are all atomic and p_{n} ↗ 1, it follows that A_{k} are all atomic.

From the above and the discussion preceding 2.6, the last part follows now trivially.

Lemma 2.7. The sequence of inclusions A_{−1} ⊂ A_{0} ⊂^{α(e_{1})} A_{1} ⊂^{α(e_{2})} A_{2} … is a Jones tower of λ-Markov inclusions.

Proof. By 2.5.5^{∘} α(e_{n+1}) belongs to A_{n+1} and by commuting squares with M_{n−1} ⊂ M_{n}, it implements the Tr-preserving expectation of A_{n} onto A_{n−1}.

By the definitions, we see that p_{n}A_{1}p_{n} is contained in \overline{sp}(p_{n+1}A_{0}p_{n+1})α(e_{1})(p_{n+1}A_{0}p_{n+1}). Since p_{n} ↗ 1, this shows that \overline{sp}A_{0}α(e_{1})A_{0} = A_{1}.

But by 2.6 the traces of the minimal projections in A_{−1} ⊂ A_{0} satisfy the conditions in [Jon83].

Thus, the basic construction A_{−1} ⊂ A_{0} ⊂^{α(e_{A_{−1}})} ⟨A_{0}, e_{A_{−1}}⟩ has a λ-Markov trace that extends Tr. Altogether, this shows that A_{0} ∋ x ↦ x ∈ A_{0} and e_{A_{−1}} ↦ α(e_{1}) extends to a trace preserving

isomorphism of A_{−1} ⊂ A_{0} ⊂^{α(e_{1})} A_{1} onto A_{−1} ⊂ A_{′} ⊂^{α(e_{A_{−1}})} ⟨A_{0}, e_{A_{−1}}⟩.

Let us summarize all the properties of the commuting square C emphasized this far. To state it, recall the following

Definition 2.8. Let Q_{i}, P_{i}, i = −1, 0 be arbitrary semifinite von Neumann algebras with inclusions

\begin{aligned}P_{−1} & ⊂ & P_{0} \\ ∪ & & ∪ \\ Q_{−1} & ⊂ & Q_{0} \end{aligned}

such that the normal semifinite faithful trace Tr on P_{0} is semifinite on each of the smaller algebras and such that the corresponding Tr-preserving expectations make the above into a commuting square with both row inclusions of finite index ≤ λ^{−1}. Let P_{−1} ⊂ P_{0} ⊂^{e} P_{1} = ⟨P_{0}, e⟩ be the basic construction for P_{−1} ⊂ P_{0}. If there exists a trace on P_{1}, Tr, which extends the trace Tr on P_{0} such that Tr(xey) = λ Tr(xy), ∀x, y ∈ P_{0} and such that Q_{0} contains an orthonormal basis for P_{−1} ⊂ P_{0}, then the above commuting square will be called a λ-Markov commuting square.

Note that λ-Markov commuting squares are non-degenerate, in the sense of [Pop94] and that each of the row inclusions is λ-Markov, as defined in [Pop95b]. Note also that if one has such a commuting square then the above projection e also implements the basic construction for Q_{−1} ⊂ Q_{0}. Moreover, the resulting system of inclusions

\begin{aligned}P_{0} & ⊂ & P_{1} \\ ∪ & & ∪ \\ Q_{0} & ⊂ & Q_{1} \end{aligned}

where Q_{1} is the algebra generated by Q_{0} and e, is itself a λ-Markov commuting square (with respect to the Tr-preserving expectations). Thus, one can iterate the basic construction and obtain from the initial λ-Markov commuting square a whole Jones tower of λ-Markov commuting squares.

Theorem 2.9. 1^{∘}. The commuting squares in the initial inclusion

C^{0} = \begin{aligned}M_{−1} & ⊂ & M_{0} \\ ∪ & & ∪ \\ B_{−1} & ⊂ & B_{0} \\ ∪ & & ∪ \\ A_{−1} & ⊂ & A_{0} \end{aligned}

of C, with its Tr-preserving expectations, are all λ-Markov.

2^{∘}. C is obtained by iterating the basic construction for C^{0}.

3^{∘}. The commuting square

M_{−1} ⊂ M_{0}
∪  ∪
B_{−1} ⊂ B_{0}

is isomorphic to the ∞-amplification of the commuting square

M_{−1} ⊂ M_{0}
∪  ∪
A_{−∞,−1} ⊂ A_{−∞,0}

i.e., it is obtained by tensoring the latter by B(ℓ^{2}(ℕ)).

4^{∘}. A_{−1} ⊂ A_{0} is an inclusion of infinite type I von Neumann algebras with discrete centers, having inclusion graph given by Γ = Γ_{M_{−1},M_{0}} and the trace Tr is given on the minimal projections of A_{−1} by s⃗ and on the minimal projections of A_{0} by t⃗ = λΓ^{t}s⃗.

5^{∘}. M′_{i} ∩ M_{j} = M′_{i} ∩ A_{j} = B′_{i} ∩ A_{j} and α gives a natural isomorphism from G_{M_{−1},M_{0}} = (M′_{i} ∩ M_{j})_{j≥i≥−1} onto (M′_{i} ∩ A_{j})_{j≥i≥−1} = (B′_{i} ∩ A_{j})_{j≥i≥−1}.

