FREE FISHER INFORMATION FOR NON-TRACIAL STATES

DIMITRI SHLYAKHTENKO

Abstract. We extend Voiculescu’s microstates-free definitions of free Fisher information and free entropy to the non-tracial framework. We explain the connection between these quantities and free entropy with respect to certain completely positive maps acting on the core of the non-tracial non-commutative probability space. We give a condition on free Fisher information of an infinite family of variables, which guarantees factoriality of the von Neumann algebra they generate.

1. Introduction.

Free entropy and free Fisher information were introduced by Voiculescu [13], [14], [16] in the context of his free probability theory [17] as analogs of the corresponding classical quantities. These quantities are usually considered in the framework of tracial non-commutative probability spaces; not surprisingly, the most striking applications of free entropy theory were to tracial von Neumann algebras (see e.g. [15], [2], [10]). Recently, however, it turned out that some type III factors associated with free probability theory [5] have certain properties in common with their type II_{1} cousins [9]. This gives rise to a speculation that there is room for free entropy to exist outside of the context of tracial non-commutative probability spaces.

The goal of this paper is to initiate the development of free Fisher information, based on Voiculescu’s microstates-free approach [16], in the non-tracial framework. The key idea is that all of the ingredients going into the definition of free Fisher information in this case must behave covariantly with respect to the modular group [11] of the non-tracial state. The principal example of a family of variables for which free Fisher information is non-trivial, and which belong to an algebra not having any traces, are semicircular generators of free Araki-Woods factors, taken with free quasi-free states [5].

We describe another route towards free Fisher information, which is based on first converting the nontracial von Neumann algebra into a larger algebra, the core (having an infinite trace), and then considering free Fisher information relative to a certain completely positive map (in the spirit of [7]). We should point out that it is this approach that is most likely to connect with the microstates free entropy (as suggested by [3]; see also [8]), since it is at present unclear what a microstates approach to free entropy in the non-tracial framework should be.

We finish the paper with a look at free Fisher information on von Neumann algebras that have traces. Our first result is that once the algebra has a trace, the free Fisher information is automatically infinite when computed with respect to a non-tracial state. It is likely that on any von Neumann algebra, free Fisher information can be finite for only very special states (however, we do not have any results in this direction in the non-tracial category). Another result of the present paper is a statement guaranteeing factoriality of a tracial von Neumann algebra, once we know that it has an infinite generating family whose free Fisher information is bounded in a certain way.

Acknowledgement.This work was supported by an NSF postdoctoral fellowship. The author is grateful to MSRI and the organizers of the operator algebras program for their hospitality.

Date: January 29, 2001.

2. Free Fisher Information for Arbitrary KMS States.

2.1. Free Brownian motion in the presence of a modular group. Let M be a von Neumann algebra, ϕ : M → ℂ be a normal faithful state on M. Denote by σ_{t}^{ϕ} the modular group of ϕ. Denote by L^{2}(M^{sa}, ϕ) ⊂ L^{2}(M, ϕ) the closure of the real subspace of self-adjoint elements M^{sa} ⊂ M.

Let X = X^{∗} ∈ M and let B ⊂ M be a subalgebra. Assume that σ_{t}^{ϕ}(B) ⊂ B for all t ∈ ℝ.

Consider the von Neumann algebra M = Γ(L^{2}(M^{sa}, ϕ) ⊂ L^{2}(M, ϕ)), taken with the free quasifree state ϕ_{M}. Consider the element Y = s(X) ∈ M (see [5] for definitions and notation). Then ϕ_{M}(Y σ_{t}^{ϕ_{M}}(Y)) = ϕ(Xσ_{t}^{ϕ}(X)), for all t ∈ ℝ.

Consider the algebra N = (M, ϕ) ∗ (M, ϕ_{M}), and denote by ϕ̂ the free product state on N. Note that σϕ̂ ; t = σ_{t}^{ϕ} ∗ σ_{t}^{ϕ_{M}}. The elements X_{ϵ} = X + \sqrt{ϵ}Y, ϵ ≥ 0 form a natural free Brownian motion, which behaves nicely under the action of the modular group. In particular, note that for all ϵ ≥ 0 and t ∈ ℝ,

ϕ̂(X_{ϵ}σϕ̂ ; t(X_{ϵ})) = ϕ(Xσ_{t}^{ϕ}(X)) · (1 + ϵ).

Furthermore, for each ϵ > 0, the distribution of X_{ϵ} is that of a free Brownian motion at time ϵ, starting at X; this is because Y is a semicircular variable, free from X.

2.2. Conjugate variables. Let B[X] denote that algebra generated by B and all translates σ_{t}^{ϕ}(X), t ∈ ℝ. Assume that {σ_{t}^{ϕ}(X)} are algebraically free over B, i.e., satisfy no algebraic relations modulo B. Denote by ∂_{X} : B[X] → N the derivation given by:

1. ∂_{X}(σ_{t}^{ϕ}(X)) = σϕ̂ ; t(Y)

2. ∂_{X}(b) = 0, b ∈ B.

