FREE FISHER INFORMATION WITH RESPECT TO A COMPLETELY POSITIVE MAP AND COST OF EQUIVALENCE RELATIONS

DIMITRI SHLYAKHTENKO

Abstract. Given a family of isometries v_{1}, …, v_{n} in a tracial von Neumann algebra M, a unital subalgebra B ⊂ M and a completely-positive map η : B → B we define the free Fisher information F^{∗}(v_{1}, …, v_{n} : B, η) of v_{1}, …, v_{n} relative to B and η. Using this notion, we define the free dimension δ^{∗}(v_{1}, …, v_{n} ≬ B) of v_{1}, …, v_{n} relative to B, id. Let R be a measurable equivalence relation on a finite measure space X. Let M be the von Neumann algebra associated to R, and let B ≅ L^{∞}(X) be the canonical diffuse subalgebra. If v_{1}, …, v_{n}, ⋯ ∈ M are partial isometries arising from a treeing of this equivalence relation, then lim_{n} δ^{∗}(v_{1}, …, v_{n}, … ≬ B) is equal to the cost of the equivalence relation in the sense of Gaboriau.

1. Introduction.

Recently [4], [3] Gaboriau introduced the notion of cost of a measurable equivalence relation R on a finite measure space X. The cost C(R) measures “how many local morphisms from X to X are required to generate R”, and is an invariant of R. If α is a free measure-preserving action of a discrete group G on a finite measure space X, he associated to it the cost C(R_{α}) of the equivalence relation R_{α} induced by the action. The cost of the group C(G) was then defined as the infimum of C(R_{α}) over all possible actions α of G. Gaboriau showed that certain groups (e.g., cyclic groups and their free products) have a fixed price — C(R_{α}) is independent of α. He also showed that the cost of the group is well-behaved under the free product operation: C(G ∗ H) = C(G) + C(H).

Let G be a (finitely generated) discrete group, and let g_{1}, g_{2}, …, g_{n} ∈ G. Then to each g_{i} one can associate a unitary u(g_{i}) in the left regular representation ℓ^{2}(G). The weak closure of the ∗-algebra generated by all such u(g_{i}) is called the von Neumann algebra of G, denoted by L(G). L(G) has a canonical trace (given by τ(u) = ⟨uδ_{e}, δ_{e}⟩, where δ_{e} ∈ ℓ^{2}(G) is the delta-function supported on the neutral element of G). To the family of unitaries u(g_{1}), …, u(g_{n}) Voiculescu [6], [9], [7] has associated two numbers, χ(u(g_{1}), …, u(g_{n})) and χ^{∗}(u(g_{1}), …, u(g_{n})), called the “microstates” and “microstatesfree” free entropy. These numbers arise naturally from Voiculescu’s free probability theory. Using microstates free entropy χ, he defined the free entropy dimension (via microstates) δ(u(g_{1}), …, u(g_{n})); replacing in his definition χ by χ^{∗} leads to the microstates-free free dimension δ^{∗}, which we use in this paper. The two notions (microstates-free and involving microstates) of free entropy dimension make sense for an arbitrary collection u_{1}, …, u_{n} of unitaries in a tracial von Neumann algebra. It is not known if the two quantities δ and δ^{∗} are different; they coincide in every example in which their values have been computed.

Research supported by MSRI.

Voiculescu has proved that δ(u(g_{1}), …, u(g_{n})) is independent of the choice of generators g_{1}, …, g_{n} of the group G, and therefore gives a non-trivial invariant of G, called the free dimension of G, δ(G). The corresponding invariance property for δ^{∗} is suspected, but so far has not been proved.

If G = K ∗ H, and k_{1}, …, k_{n} ∈ K, h_{1}, …, h_{m} ∈ H, then both δ and δ^{∗} are additive:

δ^{∗}(u(k_{1}), …, u(k_{n}), u(h_{1}), …, u(h_{n})) = δ^{∗}(u(k_{1}), …, u(k_{n}))
+δ^{∗}(u(h_{1}), …, u(h_{m}))

and similarly for δ. It is remarkable that if g is a generator of a cyclic group, then δ(u(g)) = δ^{∗}(u(g)) = C(G) = 1 − \frac{1}{|G|}. Additivity of both quantities under free products imply that a (finite) free product of cyclic groups has a sequence of generators g_{1}, g_{2}, …, g_{n} for which δ(u(g_{1}), …, u(g_{n})) = δ^{∗}(u(g_{1}), …, u(g_{n})) = C(G). There are indications that the three quantities: C(G), δ(G) and δ^{∗}(u(set of generators of G)) are actually the same.

It is tempting to suggest that there is a more general notion of “cost” (and “free dimension”) which unifies the notion of cost (free dimension) of a group, and the notion of cost of an equivalence relation. Indeed, groups and equivalence relations are both special cases of groupoids. In this paper:

1. If M is a tracial von Neumann algebra, B ⊂ M a unital subalgebra, and v_{1}, …, v_{n} ∈ M is an n-tuple of partial isometries, we define the free entropy dimension of the n-tuple v_{1}, …, v_{n} relative to (B, id : B → B), δ^{∗}(v_{1}, …, v_{n} ≬ B). For R an equivalence relation on a measure space X, a graphing (“sequence of generators”) ϕ_{1}, …, ϕ_{n}, … gives rise to a family v_{1}, …, v_{n}, … of partial isometries in W^{∗}(X, G), and hence to a number, δ^{∗}(ϕ_{1}, …, ϕ_{n}, …) = lim sup_{n} δ^{∗}(v_{1}, …, v_{n} ≬ B).

2. In certain cases (when an equivalence relation R is treeable), R has a graphing ϕ_{1}, …, ϕ_{n}, …, giving rise to partial isometries v_{1}, …, v_{n}, ⋯ ∈ W^{∗}(X, R) for which lim_{n→∞} δ^{∗}(v_{1}, …, v_{n} ≬ B) = C(R).

3. Given a sequence of generators ϕ_{1}, …, ϕ_{n} one can extend the resulting isometries v_{1}, …, v_{n} ∈ W^{∗}(X, G) to unitaries u_{1}, …, u_{n} ∈ W^{∗}(X, G) in a certain way. For a treeing ϕ_{1}, …, ϕ_{n}, …, δ^{∗}(v_{1}, …, v_{n} ≬ B) = δ^{∗}(u_{1}, …, u_{n} ≬ B).

4. If α is a free action of a discrete group on a measure space X, each element g ∈ G gives rise to a unitary w(g) in the crossed product algebra L^{∞}(X) ⋊_{α} G ≅ W^{∗}(X, R_{α}). It turns out that δ^{∗}(w(g_{1}), …, w(g_{n}) ≬ B) = δ^{∗}(u(g_{1}), …, u(g_{n})) (here u(g_{j}) ∈ B(ℓ^{2}(G)) are in the group von Neumann algebra of G), and therefore depends only on the generators g_{1}, …, g_{n} and not on the action α.

The connections between δ^{∗} and δ lead us to suspect that if ϕ_{1}, …, ϕ_{n}, … is a graphing of an equivalence relation R on X, then the number δ^{∗}(ϕ_{1}, ϕ_{2}, …) depends only on R. The fourth statement above would then imply that if R comes from a group action, δ^{∗} of any graphing of R would have the same microstates-free free dimension as any sequence of generators of the group.

The remainder of the paper is organized as follows. We start with the definition of free Fisher information F^{∗} for a family of isometries v_{1}, …, v_{n} ∈ M relative to a subalgebra B ⊂ M and a completely-positive map η : B → B. The definition involves the notion of free Fisher information for unitaries with respect to a completely positive map, and is a straightforward extension of the definitions in [8], [7] and [5]. Some proofs are direct adaptations of the arguments from these papers; we include them for the convenience of the reader. Specializing to the case η = id we get as in [8] an entropy quantity Σ^{∗}. Passing from free entropy to free dimension like in [6] gives a free dimension quantity, which is denoted by δ^{∗}(v_{1}, …, v_{n} ≬ B). Next , we study the properties of F^{∗}, Σ^{∗} and δ^{∗}. We conclude by exhibiting equality between cost and free dimension in certain cases.

Acknowledgements. This work was completed while the author was participating in the Random Matrix program at MSRI. We would like to thank the organizers for the friendly atmosphere. We would also like to acknowledge fruitful conversations with Prof. J. Feldman, Prof. S. Popa and Prof. D.-V. Voiculescu.

2. Free microstates-free Fisher information for unitaries.

The reader is referred to [10] for the basic notions from free probability theory.