The last result in this section describes the functoriality properties of the commuting squares appearing in C^{0}. To state it, recall from [Pop95a], [Pop99] that given two standard λ-lattices G^{0} = (A_{ij}^{0})_{j≥i≥−1}, G = (A_{ij})_{j≥i≥−1}, an embedding of G^{0} into G is a trace preserving isomorphism ı from ∪_{n}A_{−1,n}^{0} into ∪_{n}A_{−1,n} such that ı(A_{ij}^{0}) = A_{ij}, ∀j ≥ i ≥ −1 and such that ı takes the Jones λ-sequence of projections {e_{n}^{0}}_{n≥1} of G^{0} into a Jones sequence of projections for G, satisfying the smoothness condition

E_{A_{01}}(ı(e_{1}^{0})) = ı(E_{A_{01}^{0}}(e_{1}^{0})) (2.10.1)

Theorem 2.10. 1^{∘}. Denote by C_{M_{−1},M_{0}} the object consisting of the commuting square

M_{−1} ⊂ M_{0}
∪  ∪
A_{−1} ⊂ A_{0}

together with the fixed projection p_{0} ∈ A_{−1} is canonically associated with M_{−1} ⊂ M_{0}. 2^{∘}. The object C_{G}^{st} consisting of the commuting square

\begin{aligned}B_{−1} & ⊂ & B_{0} \\ ∪ & & ∪ \\ A_{−1} & ⊂ & A_{0} \end{aligned}

together with the fixed projection p_{0} ∈ A_{−1}, is canonically associated with the standard λ-lattice G and it is functorial in G: If G_{0} ⊂ G is a standard λ-lattice embedded in G then C_{G_{0}}^{st} is naturally nondegenerately embedded^{1} in C_{G}^{st} with commuting squares and with the corresponding projections p_{0} coinciding.

Proof. 1^{∘}. This part is clear by the construction of

\begin{aligned}B_{−1} & ⊂ & B_{0} \\ ∪ & & ∪ \\ A_{−1} & ⊂ & A_{0} \end{aligned}

as the inductive limit of the canonical commuting squares

M_{2n−1} ⊂ M_{2n}
∪  ∪
A_{−1,2n−1} ⊂ A_{−1,2n}

via embeddings which are canonical as well (being defined by using only the Jones projections in the tower e_{1}, e_{2}, …). Also, p_{0} = α̃_{0}(1) so that the position of p_{0} inside A_{−1} is canonical as well.

2^{∘}. The fact that C_{G}^{st} is canonically associated with G follows by first noticing that the extended standard lattice G̃ = (A_{ij})_{i,j∈ℤ}, associated with G as in [Pop99], is canonically constructed from G by repeated basic constructions starting from the inclusion A_{0,∞} ⊂ A_{−1,∞} (see the second paragraph in the proof of 2.2 in [Pop99]). In particular, the sequence of inclusions A_{−∞,−1} ⊂ A_{−∞,0} ⊂ …, with all the system of inclusions of higher relative commutants into it, follow canonical. From this, an argument similar to the one in part 1^{∘} ends the proof.

If G_{0} ⊂ G in an embedding of standard λ-lattices with the same Jones projections then by the definition of the embeddings in the inductive limits of 2.4, which only depends on the Jones projections, it follows that the inductive limit algebras involved in C_{G_{0}}^{st} are naturally embedded into the corresponding algebras of C_{G}^{st}, with commuting squares. To see that the embedding of the two commuting squares is non-degenerate note first that the embedding G^{0} ⊂ G implements a natural embedding between the corresponding extended standard lattices tildeG^{0}, G̃ (thus, with commuting squares!). This fact in turn is an immediate consequence of the definitions, taking into account the smoothness condition (2.10.1).

1This means that all the sides of the commuting "cube" arising from the inclusion of the two commuting squares are all non-degenerate commuting squares, with respect to the trace-preserving conditional expectations.

3. Amalgamated free products over type I algebras.

The main result of this section is the following theorem:

Theorem 3.1. Let B be a von Neumann algebra, and A ⊂ B be a subalgebra. Let E : B → A be a normal faithful conditional expectation. Assume that there exists a normal faithful semifinite trace Tr on A, so that Tr ∘E is a trace on B. Assume lastly that A is type I and has discrete center.

Let

M = (B, E) ∗_{A} (A ⊗ L(𝔽_{∞}), id ⊗τ).

If B is a of type II_{∞} and p ∈ B is a projection, Tr(p) = 1, so that there is a system of matrix units {e_{ij}} ⊂ B with e_{11} = p, ∑ e_{ii} = 1, then

M ≅ [(pBp, Tr(p·)) ∗ (L(𝔽_{∞}), τ)] ⊗ B(H);

The proof of the theorem will consist of a sequence of lemmas. The notation and assumptions of the first paragraph of the theorem remain fixed throughout this section.

It is convenient to omit the mention of the specific conditional expectations in expressions for reduced amalgamated free products. It will always be clear from the context what conditional expectations are understood. Moreover, note that all of the conditional expectations in this paper are trace-preserving.

Lemma 3.2. M is a factor iff the centers Z(A) ∩ Z(B) have trivial intersection.

Proof. By [Pop93], the relative commutant of L(𝔽_{∞}) inside M is equal to A. It follows that Z(M) ⊂ A, hence Z(M) ⊂ Z(A). Since A ⊂ B, also Z(M) ⊂ Z(A) ∩ Z(B). The other inclusion is trivial.

Let Q be a von Neumann algebra with a semifinite normal trace Tr, and let η_{i} : Q → Q be normal completely positive maps. Assume that each η_{i} is self-adjoint, i.e., Tr(η_{i}(x)y) = Tr(xη_{i}(y)) for all x, y trace-class in Q.

Define Φ(Q, η_{1}, η_{2}, …, η_{n}), n = 1, 2, … or +∞, to be the von Neumann algebra generated by Q and the Q-semicircular family X_{1}, X_{2}, …, X_{n}, so that: (i) X_{i} are free with amalgamation over Q and (ii) each X_{i} has covariance η_{i}. Denote by E_{Q} the canonical conditional expectation from Φ(Q, η_{1}, η_{2}, …, η_{n}) onto Q. By [Shl99], Tr ∘E_{Q} is a trace on Φ(Q, η_{1}, …, η_{n}). Moreover, E_{Q}(X_{i}qX_{j}) = δ_{ij}η(q), for all q ∈ Q. Recall [Shl98] that X_{i} satisfy the inequality

∥X_{i}∥ ≤ 2∥η_{i}(1)∥^{1/2}.