Notice that the range of ∂_{X} actually lies in the subspace B[X]Y B[X] ⊂ N. Note also that since ∂_{X}(σ_{t}^{ϕ}(X)) is self-adjoint, we have that for P ∈ B[X], ∂_{X}(P^{∗}) = ∂_{X}(P)^{∗}, i.e., ∂_{X} is a ∗-derivation. Observe finally that ∂_{X} is covariant with respect to the modular groups σ_{t}^{ϕ} and σϕ̂ ; t:

∂_{X}(σ_{t}^{ϕ}(P)) = σϕ̂ ; t(∂_{X}(P)), P ∈ B[X].

Define the conjugate variable J_{ϕ}(X : B) ∈ L^{2}(B[X], ϕ) to be such a vector ξ that

⟨ξ, P ⟩_{L^{2}(B[X],ϕ)} = ⟨Y, ∂_{X}(P)⟩_{L^{2}(N ,ϕ̂)}, ∀P ∈ B[X], (2.1)

if a vector ξ satisfying such properties exists. Formally, this means that ξ = ∂_{X}^{∗}(Y), where ∂_{X} : L^{2}(B[X], ϕ) → L^{2}(N , ϕ̂) is viewed as a densely defined operator.

It is clear, because of the density of B[X] in L^{2}(B[X], ϕ), that ξ is unique, if it exists.

It is convenient to talk about J_{ϕ}(X : B) even in the case that {σ_{t}(X)}_{t∈ℝ} are not algebraically free over B (such is the case, for example, when ϕ is a trace, and hence σ_{t}^{ϕ}(X) = X for all t). In this case, one can view ∂_{X} as a multi-valued map, the set of values given by the results of application of the definition of ∂_{X} in all possible ways; the definition of J_{ϕ} is then that (2.1) is valid for all values of ∂_{X}.

Note that J_{ϕ}(X : B) depends on more than just the joint distribution of X and B with respect to the state ϕ; it depends on the joint distribution of the family B ∪ {σ_{t}^{ϕ}(X) : t ∈ ℝ).

We continue to denote by σ_{t}^{ϕ} the extension of σ_{t}^{ϕ} to the Hilbert space L^{2}(M, ϕ) (this is precisely the one-parameter group of unitaries Δ_{ϕ}^{it}, where Δ_{ϕ} is the modular operator). In particular, if ϕ is a trace, then the definition of J_{ϕ} is (up to a multiple) precisely that of the conjugate variable of Voiculescu [16].

Lemma 2.1. Assume that ξ = J_{ϕ}(X : B) exists. Then ξ ∈ L^{2}(M^{sa}, ϕ), and σ_{t}^{ϕ}(J_{ϕ}(X : B)) = J_{ϕ}(σ_{t}(X) : B).

Proof. We note that, because ∂_{X} is a ∗-derivation,

⟨P^{∗}, ξ⟩ = \overline{⟨Y, ∂_{X}(P^{∗})⟩} = ⟨∂_{X}(P), Y ⟩ = \overline{⟨ξ, P ⟩}.

From this it follows that ξ is in the domain of the S operator of Tomita theory, and moreover that Sξ = ξ. Hence ξ ∈ L^{2}(M^{sa}).

One also has

⟨σ_{t}^{ϕ}(ξ), P ⟩ = ⟨Y, ∂_{X}(P)⟩
= ⟨σ_{t}^{ϕ}(Y), ∂_{σ_{t}^{ϕ}(X)}(P)⟩,

since the joint distributions of B[X] and {σϕ̂ ; t(Y)}_{t∈ℝ} is the same as B[X] and {σϕ̂ ; s+t(Y)}_{t∈ℝ}, for any s. It follows that σ_{t}^{ϕ}(J_{ϕ}(X : B)) = J_{ϕ}(σ_{t}^{ϕ}(X) : B).

Lemma 2.2. Let P, Q ∈ B[X], and assume that ξ = J_{ϕ}(X) exists and is in M. Then

ϕ(P ξQ) = ϕ̂(P Y ∂_{X}(Q)) + ϕ̂(∂_{X}(P)Y Q).

Proof. Recall that ϕ (and ϕ̂) satisfy the KMS condition: for all a, b ∈ M (or ∈ N), there exists a (unique) function f(z), analytic on the strip {z : 0 < ℑz < 1}, and so that (writing σ_{t} for either σ_{t}^{ϕ} or σϕ̂ ; t )

ϕ(aσ_{t}(b)) = f(t),
ϕ(σ_{t}(b)a) = f(t + i), t ∈ ℝ.

Fix P, Q ∈ B[X] and let f be as above, so that

ϕ(σ_{t}^{ϕ}(P)ξQ) = f(t + i),
ϕ(ξQσ_{t}^{ϕ}(P)) = f(t).