2.1. Notations. In the remainder of the paper, (M, τ) will be a tracial non-commutative probability space, i.e., M is a W^{∗}-algebra, and τ : M → ℂ is a linear functional, so that

1. τ(xy) = τ(yx) for all x, y ∈ M (traciality)

2. τ(x^{∗}x) ≥ 0 (positivity)

3. τ(x^{∗}x) = 0 iff x = 0 (faithfulness) M is a bimodule over itself under right and left multiplication. The norm ∥x∥_{2} = τ(x^{∗}x) defines a pre-Hilbert space structure on M.

We shall fix B ⊂ M a unital subalgebra, and denote by CP(B) the set of all completelypositive maps from B to B. Recall that η ∈ CP(B) if η is a linear map from B to B and its natural extension η ⊗ 1 : B ⊗ M_{n×n} → B ⊗ M_{n×n} to n × n matrices over B is positive: τ ⊗ Tr(η(m^{∗}m)) ≥ 0 for all m ∈ B ⊗ M_{n×n}. We shall be frequently concerned with the case that B is the commutative algebra B = L^{∞}(X, μ), where (X, μ) is a finite measure space, and τ(f) = ∫ f(t)dμ(t) for f ∈ B. In this case, η ∈ CP(B) if and only if η(f) is a.e. positive whenever f ∈ B is an a.e. positive function.

If η : B → B is a completely-positive map, then ⟨a ⊗ b, a′ ⊗ b′⟩ = τ(b^{∗}η(a^{∗}b)) defines a (possibly semi-definite) pre-Hilbert space structure on B ⊗ B. The Hilbert space obtained from B ⊗ B after separation and completion with respect to this inner product is denoted H^{2}(B, η). It carries the obvious structure of a B, B-bimodule.

If B ⊂ M, there exists a linear completely-positive map E_{B} : M → B, called the conditional expectation from M onto B. Sometimes we write E for E_{B} when B is understood. E satisfies E(bmb′) = bE(m)b′, b, b′ ∈ B, m ∈ M. E is determined by τ(bE(m)) = τ(bm) for all b ∈ B, m ∈ M. If η ∈ CP(B), it can be extended to all of M by composing it with E_{B}.

If η ∈ CP(B), we write H(B, η) for H^{2}(M, η ∘ E_{B}).

We call an element u ∈ M a partial isometry, i.e., uu^{∗} and u^{∗}u are projections, i.e., u^{∗}uu^{∗}u = u^{∗}u, uu^{∗}uu^{∗} = uu^{∗}. We denote by B[u] = B[u, u^{∗}] the ∗-algebra generated by B and u, u^{∗}.

2.2. Freeness with amalgamation. If A_{1}, A_{2} are subalgebras of M, we say that they are free with amalgamation over B, if whenever a_{j} ∈ W^{∗}(A_{i(j)}, B), i(1) ≠ i(2), …, i(n − 1) ≠ i(n) and E_{B}(a_{j}) = 0, one has

E_{B}(a_{1} … a_{n}) = 0.

We further say that the families (a_{1}^{(1)}, …, a_{n_{1}}^{(1)}), (a_{1}^{(2)}, …, a_{n_{2}}^{(2)}), … of elements of M are free with amalgamation over B, if the algebras A_{j} = W^{∗}(B, {a_{i}^{(j)}}_{i=1}^{n_{j}}) are free with amalgamation over B. If M and N are two tracial probability spaces, each containing B, one can form their amalgamated free product M ∗_{B} N, which is again a tracial probability space. M ∗_{B} N is generated by N and M, with the copies of B in M and N identified. M and N as subalgebras of M ∗_{B} N are free with amalgamation over B.

2.3. Free Fisher information for an isometry. Let v be a partial isometry and B ⊂ M be a subalgebra. Consider the derivation d_{u:B} : B[u, u^{∗}] → H(B[u, u^{∗}], η), determined by:

1. d_{u:B}(u) = uu^{∗} ⊗ u = (uu^{∗})1 ⊗ u(u^{∗}u), d_{u:B}(u^{∗}) = −u^{∗} ⊗ uu^{∗}

2. d_{u:B}(wv) = d_{u:B}(w) · v + w · d_{u:B}(v)

3. d_{u:B} is B-bilinear.

Definition 2.1. We call a vector ξ(u : B, η) ∈ L^{2}(B[u, u^{∗}]) conjugate to u relative to B with respect to η, if

⟨ξ(u : B, η), w⟩ = ⟨1 ⊗ 1, d_{u:B}(w)⟩ (2.1)

for all w ∈ B[u, u^{∗}].

Definition 2.2. If ξ(u : B, η) exists, we let the relative free Fisher information of u with respect to B and η to be F^{∗}(u : B, η) = ∥ξ(u : B, η)∥_{2}^{2}. If the conjugate vector does not exist, we set F^{∗}(u : B, η) = +∞. If B = ℂ and η = id, we write just F^{∗}(u).

Example 2.3. Let u = 0. Then d_{u:B} = 0, since d_{u:B}(b) = 0 for all b ∈ B, and d_{u:B}(u) = 0 · 0 ⊗ u = 0. Hence ξ(u : B, η) = 0, and F^{∗}(u : B, η) = 0.

Proposition 2.4. Let D ⊂ B be a unital subalgebra. Assume that η|_{D} ∈ CP(D). Assume that ξ(u : B, η) exists. Then ξ(u : D, η) exists and is given by E_{L^{2}(D[u,u^{∗}])}(ξ(u : B, η)).

Proof. Identical to that of [5, Porposition 3.4].

Proposition 2.5. Let D ⊂ B be a subalgebra. Assume that η = η ∘ E_{D} and u is ∗-free from B with amalgamation over D. Then ξ(u : B, η) ∈ L^{2}(D[u, u^{∗}]), and ξ(u : D, η) = ξ(u : B, η). Furthermore, F^{∗}(u : B, η) = F^{∗}(u : D, η).

Proof. If ξ(u : B, η) exists, then also ξ(u : D, η) exists. Hence we can assume that ξ(u : D, η) exists. It is sufficient to verify that it satisfies (2.1) for all w ∈ B[u, u^{∗}]. We may assume that E_{D}(w) = 0, and hence that w = b_{0}v_{0}b_{1}v_{1} … b_{m}, where b_{j} ∈ B, v_{j} ∈ D[u, u^{∗}], E_{D}(b_{j}) = E_{D}(v_{j}) = 0, except possibly b_{0} and/or b_{n} is equal to 1.

Assume first that at least one of b_{0}, b_{1} is not 1, or that m > 1. Then, since τ(ξ(u : D, η)^{∗}) = ⟨ξ(u : D, η), 1⟩ = ⟨1 ⊗ 1, 0⟩ = 0, we get by freeness that ⟨ξ(u : D, η), w⟩ = 0.

If m > 1, then d_{u:B}(w) ⊥ 1 ⊗ 1 in H(B, η). If m = 1, so that w = b_{0}v_{0}b_{1}, assume that d_{u:B}(v_{0}) = ∑ z_{i} ⊗ z′_{i}, z_{j}, z′_{j} ∈ D[u, u^{∗}]. Then d_{u:B}(w) = ∑ b_{0}z_{i} ⊗ z′_{i}b_{1}, and

⟨1 ⊗ 1, d_{u:B}(w)⟩ = ∑ τ(η(b_{0}z_{i})z′_{i}b_{1})
= ∑ τ(η(E_{D}(b_{0}z_{i}))E_{D}(z′_{i}b_{1}))
= ∑ τ(η(E_{D}(b_{0})E_{D}(z_{i}))E_{D}(z′_{i})E_{D}(b_{1})),

which is zero if (as we assumed) either E_{D}(b_{0}) or E_{D}(b_{1}) = 0.

It remains to verify (2.1) in the case b_{0} = b_{1} = 1 and m = 1, i.e., that

⟨ξ(u : D, η), v_{0}⟩ = ⟨1 ⊗ 1, d_{u:B}(v_{0})⟩.

But this follows from the definition of ξ(u : D, η) and the fact that d_{u:B}(v_{0}) = d_{u:D}(v_{0}).

Proposition 2.6. Let v ∈ B be an isometry. Assume that u = uvv^{∗} and that uu^{∗}, u^{∗}u ∈ B. Then F^{∗}(u : B, η) = F^{∗}(uv : B, η).

Proof. Let W = b_{0}u^{g(1)}b_{1} … b_{n}, and W′ = b_{0}(uvv^{∗})^{g(1)}b_{1} … b_{n}, where b_{i} ∈ B and g(j) ∈ {·, ∗}. Then W = W′.