Recall [Shl98] that if q_{i}, r_{i} ∈ Q are elements, X is Q-semicircular of covariance η, then

Y_{i} = q_{i}Xr_{i} + r_{i}^{∗}Xq_{i}^{∗}

is again Q-semicircular, of covariance

q ↦ q_{i}η(r_{i}xr_{i}^{∗})q_{i}^{∗} + r_{i}^{∗}η(q_{i}^{∗}xq_{i})r_{i}
+q_{i}η(r_{i}xq_{i})r_{i} + r_{i}^{∗}η(q_{i}^{∗}xr_{i}^{∗})q_{i}^{∗}.

In addition, {Y_{i}} are free with amalgamation over Q iff E_{Q}(Y_{i}qY_{j}) = 0 for all q ∈ Q and i ≠ j.

Lemma 3.3. M ≅ Φ(B, E, E, E, …) (infinite number of copies).

Proof. By [Shl99],

Φ(B, E, E, …) ≅ (B, E) ∗_{A} Φ(A, id, id, id, …)
≅ (B, E) ∗_{A} (A ⊗ Φ(ℂ, id, id, …))
≅ (B, E) ∗_{A} (A ⊗ L(𝔽_{∞}))
= M.

We need a slight modification of the construction Φ, which works for semifinite completely positive maps, like Tr : B → B.

Lemma 3.4. Let η_{i} : Q → Q, μ_{i} : Q → Q be normal self-adjoint c.p. maps. Assume that for each i, there exist (possibly unbounded) operators x_{i} affiliated with Q, with (possibly unbounded) inverses, so that

μ_{i}(q) = x_{i}^{∗}η_{i}(x_{i}qx_{i}^{∗})x_{i}, ∀q ∈ Q.

Then Φ(Q, η_{1}, η_{2}, …) ≅ Φ(Q, μ_{1}, μ_{2}, …) in a way that preserves Q and E_{Q}. (The equation means that μ_{i} can be obtained as the closure of the densely defined operator q ↦ x_{i}^{∗}η_{i}(x_{i}qx_{i}^{∗})x_{i}).

Proof. By definition,

Φ(Q, η_{1}, η_{2}, …) = W^{∗}(Q, X_{1}, X_{2}, …),

where X_{i} are Q-semicircular, of covariance η_{i}. We claim that x_{i}^{∗}X_{i}x_{i} ∈ Φ(Q, η_{1}, η_{2}, …) (a priori, it may not be defined, since x_{i} may be unbounded). It is sufficient, by passing to the polar decomposition x_{i} = u_{i}b_{i}, u_{i} ∈ Q unitary, to consider only the case that x_{i} are self-adjoint. Denote by x_{i}^{t} the value of the cut-off function {x ↦ x}|_{[−t,t]}applied to x_{i}. Let Y_{t} = x_{i}^{t}X_{i}x_{i}^{t}. Then Y_{t} is again Q-semicircular, of covariance

η_{i}^{t}(q) = x_{i}^{t}η_{i}(x_{i}^{t}qx_{i}^{t})x_{i}^{t}.

In particular,

∥Y_{t}∥ ≤ 2∥x_{i}^{t}η_{i}(x_{i}^{t}x_{i}^{t})x_{i}^{t}∥^{1/2}.

Since x_{t}^{t}x_{i}^{t} ≤ x_{i}^{2} we get that

x_{i}^{t}η_{i}(x_{i}^{t}x_{i}^{t})x_{i}^{t} ≤ x_{i}^{t}η(x_{i}^{2})x_{i}^{t}
= χ_{[−t,t]}(x_{i})x_{i}η(x_{i}^{2})x_{i}χ_{[−t,t]}(x_{i})
≤ x_{i}η(x_{i})^{2}x_{i} = μ(1).

Hence we that

∥Y_{t}∥ ≤ 2∥μ(1)∥^{1/2}.

Note that Y_{t} = χ_{[−t,t]}(x_{i})x_{i}^{s}X_{i}x_{i}^{s}χ_{[−t,t]}(x_{i}) if t < s. Hence Y_{t} are bounded, and moreover χ_{[−r,r]}(x_{i})Y_{t}χ_{[−r,r]}(x_{i}) does not depend on t once t > r. It follows that also the weak limit of Y_{t} exists and is bounded. We denote the limit by x_{i}X_{i}x_{i}. It is clear that x_{i}X_{i}x_{i} is Q-semicircular of covariance q ↦ x_{i}η(x_{i}qx_{i})x_{i}. Note that X_{i} ∈ W^{∗}(Q, x_{i}X_{i}x_{i}) (one simply applies the same construction, starting with x_{i}X_{i}x_{i} and using x_{i}^{−1} in the place of x_{i}).

Now,

Φ(Q, η_{1}, η_{2}, …) = W^{∗}(Q, x_{1}X_{1}x_{1}, x_{2}X_{2}x_{2}, …)
≅ Φ(Q, μ_{1}, μ_{2}, …),

since x_{i}X_{i}x_{i} has covariance q ↦ x_{i}η(x_{i}qx_{i})x_{i} = μ_{i}(x_{i}).

Definition 3.5. Φ(Q, Tr, Tr, …) = Φ(Q, η, η, …), where η is any normal c.p. map from Q to Q, so that η(q) = x^{∗} Tr(xqx^{∗})x for some x ∈ Q, having a (possibly unbounded) inverse.

It is not hard to see, from Lemma 3.4, that this definition does not depend on the choice of η. Moreover, if the trace Tr is actually finite, then this coincides with the previous definition of Φ(Q, Tr).