Then

f(t) = ⟨ξ, Qσ_{t}^{ϕ}(P)⟩
= ⟨Y, ∂_{X}(Qσ_{t}^{ϕ}(P))⟩
= ⟨Y, ∂_{X}(Q)σ_{t}^{ϕ}(P)⟩ + ⟨Y, Qσϕ̂ ; t(∂_{X}(P))⟩,

where in the last step we used the fact that ∂_{X} intertwines σ_{t}^{ϕ} and σϕ̂ ; t. Using the KMS-condition for ϕ̂, we then get

f(t + i) = ϕ̂(σϕ̂ ; t(P)Y ∂_{X}(Q)) + ϕ̂(σϕ̂ ; t(∂_{X}(P))Y Q)
= ϕ̂(σ_{t}^{ϕ}(P)Y ∂_{X}(Q)) + ϕ̂(∂_{X}(σ_{t}^{ϕ}(P))Y Q).

Since f(t + i) = ϕ(σ_{t}^{ϕ}(P)ξQ), we get, setting t = 0, that

ϕ(P ξQ) = ϕ̂(P Y ∂_{X}(Q)) + ϕ̂(∂_{X}(P)Y Q),

as claimed.

2.3. Conjugate variables as free Brownian gradients. As pointed out above, X + \sqrt{ε}Y is a natural free Brownian motion, which is covariant with respect to the appropriate modular groups. The following proposition shows that J_{ϕ}(X : B) plays the role of the free Brownian gradient of X.

Proposition 2.3. Assume that ξ = J_{ϕ}(X : B) exists and belongs to M ⊂ L^{2}(M, ϕ). Let P (Z_{1}, …, Z_{n}), be any non-commutative polynomial in n variables Z_{1}, …, Z_{n}, with coefficients from B. Write X_{t} = σ_{t}^{ϕ}(X),

Y_{t} = σϕ̂ ; t(Y), ξ_{t} = σ_{t}^{ϕ}(ξ), t ∈ ℝ.

Then for all t_{1}, …, t_{n} ∈ ℝ, we have

ϕ̂(P (X_{t_{1}} + \sqrt{ε}Y_{t_{1}}, …, X_{t_{n}} + \sqrt{ε}Y_{t_{n}})) = \frac{1}{2}ϕ(P (X_{t_{1}} + εξ_{t_{1}}, …, X_{t_{n}} + εξ_{t_{n}})
+O(ε^{2}).

Proof. We may assume, by linearity, that P is a monomial, i.e., P (Z_{1}, …, Z_{n}) = b_{0}Z_{1}b_{1} ⋯ b_{n−1}Z_{n}b_{n}, for b_{j} ∈ B. In this case, we have

ϕ̂(b_{0}(X_{t_{1}} + \sqrt{ε}Y_{t_{1}})b_{1} ⋯ b_{n}) = ϕ(P (X_{t_{1}}, …, X_{t_{n}})) + O(ε^{2})

+ε ∑_{k<l} ϕ̂(b_{0}X_{t_{1}} ⋯ b_{l}Y_{t_{l+1}}b_{l+1}X_{t_{l+2}} ⋯ b_{k}Y_{t_{k+1}} ·

·b_{k+1}X_{t_{k+2}} ⋯ X_{t_{n}}b_{n})

= ϕ(P (X_{t_{1}}, …, X_{t_{n}})) + O(ε^{2})

+\frac{1}{2}ε ∑_{l} ϕ̂(b_{0}X_{t_{1}} ⋯ X_{t_{l}}b_{l}Y_{t_{l+1}}∂_{X_{t_{l+1}}} (b_{l+1}X_{t_{l+2}} ⋯ X_{t_{n}}b_{n}))

+\frac{1}{2}ε ∑_{k} ϕ̂(∂_{X_{t_{k+1}}} (b_{0}X_{t_{1}} ⋯ X_{t_{k}}b_{k})Y_{t_{k}}b_{k+1}X_{t_{k+2}} ⋯ X_{t_{n}}b_{n})

= ϕ(P (X_{t_{1}}, …, X_{t_{n}})) + O(ε^{2})

+\frac{1}{2} ∑_{k} ϕ(b_{0}X_{t_{1}} ⋯ X_{t_{k}}b_{k}εξ_{t_{k}}b_{k+1}X_{t_{k+2}} ⋯ X_{t_{n}}b_{n}),

the last equality by Lemma 2.2. This implies the statement of the Lemma.

2.4. Examples of conjugate variables.

2.4.1. Tracial case. We have seen before that if ϕ is a trace, then the definition of J_{ϕ}(X : B) coincides with the definition of conjugate variables given by Voiculescu, up to a constant (which has to do with the fact that we choose Y so that ∥Y ∥_{L^{2}(ϕ̂)} = ∥X∥_{L^{2}(ϕ)}, and not 1). In particular,

J_{ϕ}(X : B) = J(X : B) · \frac{1}{∥X∥_{L^{2}(ϕ)}^{2}} , if ϕ is a trace.