We first claim that d_{u:B}(W) = d_{u:B}(W′) = d_{uv:B}(W′). The proof proceeds by induction on n. If n = 0, d_{u:B}(b_{0}) = d_{u:B}(b_{0}) = d_{uv:B}(b_{0}) = 0. If the equality holds for n = k, then writing W′ = b_{0}(uvv^{∗})^{g(1)}\overline{W}′, where \overline{W}′ has length k, and similarly W = b_{0}uW̄, with W̄ having length k, we have if g(1) = ·:

d_{u:B}(W) = b_{0}uu^{∗} ⊗ uW̄ + b_{0}ud_{u:B}(W̄),
d_{u:B}(W′) = b_{0}uu^{∗} ⊗ uvv^{∗}W̄′ + b_{0}uvv^{∗}d_{u:B}(W̄′),
d_{uv:B}(W′) = b_{0}uvv^{∗}u^{∗} ⊗ uvv^{∗}W̄′ + b_{0}uvv^{∗}d_{uv:B}(W̄′),

which are the same by the inductive assumption. The case that g(1) = ∗ is similar. Assume now that F^{∗}(uv : B, η) < +∞, so that ξ = ξ(uv : B, η) < +∞. Then

⟨ξ, W ⟩ = ⟨ξ, W′⟩
= ⟨1 ⊗ 1, d_{uv:B}(W′)⟩
= ⟨1 ⊗ 1, d_{u:B}(W′)⟩
= ⟨1 ⊗ 1, d_{u:B}(W)⟩.

It follows that ξ(u : B, η) = ξ.

Assuming now that F^{∗}(u : B, η) < +∞, we can apply the same reasoning as before, replacing u by (uv) and v by v^{∗} to conclude that ξ((uv)v^{∗} : B, η) = ξ(uv : B, η).

Proposition 2.7. Let v be an isometry, free with amalgamation over B from u, and assume that uu^{∗}, u^{∗}u ∈ B. Assume that u = uvv^{∗}. Assume further that F^{∗}(v : B, η) < +∞. Then F^{∗}(uv : B, η) ≤ F^{∗}(v : B, η) < +∞. In fact, ξ(uv : B, η) = E_{B[u]}(ξ(u : B, η)).

Proof. Let C = B[u]. Then

F^{∗}(v : B, η) = F^{∗}(v : C, η ∘ E_{B}) = F^{∗}(vu : C, η ∘ E_{B}),

and

ξ(v : B, η) = ξ(v : C, η ∘ E_{B}) = ξ(vu : C, η ∘ E_{B})

Hence

F^{∗}(vu : B, η) ≤ F^{∗}(vu : C, η ∘ E_{B}) = F^{∗}(v : B, η) < +∞

and

ξ(vu : B, η) = E_{B[u]}ξ(vu : C, η ∘ E_{B}) = E_{B[u]}ξ(v : B, η).

Corollary 2.8. Let u be a unitary, and p ∈ B be an isometry. Assume that upu^{∗} ∈ B. Then

F^{∗}(up : B, η) ≤ F^{∗}(u : B, η).

3. Free Fisher information for several isometries.

Definition 3.1. Let B ⊂ M, η ∈ CP(B) be as before, and let u_{1}, …, u_{n} ∈ M be isometries. Then the mutual free Fisher information of u_{1}, …, u_{n} relative to B, η is defined to be

F^{∗}(u_{1}, …, u_{n} : B, η) = ∑_{j} F^{∗}(u_{j} : B[u_{1}, …, û_{j}, …, u_{n}], η ∘ E_{B}).

Proposition 3.2. F^{∗}(u_{1}, …, u_{n}) ≤ F^{∗}(u_{1}, …, u_{k}) + F^{∗}(u_{k+1}, …, u_{n}).

Proof. If 1 ≤ j ≤ k, then

F^{∗}(u_{j} : B[u_{1}, …, û_{j}, …, u_{n}]) ≤ F^{∗}(u_{j} : B[u_{1}, …, û_{j}, …, u_{k}]),

by Proposition 2.4. Similarly, for k < j ≤ n,

F^{∗}(u_{j} : B[u_{1}, …, û_{j}, …, u_{n}]) ≤ F^{∗}(u_{j} : B[u_{k+1}, …, û_{j}, …, u_{n}]).

Proposition 3.3. If u_{1}, …, u_{k} are free with amalgamation over B from u_{k+1}, …, u_{n}, then

F^{∗}(u_{1}, …, u_{n}) = F^{∗}(u_{1}, …, u_{k}) + F^{∗}(u_{k+1}, …, u_{n}).

Proof. If 1 ≤ j ≤ n, then

F^{∗}(u_{j} : B[u_{1}, …, û_{j}, …, u_{n}]) = F^{∗}(u_{j} : B[u_{1}, …, û_{j}, …, u_{k}]),

by Proposition 2.5. Similarly, if k < j ≤ n, we get that

F^{∗}(: B[u_{1}, …, û_{j}, …, u_{n}]) = F^{∗}(u_{j} : B[u_{k+1}, …, û_{j}, …, u_{n}]).

Applying the definition of F^{∗}(u_{1}, …, u_{n}) proves the Proposition.

Corollary 3.4. If u_{j} are free with amalgamation over B, then F^{∗}(u_{1}, …, u_{n} : B, η) = ∑ F^{∗}(u_{j} : B, η).

Proposition 3.5. Let u_{1}, …, u_{n} be isometries. Then:

1. If D ⊂ B is such that η(D) ⊂ D, then

F^{∗}(u_{1}, …, u_{n} : D, η) ≤ F^{∗}(u_{1}, …, u_{n} : B, η)

2. If D ⊂ B is such that η = E_{D} ∘ η ∘ E_{D}, and u_{1}, …, u_{n} are ∗-free with amalgamation over D from B, then

F^{∗}(u_{1}, …, u_{n} : D, η) = F^{∗}(u_{1}, …, u_{n} : B, η).

Proof. Follows from 2.4 and 2.5.

3.1. F^{∗} of a single unitary. In the case that u is a unitary, B = ℂ and η = id, Voiculescu gave an explicit formula for F^{∗}(u) = F^{∗}(u : ℂ, id):

3F^{∗}(u) = −1 + ∫ p^{3}dλ

where λ is the Haar measure on 𝕋 = {z ∈ ℂ : |z| = 1}, and p is the density with respect to the Haar measure of the distribution of u. In particular, if u(t) is unitary free Brownian motion starting at identity and 0 < α < \frac{1}{2}, then there exist constants C_{1}, C_{2}, independent of t, for which F^{∗}(u(t)) ≤ C_{1}t^{−1} for 0 < t ≤ 1, and F^{∗}(u(t)) ≤ C_{2} exp(−tα).

4. Free Fisher information for normalizing isometries.

Let B ⊂ M be as before. Denote by N (B) the normalizer of B, consisting of those unitaries w ∈ M, for which wBw^{∗} = B. If w ∈ N (B), then Ad_{u}(w) = uwu^{∗} is an automorphism of B. We write GN (B) for the “full group” of the normalizer; GN (B) consists of all isometries v ∈ M, for which v^{∗}Bv ⊂ B and vBv^{∗} ⊂ B. Notice that in particular vv^{∗} and v^{∗}v are assumed to be in B.

If u ∈ N (B), then frequently F^{∗}(u : B, τ) = +∞. Indeed:

Proposition 4.1. If the conjugate variable ξ = ξ(u : B, τ) exists, and u ∈ N (B), then τ(ubu^{∗}c) = τ(b)τ(c) for all b, c ∈ B.

Proof. If ξ were to exist, we would have, for b, c ∈ B

0 = ⟨ξ, ubu^{∗}c⟩ = ⟨1 ⊗ 1, 1 ⊗ ubu^{∗}c − ubu^{∗} ⊗ c⟩
= τ(ubu^{∗}c) − τ(ubu^{∗})τ(c)
= τ(ubu^{∗}c) − τ(b)τ(c),

which would imply that τ(ubu^{∗}c) = τ(b)τ(c).

Definition 4.2. A unital subalgebra A ⊂ M is said to be Aut-independent from the algebra B, if:

1. There exist generators v_{1}, v_{2}, … of A, so that v_{j} ∈ N (B);

2. For all a ∈ A and b ∈ B, one has τ(ab) = τ(a)τ(b). Equivalently, E_{B}(A) = ℂ1. We say that unitaries a_{1}, …, a_{n} are Aut-independent from B, if W^{∗}(a_{1}, …, a_{n}) is Autindependent from B and a_{j} ∈ N (B).

Example 4.3. Let X be a finite measure space, and let α be a measure-preserving action of a discrete group G on X. Let M = L^{∞}(X) ⋊_{α} G, B = L^{∞}(X). For g ∈ G, write u_{g} = u(α, g) for the unitary in M, implementing the action of g. Then for any g_{1}, …, g_{n} ∈ G, u_{g_{1}}, …, u_{g_{n}} are Aut-independent from B. Furthermore, L(G) = W^{∗}(u_{g} : g ∈ G) ⊂ M is Aut-independent from B.

Proposition 4.4. Let A_{1}, A_{2} be two algebras, each Aut-independent from B. Assume that A_{1} and A_{2} are free with amalgamation over B. Then A_{1} ∨ A_{2} is Aut-independent from B.