Remark 3.6. The “unbounded semicircular element” of Radulescu [Răd92] (see also [DR00]) is precisely the “operator” one would get if in the construction of Φ(Q, Tr) one were to use a semifinite trace, but completely ignore the fact that Tr(1) is infinite. If η(·) = x Tr(x·x)x is as above, and X is Q-semicircular of covariance η, then Radulescu’s element would correspond to the operator x^{−1}Xx^{−1}, which does not make sense as an operator, because Tr is not a normal s.a. map from Q to itself. Note that, as used in Radulescu’s work, the finite compressions χ_{[−t,t]}(x)x^{−1}Xx^{−1}χ_{[−t,t]}(x) do make sense as operators in Φ(Q, Tr). In particular, Φ(Q, Tr) is exactly the algebra Q ∗ SX described in [DR00].

Lemma 3.7. Let N = Φ(Q, η_{1}, η_{2}, …, μ_{1}, μ_{2}, …). Denote by E_{η} : N → Φ(Q, η_{1}, η_{2}, …) = N_{η} and E_{μ} : N → Φ(Q, μ_{1}, μ_{2}, …) = N_{μ} the canonical conditional expectations. Then

N ≅ (N_{η}, E_{Q}) ∗_{Q} (N_{μ}, E_{Q}) ≅ Φ(N_{η}, μ_{1} ∘ E_{η}, μ_{2} ∘ E_{η}, …)

in a way that preserves N_{η}, Q and E_{η}, E_{Q}.

Proof. By definition,

N = W^{∗}(Q, X_{1}, X_{2}, …, Y_{1}, Y_{2}, …),

where X_{i} and Y_{i} are free over Q, and X_{i} is η_{i}-semicircular over Q, Y_{i} is μ_{i}-semicircular over Q. The claimed decomposition as an amalgamated free product follows. The second isomorphism follows from the fact that Y_{i}, being free from W^{∗}(Q, X_{1}, X_{2}, …) = N_{η} over Q, is μ_{i} ∘ E_{η}-semicircular over N_{η} (see [Shl98]).

Lemma 3.8. Assume that Q is a factor of type II_{∞}, and η_{1}, η_{2}, … are normal self-adjoint c.p. maps from Q to itself. Assume that η_{i} ≠ 0 for all i, and that for each i, there exist subalgebras A_{i}, each of type I with discrete center, so that

η_{i} = E_{A_{i}}^{Q}

Then Φ(Q, η_{1}, η_{1}, …, η_{2}, η_{2}, …) ≅ Φ(Q, Tr, Tr, Tr, …) (each η_{i} is repeated an infinite number of times), in a way that maps Q to Q, and preserves E_{Q}.

Proof. Since Φ(Q, η_{1}, …, η_{2}, …) = Φ(Q, η_{1}, η_{1}, …) ∗_{Q} Φ(Q, η_{2}, η_{2}, …) ∗_{Q} …, it is sufficient to prove the result in the case that all η_{i} are the same. We can then clearly assume that all A_{i} = A, and η_{i} = E_{A}.

Let q_{1}, q_{2}, … be the minimal central projections of A, ∑ q_{i} = 1. Then A = ∑ q_{i}Aq_{i}, and each q_{i}Aq_{i} is a type I factor; let n_{i} ∈ ℕ ∪ {+∞} be the rank (square root of the dimension) of q_{i}Aq_{i}. Let e_{st}^{i}, 1 ≤ s, t ≤ n_{i} be a system of matrix units for q_{i}Aq_{i}; that is,

e_{st}^{i}s_{t′s′}^{j} = δ_{ij}δ_{tt′}e_{ss′}^{i} e_{st}^{i} = (e_{ts}^{i})^{∗}

q_{i} = ∑_{1≤s≤n_{i}} e_{ss.}^{i}

Claim 3.9. Let P be an algebra of type II_{∞}, P̂ ⊂ P be a unital subalgebra of type II_{∞}, so that P̂ is a factor; and p ∈ P̂ be a projection of finite trace. Let ν : Q → Q be given by

ν(q) = p Tr(pqp)p, q ∈ P.

Then Φ(P, Tr, Tr, …) ≅ Φ(P, ν, ν, …) in a way that preserves P and E_{P}.

Proof. Choose matrix units f_{ij} ∈ P̂, so that f_{11} = p, ∑ f_{ii} = 1, f_{ij}f_{j′i′} = δ_{jj′}f_{ii′}, f_{ij}^{∗} = f_{ji}. Let x = ∑_{i} \frac{1}{2^{i}} f_{ii}, and let μ(p) = x Tr(xpx)x, p ∈ P be a completely-positive map from P to itself. Then Φ(P, Tr, Tr, …) ≅ Φ(P, μ, μ, …). Let X_{i} be a P -semicircular family of covariance μ; thus Φ(P, Tr, Tr, …) = W^{∗}(P, X_{1}, X_{2}, …). Let X_{ij}^{k} = ℜf_{1i}X_{k}f_{j1}, i ≤ j, Y_{ij}^{k} = ℑf_{1i}X_{k}f_{j1}, i < j. Then Φ(P, Tr, Tr, …) is generated by P and {X_{ij}^{k}}_{k,i≤j} ∪ {Y_{ij}^{k}}_{k,i<j}. A straightforward computation shows that E(X_{ij}^{k}pX_{i′j′}^{k′}) = const × δ_{ii′}δ_{jj′}δ_{kk′}p Tr(pqp), E(Y_{ij}^{k}pY_{i′j′}^{k′}) = const × δ_{ii′}δ_{jj′}δ_{kk′}p Tr(pqp), and E(X_{ij}^{k}pY_{i′j′}^{k′}) = 0. Hence, upon proper rescaling, {X_{ij}^{k}}_{k,i≤j} ∪ {Y_{ij}^{k}}_{k,i<j} form a P -semicircular family of covariance ν. Hence Φ(P, Tr, Tr, …) ≅ Φ(P, ν, ν, …), as claimed.

Claim 3.10. Φ(Q, Tr, Tr, …) ≅ Φ(Q, η, η, …, Tr, Tr, Tr, …), in a way that preserves Q and E_{Q}.