2.4.2. Free quasi-free states. Let μ be a positive finite Borel measure on ℝ, and let H_{ℝ} be the real Hilbert space L^{2}(ℝ, ℝ, μ) of μ -square-integrable real-valued functions. Denote by U_{t} the representation of ℝ on H_{ℝ}, given by

(U_{t}f)(x) = e^{2πitx}f(x), x, t ∈ ℝ.

Let h denote the vector 1 ∈ H_{ℝ}, and consider

M = Γ(H_{ℝ}, U_{t})′′, ϕ = ϕ_{U}, X = s(h) ∈ M

(see [5] for definitions and notation).

Then X = J_{ϕ}(X : ℂ). Indeed, set X_{t} = σ_{t}^{ϕ}(X) = s(U_{t}h); then we have

ϕ(X · X_{t_{1}} ⋯ X_{t_{n}}) = ∑_{k} ϕ(XX_{t_{k}})ϕ(X_{t_{1}} ⋯ X_{t_{k−1}}) · ϕ(X_{t_{k+1}} ⋯ X_{t_{n}})
= ∑_{k} ϕ̂(Y σϕ̂ ; t_{k}(Y))ϕ(X_{t_{1}} ⋯ X_{t_{k−1}}) · ϕ(X_{t_{k+1}} ⋯ X_{t_{n}})
= ∑_{k} ϕ̂(Y X_{t_{1}} ⋯ X_{t_{k−1}}Y X_{t_{k+1}} ⋯ X_{t_{n}})
= ϕ̂(Y ∂_{X}(X_{t_{1}} ⋯ X_{t_{n}})),

so that X satisfies the defining property of J_{ϕ}(X : ℂ), and hence J_{ϕ}(X : ℂ) exists and equals X.

2.5. Free Fisher information. Following [16], we define the free Fisher information Φ_{ϕ}^{∗}(X : B) to be

Φ_{ϕ}^{∗}(X : B) = ∥J_{ϕ}(X : B)∥_{2}^{2} · ∥X∥_{2}^{−2}

(the extra factor ∥X∥_{2}^{−2} comes from the fact that ∂_{X}(X) does not have unit norm in our definition). For several variables, we set

Φ_{ϕ}^{∗}(X_{1}, …, X_{n}) = ∑ Φ_{ϕ}^{∗}(X_{i} : W^{∗}(σ_{t_{1}}^{ϕ}(X_{1}), …, X̂_{i}, …, σ_{t_{n}}^{ϕ}(X_{n}) : t_{1}, …, t_{n} ∈ ℝ))

(here X̂_{i} means that X_{i} is omitted).

3. Free Fisher Information Relative to the Core.

Recall [11] that if (M, ϕ) is as above, its core is defined to be the von Neumann algebra crossed product P = M ⋊_{σ^{ϕ}} ℝ. There is a canonical inclusion M ⊂ P, and P is densely spanned by elements of the form

mU_{t}, t ∈ ℝ,

where m ∈ M, and U_{t} satisfy U_{t}mU_{t}^{∗} = σ_{t}^{ϕ}(m). The elements U_{t} : t ∈ ℝ generate a copy of the group von Neumann algebra L(ℝ) ⊂ P; the map

E^{ϕ} : mU_{t} ↦ ϕ(m)U_{t}, m ∈ M, t ∈ ℝ

extends to a normal conditional expectation from P onto L(ℝ).

For X ∈ M self-adjoint, define the completely positive map η_{X} : L(ℝ) → L(ℝ) by

η_{X}(g) = E^{ϕ}(XgX), g ∈ L(ℝ).

Identify L(ℝ) with L^{∞}(ℝ) via Fourier transform. For each t ∈ ℝ, set

η(t) = ⟨X, σ_{t}^{ϕ}(X)⟩ = E^{ϕ}(XU_{t}X).

Then η_{X}(f) = η̂ ∗ f, if f ∈ L^{∞}(ℝ) ≅ L(ℝ); here η̂ denotes Fourier transform.

Define on P an L(ℝ)-valued inner product

⟨a, b⟩_{L(ℝ)} = E^{ϕ}(a^{∗}b), a, b ∈ P.

Denote by L^{2}(P, E^{ϕ}) the L(ℝ)-Hilbert bimodule arising from the completion of P with respect to the norm induced by this inner product. Note that the restriction of ⟨·, ·⟩_{L(ℝ)} to M ⊂ P is valued in the complex field, and coincides with the inner product ⟨a, b⟩ = ϕ(a^{∗}b) on L^{2}(M).

Denote by ⟨·, ·⟩_{η} the L(ℝ)-valued inner product on P ⊗ P (algebraic tensor product) given by

⟨a ⊗ b, a′ ⊗ b′⟩_{η} = E^{ϕ}(b^{∗}η(E^{ϕ}(a^{∗}a′)b′)), a, a′, b, b′ ∈ P.