Proof. It is clear that A_{1} ∨ A_{2} have a sequence of generators from N (B). Thus we must prove that if a ∈ A_{1} ∨ A_{2}, then E_{B}(a) ∈ ℂ. Let w = a_{1}a_{2}a_{3} ⋯ a_{n} ∈ A_{1} ∨ A_{2}, so that a_{j} ∈ A_{i(j)}, i(j) ≠ i(j + 1) for all j. Let [w] be the number of j for which E_{B}(a_{j}) ≠ 0. Then we have

E_{B}(w) = E(\underbrace{(a_{1} − E_{B}(a_{1}))a_{2} … a_{n}}_{w_{1}}) + E_{B}(a_{1})E(\underbrace{a_{2} … a_{n}}_{w_{2}}).

Since E_{B}(a_{1}) ∈ ℂ by Aut-independence of A_{j(1)} and B, it is sufficient to prove that E_{B}(w_{1}) and E_{B}(w_{2}) ∈ ℂ. But [w_{1}], [w_{2}] < [w]. Hence proceeding inductively, we must prove that E_{B}(w) ∈ ℂ if [w] = 0. But in that case E_{B}(w) = 0, by the freeness with amalgamation assumption.

Proposition 4.5. Let D ⊂ B be a unital subalgebra. Assume that B is generated by D and another unital subalgebra C. Assume that C[u_{1}, …, u_{n}] is Aut-independent from D, and u_{1}, …, u_{n} ∈ N (D). Assume that η ∈ CP(B) is such that η|_{D} = id and η(C) ⊂ C. Then

F^{∗}(u_{1}, …, u_{n} : B, η) = F^{∗}(u_{1}, …, u_{n} : C, η|_{C}).

Proof. Using the definition of F^{∗} and repeatedly replacing B by B[u_{1}, …, û_{j}, …, u_{n}], C by C[u_{1}, …, û_{j}, …, u_{n}], it is sufficient to prove the proposition in the case n = 1. We write u = u_{1}.

Since η(C) ⊂ C, by Proposition 2.4, we in general have

F^{∗}(u : B, η) ≥ F^{∗}(u : C, η|_{C}).

Thus we may assume that ξ = ξ(u : C, η|_{C}) exists. We’ll show that ξ(u : C, η|_{C}) = ξ(u : B, η), which would establish the Proposition.

Let w ∈ B[u, u^{∗}]. We must show that

⟨ξ, w⟩ = ⟨1 ⊗ 1, d_{u:B}(w)⟩_{H(B,η)}.

By assumption, u^{∗}Du ⊂ D, uDu^{∗} ⊂ D. We may therefore assume that w has the form

w = c_{0}d_{0}u^{g(1)}c_{1}d_{1}u^{g(2)} ⋯ c_{n}d_{n},

where c_{j} ∈ C, d_{j} ∈ D, g(j) ∈ {·, ∗}, and also c_{j} ∈ N (D). Then

⟨ξ, w⟩ = ⟨ξ, c_{0}u^{g(1)}c_{1} ⋯ c_{n}d′_{0}d′_{1} ⋯ d′_{n}⟩,

where d′_{j} = Ad_{u^{g(n)}c_{n}}^{−1} ⋯ Ad_{u^{g(j+1)}c_{j+1}}^{−1} Ad_{u^{g(j)}c_{j}}^{−1}(d_{j}). Since ξ ∈ L^{2}(C[u, u^{∗}]) and C[u, u^{∗}] is Aut-independent from D, it follows that

⟨ξ, w⟩ = ⟨ξ, c_{0}u^{g(1)}c_{1} ⋯ c_{n}⟩ · τ(d′_{0}d′_{1} ⋯ d′_{n}).

On the other hand,

d_{u:B}(w) = ∑_{k} c_{0}d_{0}u^{g(1)} ⋯ c_{k}d_{k} \{ \begin{matrix}uu^{∗} ⊗ u & if g(k) = 1 \\ −u^{∗} ⊗ uu^{∗} & if g(k) = −1 \end{matrix} \} ×
×c_{k+1}d_{k+1}u^{g(k+1)} ⋯ c_{n}d_{n}
= d_{u:C}(w) · d′_{0}d′_{1} ⋯ d′_{n},

since a · d ⊗ b = a ⊗ d · b for all a, b ∈ B[u] and d ∈ D. Because of independence again, we find that

⟨1 ⊗ 1, d_{u:B}(w)⟩_{H(B,η)} = ⟨1 ⊗ 1, d_{u:C}(w)⟩_{H(B,η)} · τ(d′_{0}d′_{1} ⋯ d′_{n}) = ⟨ξ, w⟩,

because ξ = ξ(u : C, η|_{C}).

Corollary 4.6. Let G be a finitely-generated group and g_{1}, …, g_{n} be its generators. Let α be a measure-preserving action of G on a finite measure space X. Let u(g_{1}), …, u(g_{n}) be the unitaries in L(G) ⊂ L^{∞}(X) ⋊_{α} G, implementing the action. Then

F^{∗}(u(g_{1}), …, u(g_{n}) : L^{∞}(X), id) = F^{∗}(u(g_{1}), …, u(g_{n})).

In particular, let G be a free group, and let g_{1}, …, g_{n} be its standard generators. Then

F^{∗}(u(g_{1}), …, u(g_{n}) : L^{∞}(X), id) = F^{∗}(u(g_{1}), …, u(g_{n}))

= ∑_{i=1}^{n} F^{∗}(u(g_{i})) = 0.

Proof. It is easily seen 4.3 that L(G) is Aut-independent from L^{∞}(X) inside L^{∞}(X)⋊_{α}G. This implies the first claim.

In the case G is a free group, u(g_{i}) are free, hence

F^{∗}(u(g_{1}), …, u(g_{n})) = ∑ F^{∗}(u(g_{i})).

Since each u(g_{i}) is a Haar unitary, it follows that F^{∗}(u(g_{i})) = 0.

Proposition 4.7. Let D ⊂ B be a unital subalgebra of B, u_{1}, …, u_{n} ∈ GN (B). Let p_{i} ∈ D ⊂ B, i = 1, …, N, N ∈ ℕ ∪ {∞} be orthogonal projections, so that ∑_{i=1}^{N} p_{i} = 1.Assume that [p_{j}, B] = {0} and [u_{i}, p_{j}] = 0 for all i = 1, …, n, j = 1, …, N. Then

F^{∗}(u_{1}, …, u_{n} : B, E_{D}) = ∑ τ(p_{j})F^{∗}(p_{j}u_{1}, …, p_{j}u_{n} : p_{j}B, E_{p_{j}D}).

Proof. It is sufficient to prove the Proposition for n = 1. We write u = u_{1}. Let H = H(B[u, u^{∗}], E_{D}) be the bimodule associated to E_{D}. Then by the assumptions made, we have the direct sum decomposition (as a Hilbert space)

H = ⊕H_{i},

where

H_{i} = p_{i}Hp_{i} = p_{i}H

as bimodules. Each H_{i} can be identified with H(p_{i}Bp_{i}[p_{i}up_{i}], E_{p_{i}Dp_{i}}) by multiplying its inner product with \sqrt{τ(p_{i})^{−1}}.

Similarly, K = L^{2}(B[u]) = ⨁ K_{i}, where

K_{i} = p_{i}L^{2}(B[u]).

The space K_{i} can be identified with L^{2}(p_{i}Bp_{i}[p_{i}up_{i}]) as bimodules over p_{i}Bp_{i} by multiplying its inner product by \sqrt{τ(p_{i})^{−1}}. It is easily seen that d_{u:B} preserves these direct sum decompositions, so that d_{u:B} = ⊕d_{i}, where d_{i} : K_{i} → H_{i}. It is also clear that each d_{i} = d_{p_{i}up_{i}:p_{i}Bp_{i}}. Hence we get that ξ(u : B, E_{D}) = d_{u:B}^{∗}(1 ⊗ 1) exists if and only if all ξ(p_{i}up_{i} : p_{i}Bp_{i}, E_{p_{i}Dp_{i}}) = d_{p_{i}up_{i}:p_{i}Bp_{i}}^{∗}(p_{i}1 ⊗ 1) exist, and moreover

ξ(u : B, E_{D}) = ∑ ξ(p_{i}up_{i} : p_{i}Bp_{i}, E_{p_{i}Dp_{i}}).