Proof. Let p_{i} = e_{11}^{i} ∈ Q. Let ν_{i}(q) = p_{i} Tr(p_{i}qp_{i})p_{i}. We first notice that, in view of Claim 3.9,

Φ(Q, Tr, Tr, …) ≅ (Φ(Q, Tr, Tr, …) ∗_{Q} (Φ(Q, Tr, Tr, …) ∗_{Q} …
≅ (Φ(Q, Tr, Tr, …, ν_{1}, ν_{1}, …) ∗_{Q} (Φ(Q, Tr, Tr, …, ν_{2}, ν_{2}, …) ∗_{Q} …
≅ Φ(Q, Tr, Tr, …, ν_{1}, ν_{1}, …, ν_{2}, ν_{2}, …, …)
≅ Φ(Q, ν_{1}, ν_{1}, …, ν_{2}, ν_{2}, …, …) ∗_{Q} Φ(Q, Tr, Tr, …).

Let X_{j}^{i} be Q-semicircular variables, free with amalgamation over Q, and so that the covariance of X_{j}^{i} is ν_{i}. Note that X_{j}^{i} = e_{11}^{i}X_{j}^{k}e_{11}^{i}. Let

Y_{j} = ∑_{i} ∑_{1≤s≤n_{i}} e_{s1}X_{j}^{i}e_{1s} · \frac{1}{Tr(e_{11}^{s})^{1/2}}

This sum converges in weakly, since X is diagonal relative to the orthogonal family of projections {e_{ss}^{i}}, and

∥e_{s1}X_{j}^{i}e_{1s} · \frac{1}{Tr(e_{11}^{s})^{1/2}} ∥ ≤ 2∥ν_{i}(1)∥^{1/2} · \frac{1}{Tr(e_{11}^{s})^{1/2}} = 2.

It is not hard to see that {Y_{j}} form a Q-semicircular family of covariance E_{A} = η. Moreover, Φ(Q, ν_{1}, ν_{1}, …, ν_{2}, ν_{2}, …, …) is generated by Q and {Y_{i}}_{i}, since X_{j}^{i} is, up to a constant, e_{11}^{i}Y_{j}e_{11}^{i}. Hence Φ(Q, ν_{1}, ν_{1}, …, ν_{2}, ν_{2}, …, …) ≅ Φ(Q, η, η, …). Thus

Φ(Q, Tr, Tr, …) ≅ Φ(Q, ν_{1}, ν_{1}, …, ν_{2}, ν_{2}, …, …) ∗_{Q} Φ(Q, Tr, Tr, …)
≅ Φ(Q, η, η, …) ∗_{Q} Φ(Q, Tr, Tr, …)
≅ Φ(Q, η, η, …, Tr, Tr, …).

We now finish the proof of the lemma. By Lemma 3.7, we get that

Φ(Q, Tr, Tr, …) ≅ Φ(Q, η, η, …, Tr, Tr, …)
≅ Φ(Φ(Q, η, η, …), Tr, Tr, …).

Noticing that P = Φ(Q, η, η, …) contains a II_{∞} factor P̂ = Q, and setting p = e_{11}^{1} ∈ Q, ν(x) = p Tr(pxp)p, x ∈ Φ(Q, η, η, …), we get

Φ(Φ(Q, η, η), Tr, Tr, …) ≅ Φ(Φ(Q, η, η, …), ν, ν, …)
≅ Φ(Q, η, η, …, ν|_{Q}, ν|_{Q}, …),

the last isomorphism because ν = E_{Q}^{Φ(Q,η,η,...)} ∘ ν ∘ E_{Q}^{Φ(Q,η,η,...)}, since p ∈ Q (see Lemma 3.7).

Now, Φ(Q, η, η, …, ν|_{Q}, ν|_{Q}, …) is generated by Q and a Q-semicircular system X_{1}, X_{2}, …, Y_{1}, Y_{2}, …, where {X_{i}, Y_{i}}_{i} are free with amalgamation over Q, X_{i} has covariance η and Y_{i} has covariance ν. Note that X_{i} commutes with A (containing p = e_{11}^{1}), and Y_{i} = pY_{i}p, because of the form of ν. In particular, X_{i} = ∑_{i,1≤s≤n_{i}} e_{ss}^{i}X_{i}e_{ss}^{i}. It follows that Φ(Q, η, η, …, ν|_{Q}, ν|_{Q}, …) is generated by:

{q_{1}X_{i}q_{1}}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, {e_{11}^{1}Y_{i}e_{11}^{1}}_{i}, Q.

Furthermore, Φ(Q, η, η, …) is generated by:

{q_{1}X_{i}q_{1}}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, Q

Note that the three families {q_{1}X_{i}q_{1}}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, {pY_{i}p}_{i} are free with amalgamation over Q; this is because for all q ∈ Q,

E_{Q}(q_{1}X_{i}q_{1}q(1 − q_{1})X_{j}(1 − q_{1})) = δ_{ij}q_{1}E_{A}(q_{1}q(1 − q_{1}))(1 − q_{1}) = 0,

since q_{1} is a central projection in A.

Next, since X_{i} commutes with A, we get that

q_{1}X_{i}q_{1} = ∑_{1≤s≤n_{1}} e_{ss}^{1}X_{i}e_{ss}^{1}
= ∑_{1≤s≤n_{1}} e_{s1}^{1}e_{1s}^{1}X_{i}e_{s1}^{1}e_{1s}^{1}
= ∑_{1≤s≤n_{1}} e_{s1}^{1}pX_{i}pe_{1s}^{1}.

It follows that Φ(Q, η, η, …, ν|_{Q}, ν|_{Q}, …) is generated by:

{pX_{i}p}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, {pY_{i}p}_{i}, Q,

and the families {pX_{i}p}_{i}, {(1−q_{1})X_{i}(1−q_{1})}_{i} {pY_{i}p} are free with amalgamation over Q. Moreover, Φ(Q, η, η, …) is generated by

{pX_{i}p}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, Q.