Denote by 1 ⊗_{η} 1 the vector 1 ⊗ 1 ∈ P ⊗ P.

Let δ_{X} : B[X] · L(ℝ) → P ⊗ P be given by

δ_{X}(X) = 1 ⊗_{η} 1, δ_{X}(B · L(ℝ)) = 0

and the fact that δ_{X} is a derivation.

Theorem 3.1. Let (M, ϕ) be as above, and let P be its core. Let i : L^{2}(M, ϕ) → L^{2}(P, E^{ϕ}) be the extension of the inclusion of M ⊂ P. Then ζ = i(J_{ϕ}(X : B)) satisfies

⟨ζ, Q⟩_{L(ℝ)} = ⟨1 ⊗_{η} 1, δ_{X}(Q)⟩_{η_{X}} (3.1)

for all Q ∈ B[X] ∨ L(ℝ). Conversely, if there exists a vector ζ ∈ L^{2}(P, E^{ϕ}), so that (3.1) is satisfied, then J_{ϕ}(X : B) exists and ζ = i(J_{ϕ}(X : B)).

Proof. Assume first that J_{ϕ}(X : B) exists. Set ζ = i(J_{ϕ}(X : B)). We must verify that (3.1) holds. By linearity, and the fact that L(ℝ)BL(ℝ) ⊂ BL(ℝ), it is sufficient to consider the case when Q = b_{0}U^{s_{1}}X_{t_{1}}b_{1}U^{s_{2}} ⋯ X_{t_{n}}b_{n}U^{s_{n}}, with b_{j} ∈ B and X_{t} = σ_{t}^{ϕ}(X). Then Q = P · U^{r}, where r = ∑ s_{j} , and P = b_{0}X_{t′_{1}}b′_{1} ⋯ b′_{n}, with b′_{j} = σ_{s_{j−1}}^{ϕ} ∘ ⋯ σ_{s_{1}}^{ϕ}(b_{j}), t′_{j} = s_{j−1} + ⋯ + s_{1} + t_{j}. Note that for x, y, x′, y′ ∈ P, ⟨x ⊗ y, x′ ⊗ Uy′⟩_{η} = ⟨x ⊗ y, x′U ⊗ y⟩_{η}, and ⟨x ⊗ y, U^{r}(x′ ⊗ y′)U^{s}⟩ = ⟨U^{−r}x ⊗ yU^{−s}⟩. Using this , we get

⟨ζ, Q⟩_{L(ℝ)} = ⟨ζ, P ⟩_{L(ℝ)}g
= ⟨ζ, P ⟩_{L^{2}(M,ϕ)} · U^{r}
= ϕ̂(Y ∂_{X}(P)) · U^{r}
= ∑_{j} ϕ(b′_{0}X_{t′_{1}} ⋯ X_{t′_{j}}b′_{j})ϕ(b′_{j+1}X_{t′_{j+2}} ⋯ X_{t′_{n}}b′_{n}) · ϕ̂(Y Y_{t_{j}})U^{r}
= ∑_{j} ϕ(b′_{0}X_{t′_{1}} ⋯ X_{t′_{j}}b′_{j})ϕ(b′_{j+1}X_{t′_{j+2}} ⋯ X_{t′_{n}}b′_{n}) · ϕ(XU^{t′_{j}}XU^{−t′_{j}})U^{r}
= ∑_{j} ϕ(b′_{0}X_{t′_{1}} ⋯ X_{t′_{j}}b′_{j})ϕ(b′_{j+1}X_{t′_{j+2}} ⋯ X_{t′_{n}}b′_{n}) · E^{ϕ}(XU^{t′_{j}}XU^{−t′_{j}}U^{r})
= ∑_{j} ϕ(b′_{0}X_{t′_{1}} ⋯ X_{t′_{j}}b′_{j})ϕ(b′_{j+1}X_{t′_{j+2}} ⋯ X_{t′_{n}}b′_{n}) · η_{X}(U^{t′_{j}})U^{r−t′_{j}}
= ∑_{j} η_{X} ∘ E^{ϕ}(b′_{0}X_{t′_{1}} ⋯ X_{t′_{j}}b′_{j}U^{t′_{j}}) · E^{ϕ}(U^{r−t′_{j}}b′_{j+1}X_{t′_{j+2}} ⋯ X_{t′_{n}}b′_{n})
= ∑_{j}⟨1 ⊗_{η} 1, b′_{0}X_{t′_{1}} ⋯ X_{t′_{j}}b′_{j}U^{t′_{j}} ⊗ U^{r−t_{j}}b′_{j+1}X_{t′_{j+2}} ⋯ X_{t′_{n}}b′_{n}⟩_{η}
= ∑_{j}⟨1 ⊗_{η} 1, b_{0}U^{s_{1}}X_{t_{1}} ⋯ X_{t_{j}}b_{j}U^{s_{j}}U^{t′_{j}} ⊗ U^{r−t_{j}}U^{s_{j+1}}b_{j+1}X_{t_{j+2}} ⋯ X_{t_{n}}b_{n}U^{s_{n}}U^{−r}⟩_{η}
= ∑_{j}⟨1 ⊗_{η} 1, b_{0}U^{s_{1}}X_{t_{1}} ⋯ X_{t_{j}}b_{j}U^{s_{j}} ⊗ U^{s_{j+1}}b_{j+1}X_{t_{j+2}} ⋯ X_{t_{n}}b_{n}U^{s_{n}}⟩_{η}
= ⟨1 ⊗_{η} 1, δ_{X}(Q)⟩_{η}.