It follows that

F^{∗}(u : B, E_{D}) = ∥ξ(u : B, E_{D})∥_{2}^{2} = ∑ ∥ξ(p_{i}up_{i} : p_{i}Bp_{i}, E_{p_{i}Dp_{i}})∥_{K_{i}}^{2}
= ∑ τ(p_{i})∥ξ(p_{i}up_{i} : p_{i}Bp_{i}, E_{p_{i}Dp_{i}})∥_{L^{2}(p_{i}Bp_{i}[p_{i}up_{i}])}^{2}
= ∑ τ(p_{i})F^{∗}(p_{i}up_{i} : p_{i}Bp_{i}, E_{p_{i}Dp_{i}}).

Remark 4.8. The same proof shows that Proposition 4.7 still holds, if rather than dealing with a discrete family of projections p_{i}, we are dealing with a projection-valued measure μ on X, μ(X) = 1. In that case, we get that F^{∗}(u : B, E_{D}) = ∫_{X} F^{∗}(u(t) : B(t), E_{dμ(t)D})τ(dμ(t)), where u(t) and B(t) are the disintegrations of u and B over the commutative algebra W^{∗}(μ(A) : A ⊂ X).

The following proposition is essentially the basis of all computations of F^{∗} in this paper. Indeed, it reduced the computation of F^{∗} of several unitaries relative to a subalgebra to the computation of F^{∗} of a single unitaries relative to ℂ, for which there is an explicit formula (see Section 3.1).

We mention that the situation as in the hypothesis of the proposition occurs in the case of a treeing of an equivalence relation (see later in the paper).

Proposition 4.9. Let u_{1}, u_{2}, … be unitaries in M, B ⊂ M unital subalgebra, so that: 1. u_{j} ∈ N (B) for all j

2. For each j, there exist central projections p_{1}^{(j)}, …, p_{n(j)}^{(j)} ∈ B, so that [p_{k}^{(j)}, u_{j}] = 0

for all k, ∑ p_{k}^{(j)} = 1, and for all k, p_{k}^{(j)}u_{j}p_{k}^{(j)} is Aut-independent from p_{k}^{(j)}Bp_{k}^{(j)}.

3. u_{1}, u_{2}, … are free with amalgamation over B. Then

F^{∗}(u_{1}, …, u_{n} : B, id) = ∑_{j=1}^{n} ∑_{i=1}^{n(j)} τ(p_{i}^{(j)})F^{∗}(p_{i}^{(j)}u_{j}p_{i}^{(j)})

and

lim_{n→∞} F^{∗}(u_{1}, …, u_{n} : B, id) = ∑_{j=1}^{∞} ∑_{i=1}^{n(j)} F^{∗}(p_{i}^{(j)}u_{j}p_{i}^{(j)}).

Proof. Since u_{1}, …, u_{n} are free with amalgamation over B, it follows that

F^{∗}(u_{1}, …, u_{n} : B, id) = ∑_{j=1}^{n} F^{∗}(u_{j} : B, id).

Next, we get that

F^{∗}(u_{j} : B, id) = ∑ τ(p_{i}^{(j)})F^{∗}(p_{i}^{(j)}u_{j}p_{i}^{(j)} : p_{i}^{(j)}B, id).

Since p_{i}^{(j)}B is independent from p_{i}^{(j)}u_{j}p_{i}^{(j)}, it follows that

F^{∗}(p_{i}^{(j)}u_{j}p_{i}^{(j)} : p_{i}^{(j)}B, id) = F^{∗}(p_{i}^{(j)}up_{i}^{(j)}).

This implies the statement.

5. Free dimension for normalizing isometries.

5.1. Free Brownian motion. Let u_{1}, …, u_{n} ∈ M, B ⊂ M be as before. Consider multiplicative free Brownian motion w_{1}(t), …, w_{n}(t), such that:

1. w_{1}(t), …, w_{n}(t) are free among each other with amalgamation over B

2. w_{1}(t), …, w_{n}(t) are free from u_{1}, …, u_{n} with amalgamation over B

3. [w_{j}(t), B] = {0}, and {w_{j}(t)}_{j} is Aut-independent from B. To construct such a family, take w(t) to be free multiplicative Brownian motion [1] in an algebra C. Let C_{1}, …, C_{n} be n isomorphic copies of C. Consider ∗_{B}(C_{i} ⊗ B), and let w_{i}(t) be w(t) ⊗ 1 in the i-th copy C_{i} of C.

Proposition 5.1. Let u_{1}, …, u_{n}, w_{1}(t), …, w_{n}(t) be as in Section 5.1. Then, given 0 < α < \frac{1}{2}, there exist constants C_{1}, C_{2}, independent of t, so that :

(a) F^{∗}(w_{j}(t) : B, id) < +∞; more precisely,

F^{∗}(w_{j}(t) : B, id) ≤ \frac{C_{1}}{t} , 0 < t ≤ 1,

F^{∗}(w_{j}(t) : B, id) ≤ C_{2}e^{−tα}, 1 ≤ t.

(b)F^{∗}(uw_{1}(t), …, uw_{n}(t)) < +∞; more precisely,

F^{∗}(uw_{1}(t), …, uw_{n}(t)) ≤ nF^{∗}(w_{1}(t)) ≤ \{ \begin{matrix}n\frac{C_{1}}{t} , & 0 < t ≤ 1 \\ nC_{2}e^{−tα}, & 1 ≤ t \end{matrix} .

Proof. By Proposition 3.2,

F^{∗}(uw_{1}(t), …, uw_{n}(t) : B, id) ≤ ∑ F^{∗}(uw_{j}(t) : B, id).

By Proposition 2.7, we get that

F^{∗}(uw_{j}(t) : B, id) ≤ F^{∗}(w_{j}(t) : B, id).

Since w_{j}(t) is Aut-independent from B, we get by Proposition 4.5 that F^{∗}(w_{j}(t) : B, id) = F^{∗}(w_{j}(t)). The estimates now follow from Voiculescu’s results for a single unitary (see Section 3.1).

Definition 5.2. Let v_{1}, …, v_{n} ∈ M be partial isometries. The free entropy of v_{1}, …, v_{n} ∈ M relative to B, id is defined to be

Σ^{∗}(v_{1}, …, v_{n} ≬ B) = \frac{1}{2} ∫_{0}^{∞} F^{∗}(v_{1}w_{1}(t), …, v_{n}w_{n}(t))dt.

By [8, Corollary 10.9], we get that if u is a single unitary, then

Σ^{∗}(u ≬ ℂ) = Σ(u) = ∫∫ log |z_{1} − z_{2}|dμ(z_{1})dμ(z_{2}),

where μ is the distribution of u, supported on the unit circle.

Definition 5.3. Fix a an element ψ of the β-compactification of I = (0, 1], which is not in I. The free dimension of u_{1}, …, u_{n} relative to B, id is defined to be

δ^{∗}(u_{1}, …, u_{n} ≬ B) = ∑ τ(u_{j}^{∗}u_{j}) − lim_{t→ψ} \frac{Σ^{∗}(u_{1}w_{1}(t), …, u_{n}w_{n}(t) ≬ B)}{log t^{1/2}}.

In the case that B = ℂ, we write simply δ^{∗}(u_{1}, …, u_{n}).

5.2. Properties of δ^{∗}. The properties of F^{∗}, together with Proposition 4.4 immediately give rise to the following properties of δ^{∗}(⋯ ≬ B):

Proposition 5.4. Let u_{1}, …, u_{n}, B be as above. Then we have:

1.

0 ≤ δ^{∗}(u_{1}, …, u_{n}) ≤ ∑_{i=1}^{n} τ(u_{i}u_{i}^{∗})

2. If u_{1}, …, u_{k} are free from u_{k+1}, …, u_{n} with amalgamation over B, then

δ^{∗}(u_{1}, …, u_{n}) = δ^{∗}(u_{1}, …, u_{k}) + δ^{∗}(u_{k+1}, …, u_{n})

3.

δ^{∗}(u_{1}, …, u_{n}) ≤ δ^{∗}(u_{1}, …, u_{k}) + δ^{∗}(u_{k+1}, …, u_{n})

4. If D ⊂ B is a unital subalgebra, then

δ^{∗}(u_{1}, …, u_{n} ≬ B) ≤ δ^{∗}(u_{1}, …, u_{n} ≬ D) ≤ δ^{∗}(u_{1}, …, u_{n}).

5. If D ⊂ B is a unital subalgebra, so that B = D ∨ C, and C[u_{1}, …, u_{n}] is Autindependent from B, and u_{1}, …, u_{n} ∈ GN (D), then

δ^{∗}(u_{1}, …, u_{n} ≬ B) = δ^{∗}(u_{1}, …, u_{n} ≬ C).

In particular, if u_{1}, …, u_{n} are Aut-independent from B, then

δ^{∗}(u_{1}, …, u_{n} ≬ B) = δ^{∗}(u_{1}, …, u_{n}).