Now, {pX_{i}p}_{i} are free with amalgamation over Q, and pX_{i}p is Q-semicircular with covariance

q ↦ E_{Q}(pX_{i}pqpX_{i}p) = pE_{A}(pqp)p = const × p Tr(pqp)p = const × ν(q).

It follows that {pX_{i}p}_{i} (upon rescaling by some non-zero constant) form an infinite Q-semicircular family of covariance ν|_{Q}. Hence, by renumbering, we can joint {pX_{i}p}_{i} ∪ {pY_{i}p}_{i} into a single semicircular family of covariance ν. It follows that the algebras

W^{∗}({pX_{i}p}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, {pY_{i}p}_{i}, Q)

and

W^{∗}({pX_{i}p}_{i}, {(1 − q_{1})X_{i}(1 − q_{1})}_{i}, Q)

are isomorphic to each other, in a way that maps Q to Q, and preserves E_{Q}. But we saw before that the first of these algebras is Φ(Q, η, η, …, ν|_{Q}, ν|_{Q}, …), while the second is Φ(Q, η, η, …).

Lemma 3.11. If B is a of type II_{∞} and p ∈ B is a projection, Tr(p) = 1, so that there is a system of matrix units {e_{ij}} ⊂ B with e_{11} = p, ∑ e_{ii} = 1, then

Φ(B, Tr) ≅ [(pBp, Tr(p·)) ∗ (L(𝔽_{∞}), τ)] ⊗ B(H);

In particular, Φ(B, Tr, Tr, …) is a II_{∞} factor.

Proof. Let p_{i} = e_{ii} be orthogonal projections in B, Tr(p_{i}) = 1, ∑ p_{i} = 1. Let x = ∑ \frac{1}{2^{n}} p_{n}, η : B → B be given by η(b) = x Tr(xbx)x. Then Φ(B, η, η, …) ≅ Φ(B, Tr, Tr, …), by definition. Hence Φ(B, Tr, Tr, …) ≅ W^{∗}(B, X_{1}, X_{2}, …), where X_{i} are B-semicircular, each of covariance η. Then

p_{1}Φ(B, Tr, Tr, …)p_{1} ≅ W^{∗}(p_{1}Bp_{1}, {X_{ij}^{r}}_{r,i,j}),

where X_{ij}^{r} = e_{1i}X_{r}e_{j1}. It is not hard to see that {X_{ii}^{r}} ∪ {ℜX_{ij}^{r} : i > j} ∪ {ℑX_{ij}^{r} : i > j} are free over p_{1}Bp_{1} and are again a p_{1}Bp_{1}-semicircular family, each having covariance \frac{1}{2^{i+j}} · Tr(p_{1} · p_{1}). Denoting p = p_{1} and τ(·) = Tr(p · p), we get (see [Shl99])

pΦ(B, Tr, Tr, …)p ≅ Φ(pBp, τ, τ, …) ≅ (B, τ) ∗ L(𝔽_{∞}).

The following corollary, together with Lemma 3.11, implies Theorem 3.1.

Corollary 3.12. Let M be as in Theorem 3.1. Then M ≅ Φ(B, Tr, Tr, Tr, …).

Proof. Let F : Φ(B, E, E, …) → A denote the composition of E : B → A and E_{B} : Φ(B, E, E, …) → B. Let N = Φ(B, Tr, Tr, …), and denote by G : N → A the composition of E_{B} : N → B and E : B → A. Note that F, G and E all satisfy the hypothesis of Lemma 3.8; moreover, by 3.11, N is a factor. We have

M ≅ Φ(B, E, E, …)
≅ Φ(Φ(B, E, E, …), F, F, F, …)
≅ Φ(M, F, F, …)
≅ Φ(M, Tr, Tr, …)
≅ Φ(Φ(B, E, E, …), Tr, Tr, …)
≅ Φ(B, E, E, E, …, Tr, Tr, …)
≅ Φ(B, Tr, Tr, …, E, E, …)
≅ Φ(Φ(B, Tr, Tr, …), G, G, …)
≅ Φ(N, G, G, …)
≅ Φ(N, Tr, Tr, …)
≅ Φ(Φ(B, Tr, Tr, …), Tr, Tr, …)
≅ Φ(B, Tr, Tr, …).

This completes the proof.

4. Constructions of subfactors.

Let us begin by recording the following general result (which is well-known for semifinite inclusions with trace-preserving conditional expectations).

Proposition 4.1. Let

\begin{matrix}P_{−1} & ⊂^{F_{0}} & P_{0} \\ ∪E_{−1} & & ∪E_{0} \\ Q_{−1} & ⊂^{F_{−1}} & Q_{0} \end{matrix} (4.1.1)

be a commuting square, and assume that E_{i} are faithful normal conditional expectations. Let Q be a diffuse finite von Neumann algebra with a normal finite faithful trace τ, and set

M_{i} = ((Q_{i} ⊗ Q), id ⊗τ) ∗_{Q_{i}} (P_{i}, E_{i}).

Then

\begin{matrix}M_{−1} & ⊂^{F̂} & M_{0} \\ ∪ & & ∪ \\ P_{−1} & ⊂^{F_{0}} & P_{0} \\ ∪E_{−1} & & ∪E_{0} \\ Q_{−1} & ⊂^{F_{−1}} & Q_{0} \end{matrix} (4.1.2)

forms a commuting diagram of inclusions of von Neumann algebras. Moreover, M′_{−1} ∩ M_{0} = P′_{−1} ∩ Q_{0}.

(b) Assume that (4.1.1) forms a commuting square, and F_{i} are finite-index conditional expecta-

tions. Assume also that (4.1.1) is non-degenerate, i.e., any ONB {m_{i}} for the inclusion Q_{−1} ⊂^{F_{1}} Q_{0}

forms an ONB for the inclusion P_{−1} ⊂^{F_{1}} P_{0} (equivalently, \overline{sp}(Q_{0}P_{−1}) = P_{0}).