Conversely, assume that ζ satisfying (3.1) exists. Since the argument above is reversible, it is sufficient to prove that ζ is in the image of i : L^{2}(M) → L^{2}(P, E^{ϕ}). Let θ_{t} be the dual action of ℝ on P, given by θ_{t}(U_{s}) = exp(2πist), θ_{t}(m) = m, m ∈ M. It is sufficient to prove that θ_{t}(ζ) = ζ, since i(L^{2}(M)) consists precisely of those vectors, which are left fixed by θ. It is sufficient to prove that θ_{s}(E^{ϕ}(ζmU_{t})) = exp(2πist) if m ∈ M. Since ζ is assumed to be in the closure of B[X] ∨ L(ℝ), it is sufficient to check this for m ∈ B[X]. But then by (3.1),

E^{ϕ}(ζmU_{t}) = ⟨ζ, mU_{t}⟩_{L(ℝ)}
= ⟨1 ⊗_{η} 1, δ_{X}(mU)⟩_{η}
= ⟨1 ⊗_{η} 1, δ_{X}(m)⟩_{η}U
∈ M · U_{t},

which gives the desired result, since θ_{s} acts trivially on M.

Note that (3.1) means that ζ is equal to J(X : B ∨ ℝ, η) in the notation of [7]. (This is strictly speaking incorrect, since the setting of [7] presumes the existence of a finite trace on B[X] ∨ L(ℝ); however, it is not hard to check that the arguments in [7] go through also in the case of a semifinite trace, which exists in our case).

This fact has many consequences for the conjugate variables J_{ϕ}(X : B), coming from the properties of J(X : B ∨ L(ℝ), η). Note in particular that if X is free from B with amalgamation over D ⊂ B with respect to some conditional expectation E : B → D, and E is ϕ-preserving, then X is free from B ∨ L(ℝ) with amalgamation over D ∨ L(ℝ) (see [12], [6]). We record this as

Theorem 3.2. Assume that E : B → D is a ϕ-preserving conditional expectation. If X is free from B over D, then

J_{ϕ}(X : B) = J_{ϕ}(X : D).

In a similar way, one can generalize to J_{ϕ}(X : B) all the properties of the conjugate variable J(X : B, η) proved in [7].

Reformulating gives the following properties of Φ_{ϕ}, which we list for reader’s convenience, since they are needed in the rest of the paper:

Theorem 3.3. Let ϕ be a normal faithful state on M, B ⊂ M be globally fixed by the modular group (i.e., σ_{t}^{ϕ}(B) = B for all t), and X_{i} ∈ M. Then:

(a) Φ_{ϕ}^{∗}(λX_{1}, …, λX_{n} : B) = λ^{−2}Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : B) for all λ ∈ ℝ \ {0}

(b) If B ⊂ A ⊂ M and A is globally fixed by σ^{ϕ}, then Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : A) ≥ Φ^{∗}(X_{1}, …, X_{n} : B).

(c) If C ⊂ M is globally fixed by σ^{ϕ}, and W^{∗}(X_{1}, …, X_{n}) and B are free with amalgamation over C (with respect to the unique ϕ-preserving conditional expectation from M onto C), then Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : B ∨ C) = Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : B).

(d) If Y_{i} ∈ M are self-adjoint, D ⊂ B, D ⊂ C subalgebras of M, which are globally fixed by σ^{ϕ}, and B[X_{1}, …, X_{n}] is free from C[X_{1}, …, X_{n}] over D (with respect to the unique ϕ-preserving conditional expectation from M onto D), then Φ_{ϕ}^{∗}(X_{1}, …, X_{n}, Y_{1}, …, Y_{m} : B ∨ C) = Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : B) + Φ_{ϕ}^{∗}(Y_{1}, …, Y_{n} : C).

(e) Φ_{ϕ}^{∗}(X_{1}, …, X_{n}, Y_{1}, …, Y_{n} : B) ≥ Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : B) + Φ_{ϕ}^{∗}(Y_{1}, …, Y_{m} : B).