6. If p_{i}, i = 1, …, N, N ∈ ℕ ∪ {∞} are orthogonal central projections in B, so that ∑ p_{i} = 1, and so that [p_{i}, u_{j}] = 0 for all i, j, then

δ^{∗}(u_{1}, …, u_{n} ≬ B) = ∑_{i} τ(p_{i})δ^{∗}(p_{i}u_{1}p_{i}, …, p_{i}u_{n}p_{i} ≬ p_{i}Bp_{i}).

The same conclusion holds if μ is a projection-valued measure on X, μ(X) = 1, so that μ(Y) ∈ B is in the center of B[u_{1}, …, u_{n}] for all Borel subsets Y ⊂ X:

δ^{∗}(u_{1}, …, u_{n}) = ∫_{X} δ^{∗}(u_{1}(t), …, u_{n}(t) ≬ B(t))τ(dμ(t)),

where u_{j}(t), B(t) is the disintegration of u_{j}, B over W^{∗}(μ(A) : A ⊂ X).

6. Free dimension for one variable.

Proposition 6.1. Let u ∈ M be a unitary, and let μ be its distribution, supported on the circle T = {z : |z| = 1}. Let w(t) be multiplicative free Brownian motion, free from u. Let u_{t} = uw(t), and let μ_{t} be the distribution of u_{t}. Then

2 lim_{t→0} \frac{∫∫ log |z − w|dμ_{t}(z)dμ_{t}(w)}{| log t|} = − ∑_{t∈T}(μ\{t\})^{2}.

Proof. We reduce the problem to the form dealt with in Voiculescu’s proof of [6, Proposition 6.3]. We know by [8, Corollary 1.7] that μ_{t} is absolutely continuous with respect to Haar measure on T. Denote by p_{t} its density; then we also know that ∥p_{t}∥_{∞} ≤ 2/\sqrt{t} for t sufficiently small. We have that ∥w(t) − 1∥_{∞} ≤ Kt^{1/2} for some constant K. We also have that

μ_{t^{2}}(−π, b − Kt) ≤ μ(−π, b) ≤ μ_{t^{2}}(−π, b + Kt); (6.1)

this follows from the fact that μ_{t} is the distribution of uw(t), and ∥w(t^{2}) − 1∥ ≤ Kt, so that the spectrum of w(t^{2}) is contained in the arc −Kt, Kt.

The quantity ∫∫ log |z−w|dμ_{t}(z)dμ_{t}(w) does not change if we replace μ_{t}(w) by μ_{t}(rw) for some fixed r ∈ T. Hence we may assume that there are no atom s of μ in some arc around −1. Let −π ≤ θ ≤ π be the polar coordinate on T. We’ll from now on write dμ_{t}(θ) to denote the density in [−π, π] of the push-forward via θ of the measure μ_{t}.

Notice that

log |e^{iθ} − e^{iψ}| = \frac{1}{2} log(2 − 2 cos(θ − ψ)) = \frac{1}{2} log(1 − cos(θ − ψ)) + \frac{1}{2} log 2.

It will thus suffice to prove that

lim_{t→0} \frac{∫∫ log(1 − cos(θ − ψ))dμ_{t}(θ)dμ_{t}(ψ)}{| log t|} = − ∑_{t∈T}(μ\{t\})^{2}.

There is a δ > 0 so that for |α| < δ, \frac{α^{2}}{4} ≤ 1 − cos α ≤ \frac{α^{2}}{2} . It follows that

\frac{2}{| log t|} ∫∫ _{|θ−ψ|<δ} log |θ − ψ|dμ_{t}(θ)dμ_{t}(ψ) − \frac{log 4}{| log t|} ∫∫ _{|θ−ψ|<δ} 1dμ_{t}(θ)dμ_{t}(ψ)

≤ \frac{1}{| log t|} ∫∫ _{|θ−ψ|<δ} log(1 − cos(θ − ψ))dμ_{t}(θ)dμ_{t}(ψ)

≤ \frac{2}{| log t|} ∫∫ _{|θ−ψ|<δ} log |θ − ψ|dμ_{t}(θ)dμ_{t}(ψ) − \frac{log 2}{| log t|} ∫∫ _{|θ−ψ|<δ} 1dμ_{t}(θ)dμ_{t}(ψ).

When 2π−δ ≥ |θ−ϕ| ≥ δ, 0 ≥ log(1−cos(θ−ψ)) ≥ log(1−cos δ) and when |θ−ϕ| ≥ δ, 0 ≥ log |θ − ψ| ≥ log δ. Hence as t → 0, the following expressions

| log t|^{−1} ∫∫ log(1 − cos(θ − ψ))dμ_{t}(θ)dμ_{t}(ψ),
| log t|^{−1} ∫∫ _{|θ−ψ|<δ} log(1 − cos(θ − ψ))dμ_{t}(θ)dμ_{t}(ψ)+
| log t|^{−1} ∫∫ _{|θ−ψ|≥2π−δ} log(1 − cos(θ − ψ))dμ_{t}(θ)dμ_{t}(ψ),
| log t|^{−1} ∫∫ _{|θ−ψ|<δ} 2 log |θ − ψ|dμ_{t}(θ)dμ_{t}(ψ)+
| log t|^{−1} ∫∫ _{|θ−ψ|≥2π−δ} 2 log |2π − (θ − ψ)|dμ_{t}(θ)dμ_{t}(ψ) and
| log t|^{−1} ∫∫ 2 log |θ − ψ|dμ_{t}(θ)dμ_{t}(ψ)+
| log t|^{−1} ∫∫ 2 log |2π − (θ − ψ)|dμ_{t}(θ)dμ_{t}(ψ)

all have the same limit, if any.

It is therefore sufficient to show that

lim_{t→0} 2\frac{∫∫ log |θ − ψ|dμ_{t}(z)dμ_{t}(w)}{| log t|} = − ∑_{t∈T}(μ\{t\})^{2},

2 lim_{t→0} \frac{∫∫ log |2π − (θ − ψ)|dμ_{t}(θ)dμ_{t}(ψ)}{| log t|} = 0, i.e.,

lim_{t→0} \frac{∫∫ log |θ − ψ|dμ_{t}(z)dμ_{t}(w)}{| log t^{1/2}|} = − ∑_{r∈T}(μ\{r\})^{2},

lim_{t→0} \frac{∫∫ log |2π − (θ − ψ)|dμ_{t}(θ)dμ_{t}(ψ)}{| log t^{1/2}|} = 0.

Let t = s^{2}/K^{2}, and write ν_{t} for μ_{t^{2}/K^{2}}. Then we must prove that

lim_{s→0} \frac{∫∫ log |θ − ψ|dν_{s}(z)dν_{s}(w)}{| log s|} = − ∑_{t∈T}(μ\{t\})^{2},

lim_{t→0} \frac{∫∫ log |(θ − ψ)|dν_{s}(θ − π)dν_{s}(ψ + π)}{| log s|} = 0.

Let v_{s} be the density of ν_{s}. We now proceed exactly as in the proof of [6, Proposition 6.3], noticing that the only estimates on ν_{s} needed in that proof are that: (a) ∥v_{s}∥_{2} ≤ Cs^{−1/2} and (b) ν_{s}(−π, b − s) ≤ μ(−π, b) ≤ ν_{s}(−π, b + s). Voiculescu’s proof then shows that

lim_{s→0} \frac{∫∫ log |θ − ψ|dν_{s}(z)dν_{s}(w)}{| log s|} − μ ⊗ μ(Δ) = 0,

lim_{t→0} \frac{∫∫ log |(θ − ψ)|dν_{s}(θ − π)dν_{s}(ψ + π)}{| log s|} − μ^{−π} ⊗ μ^{π}(Δ) = 0,

where Δ = {(t, t) : t ∈ [−π, π]} and μ^{r} denotes the translation of μ by r. Since μ has no atoms in around π and −π, μ^{−π} = μ^{−π} = 0, while μ ⊗ μ(Δ) = ∑_{t∈T} (μ{t})^{2}.

It remains to prove estimates (a) and (b). To prove estimate (a), it is sufficient to prove that ∥p_{t}∥_{2} ≤ Ĉt^{− \frac{1}{4}} . Since ∥p_{t}∥_{∞} ≤ Kt^{−1/2} and p_{t} is supported on an arc of length Ct^{1/2}, it follows that

∥p_{t}∥_{r}^{r} = ∫_{supp p_{t}} |p_{t}(θ)|^{r}dθ ≤ Ct^{1/2} · K^{r}t^{−r/2} = CK^{r}t^{(1−r)/2}.

Hence ∥p_{t}∥_{r} ≤ Ĉt^{\frac{1}{2} (−1+1/r)}. Setting r = 2 gives the estimate ∥p∥_{2} ≤ Ĉt^{− \frac{1}{4}} . Estimate (b) follows by substituting the definition of ν_{s} and s into (6.1).