Then all the commuting squares in (4.1.2) are non-degenerate. In particular, the index of M_{−1} ⊂^{F̂} M_{0} is given by ∑ m_{j}m_{j}^{∗}.

(c) Assume that

\begin{aligned}P_{−1}^{−1} & ⊂ & P_{0}^{−1} \\ ∪ & & ∪ \\ Q_{−1}^{−1} & ⊂ & Q_{0}^{−1} \end{aligned} ⊂ \begin{aligned}P_{−1}^{0} & ⊂ & P_{0}^{0} \\ ∪ & & ∪ \\ Q_{−1}^{0} & ⊂ & Q_{0}^{0} \end{aligned}

is a non-degenerate inclusion of non-degenerate commuting squares (non-degeneracy here means that all of the 6 commuting squares obtained by combining the inclusions of P_{i}^{j} and Q_{i}^{j}, are nondegenerate). Then

M_{−1}^{0} ⊂ M_{0}^{0}
∪  ∪
M_{−1}^{−1} ⊂ M_{0}^{−1}

is again a non-degenerate commuting square.

Proof. (a) Note that the algebra generated by P_{−1} and Q inside M_{0} is isomorphic to M_{1}; this is because Q and P_{−1} are free with amalgamation over Q_{−1}, and the conditional expectations involved in the amalgamated free products are faithful.

(b) By the nondegeneracy and commuting square condition, an orthonormal basis {m_{i}} for Q_{−1} ⊂ Q_{0} "pulls out" to become an orthonormal basis for M_{−1} ⊂ M_{0}.

(c) By arguing as in part (a), we get the vertical inclusions in

\begin{aligned}M_{−1}^{0} & ⊂ & M_{0}^{0} \\ ∪ & & ∪ \\ M_{−1}^{−1} & ⊂ & M_{0}^{−1} \end{aligned}.

Using the commuting square conditions and non-degeneracy, we see that an ONB for M_{−1}^{−1} ⊂ M_{0}^{−1} (coming from an ONB for Q_{−1}^{−1} ⊂ Q_{0}^{−1}) is an ONB for M_{−1}^{0} ⊂ M_{0}^{0}.

We now come to the functorial construction of subfactors in L(𝔽_{∞}). We denote by 𝔾 the category whose objects are standard lattices. The morphisms in this category are by definition embeddings of standard lattices with the same index (i.e., embeddings of λ-lattices with the same λ), satisfying the smoothness condition (2.10.1).

Theorem 4.2. Let 𝔾 be the category of standard lattices, with (index-preserving) embeddings as morphisms. Let 𝕊 = 𝕊(L(𝔽_{∞})) be the category of subfactors (P_{−1} ⊂ P_{0}), P_{0} = L(𝔽_{∞}), P_{−1} ≅ P_{0} of L(𝔽_{∞}) with morphisms ı : (P_{−1} ⊂ P_{0}) → (Q_{−1} ⊂ Q_{0}) given by commuting square inclusions

\begin{aligned}Q_{−1} & ⊂ & Q_{0} \\ ∪ & & ∪ \\ P_{−1} & ⊂ & P_{0} \end{aligned} .

Denote by G the functor G : 𝕊 → 𝔾, assigning to an inclusion its standard lattice,

G(P_{−1} ⊂ P_{0}) = G_{P_{−1}⊂P_{0}}.

Then there exists a functor F : 𝔾 → 𝕊, which is a right inverse for G:

G ∘ F = id .

Proof. Let G_{1}, G_{2} ∈ 𝔾 be standard λ-lattices and assume G_{1} ⊂ G_{2}. Associated to them, consider the inclusion of λ-Markov squares

\begin{aligned}B_{−1}^{(1)} & ⊂ & B_{0}^{(1)} \\ ∪ & & ∪ \\ A_{−1}^{(1)} & ⊂ & A_{0}^{(1)} \end{aligned} ⊂ \begin{aligned}B_{−1}^{(2)} & ⊂ & B_{0}^{(2)} \\ ∪ & & ∪ \\ A_{−1}^{(2)} & ⊂ & A_{0}^{(2)} \end{aligned}

as in Theorems 2.9 and 2.10. Note that A_{−1,0}^{(1),(2)} are type I with discrete center. Moreover, B_{−1,0}^{(1),(2)} are hyperfinite and each is isomorphic to an algebra of the form R ⊗ B(H), where R is hyperfinite type II_{1} (possibly with non-trivial center). Let Q = L(𝔽_{∞}). Denote by F(G) the compression of the inclusion

(Q ⊗ A_{−1}^{(1)}) ∗_{A_{−1}^{(1)}} B_{−1}^{(1)} ⊂ (Q ⊗ A_{0}^{(1)}) ∗_{A_{0}^{(1)}} B_{0}^{(1)}

to the canonical trace 1 projection in A_{−1}^{(1)} (denoted by p_{0} in Theorem 2.9). By Proposition 4.1, we get that the standard lattice of this inclusion is G_{1}, i.e., G ∘ F = G. Also, by Proposition 4.1, the inclusions

(Q ⊗ A_{−1}^{(2)}) ∗_{A_{−1}^{(2)}} B_{−1}^{(2)} ⊂ (Q ⊗ A_{0}^{(2)}) ∗_{A_{0}^{(2)}} B_{0}^{(2)}
∪  ∪
(Q ⊗ A_{−1}^{(1)}) ∗_{A_{−1}^{(1)}} B_{−1}^{(1)} ⊂ (Q ⊗ A_{0}^{(1)}) ∗_{A_{0}^{(1)}} B_{0}^{(1)}

form a commuting square, which means that F is a functor between the categories 𝔾 and 𝕊. Lastly, Theorem 3.1 implies that each of the algebras involved is isomorphic to an algebra of the form (R ∗ L(𝔽_{∞})) ⊗ B(H) ≅ L(𝔽) ⊗ B(H), where R is a type II_{1} hyperfinite algebra (the last isomorphism follows from the results of Ken Dykema, [Dyk93]). It follows that the compressed inclusion F(G_{i}) consists of algebras isomorphic to L(𝔽_{∞}).