(f) Φ_{ϕ}^{∗}(X_{1}, …, X_{n} : B) · ϕ(∑ X_{i}^{∗}X_{i})^{2} ≥ n^{2}. Equality holds iff {σ_{t_{1}}^{ϕ}(X_{1}), …, σ_{t_{n}}^{ϕ}(X_{n}) : t_{1}, …, t_{n} ∈ ℝ} have the same distribution as the semicircular family {κs(σ_{t_{1}}^{ϕ}(X_{1})), …, κs(σ_{t_{n}}^{ϕ}(X_{n})) : t_{1}, …, t_{n} ∈ ℝ} with respect to the free quasi-free state, κ > 0.

We mention that all of the statements in sections 3 and 4 of [7] remain valid for Φ_{ϕ}^{∗}; we leave details to the reader.

One can also define and study free entropy χ_{ϕ}^{∗}(X_{1}, …, X_{n}) by setting X_{i}^{ϵ} = X_{i} + Y_{i} to be the free Brownian motion described in the beginning of the paper, and letting

χ_{ϕ}^{∗}(X_{1}, …, X_{n}) = \frac{1}{2} ∫_{0}^{∞} (\frac{n}{1 + t} − Φ_{ϕ}^{∗}(X_{1}^{ϵ}, …, X_{n}^{ϵ})) dt.

The properties of χ^{∗}(⋯, η) once again generalize to χ_{ϕ}^{∗} (compare section 8 of [7]).

4. States on a II_{1} factor.

4.1. Φ_{ϕ}^{∗} vs. Φ_{τ}^{∗}. The following theorem is somewhat surprising, since it shows that Φ_{ϕ}^{∗} is identically infinite for most states ϕ on a II_{1} factor (the analogy with classical Fisher information would instead suggest that ϕ ↦ Φ_{ϕ}^{∗} would have some nice convexity properties). This, on the other hand, goes well with the “degenerate convexity” property of the microstates free entropy χ [15] (which is reflected in that it is identically −∞ on generators of any von Neumann algebra which more than one unital trace).

Theorem 4.1. Let M be a tracial von Neumann algebra, ϕ a faithful normal state on M, B ⊂ M a subalgebra so that σ_{t}^{ϕ}(B) = B for all t, and X = X^{∗} ∈ M. Then if J_{ϕ}(X : B) exists, the modular group of ϕ must fix X.

Proof. Let d ∈ M be a positive element, so that ϕ(x) = τ(dx), where τ is a normal faithful trace on M, and d is an unbounded operator on L^{2}(M, τ), affiliated to M. The modular group of ϕ is then given by σ_{t}^{ϕ}(x) = d^{it}xd^{−it}, x ∈ M. Denoting by X_{t} the element σ_{t}^{ϕ}(X), we then get

X = X_{0} = d^{−it}X_{t}d^{it}, t ∈ ℝ.

Consider

ϕ(X_{0}^{2}) = ϕ(J_{ϕ}(X : B) · X_{0}) = ϕ(J_{ϕ}(X : B)d^{−it}X_{t}d^{it}).

Let b_{1} and b_{2} be two elements in the domain of ∂_{X}, so that b_{1} = b_{2}^{∗}. Then we get, writing Y_{t} = σϕ̂ ; t(Y):

ϕ(J_{ϕ}(X : B)b_{1}X_{t}b_{2}) = ϕ̂(Y_{0}b_{1}Y_{t}b_{2})
+ϕ̂(Y_{0}∂_{X}(b_{1})X_{t}b_{2}) + ϕ̂(Y_{0}b_{1}X_{t}∂_{X}(b_{2}))
= ϕ(b_{1})ϕ(b_{2})ϕ̂(Y_{0}Y_{t})
+ϕ̂(Y_{0}∂_{X}(b_{1})X_{t}b_{2}) + ϕ̂([∂_{X}(b_{2}^{∗})X_{t}b_{1}^{∗}Y_{0}]^{∗})
= ϕ(b_{1})ϕ(b_{1}^{∗})ϕ̂(Y_{0}Y_{t})
+ϕ̂(Y_{0}∂_{X}(b_{1})X_{t}b_{1}^{∗}) + ϕ̂([∂_{X}(b_{1})X_{t}b_{1}^{∗}Y_{0}]^{∗}).

Now, for all m, n ∈ M, we have

ϕ̂(Y_{0}mY_{0}n) = ϕ(m)ϕ(n) = ϕ̂(mY_{0}nY_{0}),

so that

ϕ̂(Y_{0}∂_{X}(b_{1})X_{t}b_{1}^{∗}) + ϕ̂([∂_{X}(b_{1})X_{t}b_{1}^{∗}Y_{0}]^{∗}) = ϕ̂(Y_{0}∂_{X}(b_{1})X_{t}b_{1}^{∗}) + ϕ̂([Y_{0}∂_{X}(b_{1})X_{t}b_{1}^{∗}]^{∗})
= ϕ̂(Y_{0}∂_{X}(b_{1})X_{t}b_{1}^{∗}) + \overline{ϕ̂(Y_{0}[∂_{X}(b_{1})X_{t}b_{1}^{∗}])}
∈ ℝ.