Corollary 6.2. Let u ∈ M be a unitary, and let μ be its distribution, supported on the circle T = {z : |z| = 1}. Then δ^{∗}(u) = 1 − ∑_{t∈T} μ({t})^{2}.

Corollary 6.3. Let G be a cyclic singly-generated group, and g ∈ G be its generator. Let u ∈ L(G) be the unitary implementing g. Then δ^{∗}(u) = 1 − \frac{1}{|G|}.

Proof. If |G| = ∞, the distribution of u is a Haar measure. It follows that δ^{∗}(u) = 1.

If |G| = n, then the distribution of u is equal to \frac{1}{n} ∑ δ_{ω^{n}}, where ω^{n} is a primitive n-th root of unity. We have that μ(\{ω^{n}\})^{2} = \frac{1}{n^{2}} , so that ∑_{t∈T} μ(\{t\})^{2} = n\frac{1}{n^{2}} = \frac{1}{n}.

Proposition 6.4. Let v be an isometry, v ∈ GN (B). Let w be a Haar unitary, independent from B and free from v with amalgamation over B. Assume that the B-valued distributions of v and vw are the same. Then δ^{∗}(v ≬ B) = τ(vv^{∗}).

Proof. Since v ∈ GN (B), one has in particular vv^{∗}, v^{∗}v ∈ B. Then δ^{∗}(v) ≤ τ(v^{∗}v) = τ(vv^{∗}). Let w be a Haar unitary, independent from B and free from v with amalgamation over B. Assume that vw has the same B-valued distribution as v. Then F^{∗}(w : B, id) = F^{∗}(w) = 0 because w is independent from B; and by Proposition 2.4, we get that 0 ≤ F^{∗}(vw : B, id) ≤ F^{∗}(w : B, id) = 0. Hence F^{∗}(vw : B, id) = 0. Let w(t) be multiplicative unitary free Brownian motion, which is independent from B and free from w, v with amalgamation over B. Then the B-valued distributions of vw(t) and vww(t) coincide. Since ww(t) is free from v with amalgamation over B and has the same B-valued distribution as w, it follows that the B-valued distributions of vw(t) and vw are the same. Hence we get that

δ^{∗}(v ≬ B) = τ(v^{∗}v) − lim_{t→ψ} \frac{F^{∗}(vw(t) : B, id)}{F^{∗}(w(t))} = τ(v^{∗}v) − 0.

Problem 6.5. Let u_{1}, …, u_{n} ∈ GN (B). Is δ^{∗}(u_{1}, …, u_{n} ≬ B) is an invariant for the pair (W^{∗}(u_{1}, …, u_{n} ≬ B), B)?

7. Free dimension and cost.

7.1. Local morphisms. Let (X, μ) be a finite measure space. If A ⊂ X is a subset, a local morphism ϕ : A → ϕ(A) is a measure-preserving measurable monomorphism from its domain A to its range ϕ(A). Let R be a measurable equivalence relation on X. We say that ϕ is consistent with R, if ϕ(x) ∼_{R} x for almost all x ∈ A.

If ϕ, ψ are local morphisms, then so are ϕ ∘ ψ and ϕ^{−1}; these are defined on the subsets {x ∈ dom ψ : ψ(x) ∈ dom ϕ} and ran ϕ, respectively.

7.2. Graphings and treeings. Following Gaboriau [3], we say that a set {ϕ_{i}}_{i∈I} of local morphisms is a graphing of an equivalence relation R, if a.e. in X, x ∼_{R} y iff x = ϕ_{i_{1}}^{g(1)}(ϕ_{i_{2}}^{g(2)}(… ϕ_{i_{n}}^{g(n)}(y) …)) for some i_{1}, …, i_{n} ∈ I and g(1), …, g(n) ∈ {±1} . A graphing is a treeing, if for any i_{1} ≠ i_{2}, i_{2} ≠ i_{3},… and g(1), …, g(n) ∈ ±ℕ, the set {x : ϕ_{i_{1}}^{g(1)}(ϕ_{i_{2}}^{g(2)}(… ϕ_{i_{n}}^{g(n)}(x) …)) = x} has measure zero.

7.3. Cost of an equivalence relation. The cost C(R) of R was defined by Gaboriau to be the infimum of ∑_{i∈I} μ(dom ϕ_{i}) over all graphings {ϕ_{i}}_{i∈I} of R. If {ϕ_{i}}_{i∈I} is a treeing, then C(R) is given by ∑_{i∈I} μ(dom ϕ_{i}).

7.4. Free equivalence relations. Two equivalence relations R_{1} and R_{2} are called free, if for each n ≥ 2, the set of all x ∈ X for which x ∼_{R_{i_{1}}} x_{1} ∼_{R_{i_{2}}}∼_{R_{i_{3}}} ⋯ ∼_{R_{in}} x with i_{1} ≠ i_{2}, i_{2} ≠ i_{3}, …, i_{n−1} ≠ i_{n} has measure zero. Gaboriau proved that C(R_{1} ∨ R_{2}) = C(R_{1})+C(R_{2}), if R_{1} ∨R_{2} is the equivalence relation generated by R_{1} and R_{2}, and R_{1}, R_{2} are free.

7.5. The von Neumann algebra of an equivalence relation. Feldman and Moore [2] associated to a measurable equivalence relation R on X a von Neumann algebra W^{∗}(X, R). We summarize some properties of W^{∗}(X, R):

1. W^{∗}(X, R) contains L^{∞}(X) as a unital subalgebra;

2. Every local morphism ϕ consistent with R defines a partial isometry v_{ϕ} ∈ M, so that v_{ϕ}^{∗}v_{ϕ} = χ_{dom ϕ}, v_{ϕ}v_{ϕ}^{∗} = χ_{ran ϕ}, and v_{ϕ}fv_{ϕ}^{∗} = f ∘ ϕ for all f ∈ L^{∞}(dom ϕ) ⊂ L^{∞}(X) (χ denotes the characteristic function of a set). Furthermore, v_{ϕ}v_{ψ} = v_{ϕψ}, v_{ϕ}^{∗} = v_{ϕ^{−1}}.

3. If ϕ is a local morphism, and C ⊂ dom ϕ, then v_{ϕ|_{C}} = v_{ϕ}χ_{C}.

4. M is generated by all v_{ϕ}, where ϕ ranges over all possible local morphisms, consistent with R.

5. There is a conditional expectation E from M onto L^{∞}(X), so that τ = μ ∘ E is a trace on M. Moreover, if ϕ is a local morphism, then E(v_{ϕ}) = χ_{{x∈X:ϕ(x)=x}}

7.6. Extensions of local morphisms. Let ϕ : A → G be a local morphism. Let A_{∞} = {x ∈ X : x ∈ dom ϕ^{−n}, ∀n}. Then ϕ(A) · A_{∞} = ϕ^{−1}(A) · A_{∞} = A_{∞}. Let A^{∞} = {x ∈ X : x ∈ dom ϕ^{n}, ∀n}. Let C = dom ϕ ∪ ran ϕ. Then ϕ(A) · C = ϕ^{−1}(A) · C = C. Let D = C \ A_{∞} \ A^{∞}. Then for all x ∈ D, there exists a number n(x), so that x ∈ dom ϕ^{−n}, but x ∉ dom ϕ^{−(n+1)}. Let A_{n} = {x : x(n) = n}. Now define Θ_{ϕ}(x) = ϕ(x) if x ∈ A_{∞}, Θ_{ϕ}(x) = ϕ^{−1}(x) if x ∈ A^{∞} \ A_{∞}, Θ_{ϕ}(x) = x if x ∉ C, and Θ_{ϕ}(x) = ϕ^{−1}(x) if x ∈ A_{n}, n > 0, Θ_{ϕ}(x) = ϕ^{m(x)}(x) if x ∈ A_{0}, where m(x) = max{p : x ∈ dom ϕ^{p}}.

Proposition 7.1. Let ϕ : A → G be a local morphism, compatible with an equivalence relation R. Then Θ_{ϕ} is an automorphism of X, compatible with R.

Proof. We must check that Θ_{ϕ}(x) ≠ Θ_{ϕ}(y) if x ≠ y. This is clear if at least one of (x, y) is not in C, or if x ∈ A_{n}, y ∈ A_{m} for n = m. Assume that x ∈ A_{n}, y ∈ A_{m}, but n ≠ m. For definiteness, assume that n > 0. Then Θ_{ϕ}(x) ∈ A_{n−1}. If m > 0, then Θ_{ϕ}(y) ∈ A_{m−1}, so that Θ_{ϕ}(x) ≠ Θ_{ϕ}(y). If m = 0, Θ_{ϕ}(y) = Θ_{ϕ}(x) would imply that ϕ^{m(y)}(y) = Θ_{ϕ}(x) ∈ dom ϕ.