Corollary 4.3. Let G be any standard lattice. Then there exists an inclusion P_{−1} ⊂ P_{0}, having G as its system of higher relative commutants, and so that P_{−1} ≅ P_{0} ≅ L(𝔽_{∞}).

Theorem 4.4. Let M_{−1} ⊂ M_{0} be an inclusion of II_{1} factors with finite index. Then there exists an inclusion M̂_{−1} ⊂ M̂_{0}, with M̂_{−1} ≅ M_{−1} ∗ L(𝔽_{∞}), M̂_{0} ≅ M_{0} ∗ L(𝔽_{∞}), so that M̂_{−1} ⊂ M̂_{0} has the same index and the same standard lattice of higher relative commutants as M_{−1} ⊂ M_{0}.

Proof. By Theorem 2.9, there exists a non-degenerate commuting square

M_{−1} ⊗ B(H) ⊂ M_{0} ⊗ B(H)
∪  ∪
A_{−1}  ⊂  A_{0}

with A_{−1} and A_{0} type I with discrete centers. Letting M_{−1} = (M_{−1} ⊗B(H))∗_{A_{−1}} (L(𝔽_{∞} ⊗A_{−1}), M_{0} = (M_{0} ⊗ B(H)) ∗_{A_{0}} (L(𝔽_{∞} ⊗ A_{0}), we see that M_{−1} ⊂ M_{0} has the same higher relative commutants as M_{−1} ⊂ M_{0}. Compressing by a finite projection and noticing that in view of Theorem 3.1, M_{−1} ≅ (M_{−1}∗L(𝔽_{∞}))⊗B(H), M_{0} ≅ (M_{0}∗L(𝔽_{∞}))⊗B(H), gives the result.

5. Further constructions of subfactors of L(𝔽_{∞})

Returning to the setting and notations of Section 2, let us note that one can associate to a given standard λ-lattice G = (A_{ij})_{j≥i≥−1} a canonical commuting square of type I algebras as follows:

Notation 5.1. Let (A_{−1}^{0} ⊂ A_{0}^{0}) =^{def} \overline{⋃_{n} α̃_{n}(A_{−2,2n−1} ⊂ A_{−2,2n})}. Note that Tr is semifinite on the algebras A_{i}^{0}, i = 0, −1. Note also that if we re-denote (A_{−1} ⊂ A_{0}) as (A_{−1}^{−1} ⊂ A_{0}^{−1}) then one has the following inclusions with commuting squares (for the Tr-preserving expectations):

\begin{aligned}B_{−1} & ⊂ & B_{0} \\ ∪ & & ∪ \\ A_{−1}^{0} & ⊂ & A_{0}^{0} \\ ∪ & & ∪ \\ A_{−1}^{−1} & ⊂ & A_{0}^{−1} \end{aligned} (5.1.1)

We denote by C_{G} the pair consisting of the commuting square

\begin{aligned}A_{−1}^{0} & ⊂ & A_{0}^{0} \\ ∪ & & ∪ \\ A_{−1}^{−1} & ⊂ & A_{0}^{−1} \end{aligned} (5.1.2)

and the projection p_{0} ∈ A_{−1}^{−1}(= A_{−1}).

Theorem 5.2. 1^{∘}. The commuting square (5.1.2) is λ-Markov and if

A_{−1}^{0} ⊂ A_{0}^{0} ⊂ A_{1}^{0} ⊂ …
∪ ∪ ∪
A_{−1}^{−1} ⊂ A_{0}^{−1} ⊂ A_{1}^{−1} ⊂ …

is its basic construction, then

(A_{i}^{0′} ∩ A_{j}^{−1})_{j≥i≥−1} ≃ (A_{ij})_{j≥i≥−1}.

2^{∘}. C_{G} is canonically associated with G and G ↦ C_{G} is a functor from the category of standard λ-lattices to the category of λ-Markov commuting squares of type I algebras with discrete centers.

Proof. 1^{∘}. Since the commuting square formed by the top and bottom rows of (5.1.1) is λ-Markov (cf. 2.8), the commuting square (5.1.2) follows λ-Markov as well.

Moreover, since A′_{−2,i} ∩ A_{−1,j} = A_{ij}, it follows that A′_{−2,i+2n} ∩ A_{−1,j+2n} = σ^{′n}(A_{ij}) so that α̃_{n}(A_{−2,i+2n})′ ∩ α̃_{n}(A_{−1,j+2n}) = α̃_{n}(σ^{′n}(A_{ij})) which gives p_{n}A_{i}^{0}p′_{n} ∩ p_{n}A_{j}^{−1}p_{n} = α(A_{ij})p_{n}. Since p_{n} ↗ 1, this proves 1^{∘}.

Part 2^{∘} is trivial, by the definition of C_{G}, since the construction of A_{−1}^{0} ⊂ A_{0}^{0} only depends on the Jones projections in G. Also, the commuting square conditions involved in the embedding G_{0} ⊂ G and the definition of the inductive limit, shows that C_{G_{0}} sits inside C_{G} with non-degenerate commuting squares.

One can then prove (utilizing essentially the same techniques as Radulescu in [Răd94] or Dykema in [Dyk95]), the following:

Theorem 5.3. Let A ⊂ C be a trace-preserving inclusion of type I von Neumann algebras, each having discrete center. Then

(A ⊗ L(𝔽_{∞})) ∗_{A} C ≅ L(𝔽_{∞}) ⊗ B(H).

Combining the two theorems together and compressing to the canonical projection p_{0} ∈ A_{−1} gives an alternative proof of Theorem 4.2.

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Department of Mathematics, UCLA, Los Angeles, CA 90095, USA

  E-mail address: popa@math.ucla.edu
  E-mail address: shlyakht@math.ucla.edu