Given a graphing {ϕ_{i}} of an equivalence relation R, we can associate to it unitaries u_{i} ∈ W^{∗}(X, R) by taking the unitaries u_{ϕ_{i}} associated to bisections Θ_{ϕ_{i}}.

Proposition 7.2. Let {ϕ_{i}}_{i∈I} be a graphing of an equivalence relation R on a finite measure space X. Then {Θ_{ϕ_{i}}}_{i∈I} is also a graphing of R. In particular, the von Neumann algebra W^{∗}(X, R) is generated by L^{∞}(X) and {v_{ϕ_{i}}}_{i∈I}.

Proof. Clearly, only the first statement needs to be proved. It is clearly sufficient to prove that if x = ϕ_{i}(y), then x and y are equivalent in the equivalence relation generated by Θ_{ϕ_{i}}. From the definition of Θ, we have that either Θ_{ϕ_{i}}(x) = y, or Θ_{ϕ_{i}}(y) = x, unless y ∉ dom ϕ_{i}^{−1}, and ϕ_{i}^{n}(y) ∉ dom ϕ_{i} for some n ∈ ℕ. In that case, we have that Θ_{ϕ_{i}}(x) = y.

Lemma 7.3. A treeing of R is a graphing {ϕ_{i}}, such that R = ∗R_{i}, where R_{i} is the equivalence relation generated by ϕ_{i}, and so that ϕ_{i}^{n}(x) ≠ x for all x ∈ X and n ∈ ±ℕ.

Conjecture 7.4. Let {ϕ_{i}} be a graphing of an equivalence relation R. Let u_{i} be the unitary in W^{∗}(X, R) associated to Θ_{ϕ_{i}}, and let v_{i} ∈ W^{∗}(X, R) be the partial isometry associated to ϕ_{i}. Then

δ^{∗}({ϕ_{i}}) = lim_{n→∞} δ^{∗}(u_{1}, u_{2}, …, u_{n} ≬ L^{∞}(X)) = lim_{n→∞} δ^{∗}(v_{1}, v_{2}, …, v_{n} ≬ L^{∞}(X))

is independent of the graphing {ϕ_{i}}.

Proposition 7.5. Let R_{1}, R_{2} be two equivalence relations, which are free. Then the algebras W^{∗}(X, R_{1}) and W^{∗}(X, R_{2}) are free in W^{∗}(X, R) with amalgamation over L^{∞}(X).

Proof. Let ϕ_{i}^{(1)}, i ∈ I be local morphisms compatible with R_{1}, and ϕ_{i}^{(2)}, i ∈ I be local morphisms compatible with R_{2}. Then W^{∗}(X, R_{1} ∨ R_{2}) = W^{∗}(X, R_{1}) ∨ W^{∗}(X, R_{2}) is densely linearly spanned by L^{∞}(X) and words of the form w = v_{ϕ_{i_{1}}^{j_{1}}} … v_{ϕ_{in}^{jn}}, where

j_{1} ≠ j_{2}, j_{2} ≠ j_{3}, etc., and E_{L^{∞}(X)}(v_{ϕ_{i_{k}}^{j_{k}}}) = 0. We must prove that E_{L^{∞}(X)}(w) = 0. But E_{L^{∞}(X)} = χ_{{x∈X:ϕ_{i_{1}}^{j_{1}}∘...ϕ_{in}^{jn}(x)=x}} = 0 by the assumption that R_{1} and R_{2} are free.

Proposition 7.6. Let {ϕ_{i}} be a treeing of an equivalence relation R. Let u_{ϕ_{i}} be the unitary associated to the bisection Θ_{ϕ_{i}}. Then

C(R) = lim_{n→∞} δ^{∗}(u_{ϕ_{1}}, u_{ϕ_{2}}, …, u_{ϕ_{n}} ≬ L^{∞}(X)).

Proof. In view of additivity of both C(R) and δ^{∗} under free products, it is sufficient to verify the equality for a single ϕ. Let A = dom ϕ, B = ran ϕ, C = A ∪ B. R be the equivalence relation generated by ϕ, and let

A_{n} = {x ∈ X : #{y : y ∼_{R} x} = n}.

Then C(R) = μ(A_{∞}) + ∑ μ(A_{n})(1 − \frac{1}{n}), since A_{n} are disjoint, and R is cyclic of order n on A_{n}.

On the other hand, let p_{n} be the characteristic function of A_{n}. Then [u_{ϕ}, p_{n}] = 0. Moreover, u_{n} = p_{n}u_{ϕ}p_{n} implements a free action of a cyclic group of order n on A_{n}. Therefore,

δ^{∗}(u_{ϕ} ≬ L^{∞}(X)) = τ(p_{∞})δ^{∗}(u_{∞} ≬ L^{∞}(A_{∞})) + ∑ τ(p_{n})δ^{∗}(u_{n} ≬ L^{∞}(A_{n})).

Since u_{n} is Aut-independent from L^{∞}(A_{n}), it follows that δ^{∗}(u_{n} ≬ L^{∞}(A_{n})) = δ^{∗}(u_{n}) = 1 − \frac{1}{n}. Thus C(R) = δ^{∗}(u ≬ L^{∞}(X)).

If conjecture 7.4 is correct, then δ^{∗}({ϕ_{i}}) ≤ C(R). Indeed, assume that each ϕ_{i} by itself is a treeing. Let R_{ϕ_{i}} be the equivalence relation generated by ϕ_{i}.

δ^{∗}({ϕ_{i}}) = lim_{n→∞} δ^{∗}(u_{ϕ_{1}}, u_{ϕ_{2}}, …, u_{ϕ_{n}} ≬ L^{∞}(X))
≤ lim_{n→∞} ∑ δ^{∗}(u_{ϕ_{i}} ≬ L^{∞}(X)) = ∑ C(R_{ϕ_{i}}) ≤ C(R),

the last inequality because given a graphing {ϕ_{i}} we can always assume that each ϕ_{i} is a treeing by itself by shrinking the domains of ϕ_{i}.

Proposition 7.7. Let {ϕ_{i}} be a treeing of an equivalence relation R. Let v_{ϕ_{i}} be the partial isometry in W^{∗}(X, R) associated to ϕ_{i}. Then

C(R) = lim_{n→∞} δ^{∗}(v_{ϕ_{1}}, …, v_{ϕ_{n}} ≬ L^{∞}(X)) = ∑_{j=1}^{∞} δ^{∗}(v_{ϕ_{i}} ≬ L^{∞}(X)).

Proof. Since {ϕ_{i}} is a treeing, v_{ϕ_{j}}, j = 1, 2, … are free with amalgamation over L^{∞}(X). Hence

δ^{∗}(v_{ϕ_{1}}, …, v_{ϕ_{n}} ≬ L^{∞}(X)) = ∑_{j=1}^{n} δ^{∗}(v_{ϕ_{i}} ≬ L^{∞}(X)).

Since each ϕ_{i} is a treeing, the L^{∞}(X)-valued distribution of v_{ϕ_{i}} remains the same if v_{ϕ_{i}} is replaced by v_{ϕ_{i}}w, where w is a Haar unitary, independent from L^{∞}(X) and free from v_{ϕ_{i}} with amalgamation over L^{∞}(X). From Proposition 6.4 we get that δ^{∗}(v_{ϕ_{i}}) = τ(v_{ϕ_{i}}^{∗}v_{ϕ_{i}}) = μ(dom ϕ_{i}). Since C(R) = ∑_{j=1}^{∞} μ(dom ϕ_{i}), C(R) = lim_{n} δ^{∗}(v_{ϕ_{1}}, …, v_{ϕ_{n}}).

Problem 7.8. Choose any graphing {ϕ_{i}} of R, let u_{ϕ_{i}} be the unitary in W^{∗}(X, R) associated to the bisection Θ_{ϕ_{i}}, and let v_{ϕ_{i}} be the partial isometry in W^{∗}(X, R) associated to ϕ_{i}. Is it true that C(R) = lim_{n→∞} δ^{∗}(u_{ϕ_{1}}, …, u_{ϕ_{n}} ≬ L^{∞}(X)) = lim_{n→∞} δ^{∗}(v_{ϕ_{1}}, …, v_{ϕ_{n}})?

If this question is answered in the affirmative, then all groups have fixed cost. Indeed, if ϕ_{i} = α_{g_{i}} comes from the free action of a group G with some fixed set of generators g_{1}, g_{2}, …, then δ^{∗}(u_{ϕ_{1}}, …, u_{ϕ_{n}} ≬ L^{∞}(X)) = δ^{∗}(u_{ϕ_{1}}, …, u_{ϕ_{n}}), the latter depending only on g_{1}, g_{2}, ⋯ ∈ G, and independent of the action α.

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Department of Mathematics, UCLA, Los Angeles, CA 90095

 E-mail address: shlyakht@member.ams.org