It follows that

ℑϕ(J_{ϕ}(X : B)b_{1}X_{t}b_{1}^{∗}) = ℑϕ̂(Y_{0}b_{1}Y_{t}b_{1}^{∗}).

Now fix t ∈ ℝ and choose a_{n} in the domain of ∂_{X}, ∥a_{n}∥ ≤ 1, so that

a_{n} → d^{it}, a_{n}^{∗} → d^{−it} strongly.

One can choose a_{n}, for example, to be elements of the algebra B[X]. Then

0 = ℑϕ(X_{0}^{2}) = ℑϕ(J_{ϕ}(X : B) · X_{0})
= ℑϕ(J_{ϕ}(X : B)d^{−it}X_{t}d^{it})
= lim_{n→∞} ℑϕ(J_{ϕ}(X : B)a_{n}X_{t}a_{n}^{∗})
= lim_{n→∞} ℑϕ̂(Y_{0}a_{n}Y_{t}a_{n}^{∗})
= lim_{n→∞} ℑ(ϕ(a_{n})ϕ(a_{n}^{∗})ϕ̂(Y_{0}Y_{t})
= lim_{n→∞} ϕ(a_{n})ϕ(a_{n}^{∗})ℑϕ̂(Y_{0}Y_{t})
= ϕ(d^{it})ϕ(d^{−it})ℑϕ̂(Y_{0}Y_{t}).

Since ϕ̂(Y_{0}Y_{t}) = ϕ(X_{0}X_{t}), for t sufficiently close to zero (so that ϕ(d^{it}) ≠ 0), we get that

ϕ(XX_{t}) ∈ ℝ.

Thus

0 = τ(dXX_{t}) − τ(d(XX_{t})^{∗})
= τ(dXd^{it}Xd^{−it} − d^{it}Xd^{−it}Xd)
= τ((dX − Xd)d^{it}Xd^{−it})
= τ([d, X]d^{it}Xd^{−it}).

Differentiating this in t, and noting that (d/dt)_{t=0}(d^{it}Xd^{−it}) = i[d, X] gives

iτ([d, X]^{2}) = 0.

Since [d, X] is anti-self-adjoint, this implies that τ(|[d, X]|^{2}) = 0, so that [d, X] = 0, because τ is faithful. This means that σ_{t}^{ϕ}(X) = X for all t.

Corollary 4.2. Suppose that X_{1}, …, X_{n} are self-adjoint generators of a II_{1} factor M. Let ϕ be a normal faithful state on M, and denote by τ the unique faithful normal trace on M. Then Φ_{ϕ}^{∗}(X_{1}, …, X_{n}) < +∞ implies that:

(1) Φ_{τ}^{∗}(X_{1}, …, X_{n}) < ∞ and

(2) ϕ is a multiple of the trace τ on M.

Proof. Clearly, the second statement implies the first. To get the second statement, write ϕ(·) = τ(d·) and apply the theorem to conclude that [d, X_{i}] = 0. Since X_{1}, …, X_{n} generate M, d must be in the center of M, which must consist of multiples of identity, since M is a factor. But then d is a scalar multiple of identity, so that ϕ and τ are proportional.

4.2. Factoriality. Voiculescu showed [15] that for his microstates entropy χ the following implication holds:

χ(X_{1}, …, X_{n}) > −∞ ⇒ W^{∗}(X_{1}, …, X_{n}) is a factor.

In fact, the conclusion is stronger: not only is the center of W^{∗}(X_{1}, …, X_{n}) is trivial, but so is its asymptotic center. Unfortunately, we don’t know if the same implication holds for the non-microstates free entropy χ^{∗} introduced by Voiculescu in [16], or even under the stronger assumption that Φ^{∗}(X_{1}, …, X_{n}) is finite. We prove a weaker version of the assertion above for Φ^{∗} = Φ_{τ}^{∗}. We first need a technical lemma:

Lemma 4.3. Let ϕ be a normal faithful state on M. Let X ∈ M be self-adjoint and B ⊂ M be a subalgebra, so that σ_{t}^{ϕ}(B) = B for all t. Assume that p ∈ B is a self-adjoint projection, ϕ(p) = α, and so that σ_{t}^{ϕ}(p) = p for all p. Assume that ∥[X, p]∥_{2} < δ. Then

Φ_{ϕ}^{∗}(X : B) > 4\frac{α^{2}(1 − α)^{2}}{δ^{2}} .

Proof. Let (A, τ) be a copy of L(𝔽_{2}), free from B[X]. Since Φ_{ϕ}^{∗}(X : B) = Φ_{ϕ}^{∗}(X : B ∨ A), and since the centralizer of {σ_{t}^{ϕ}(B)}_{t}∨A is a factor [1], we can find a projection q ∈ B∨A, which is fixed by the modular group, and so that ∥[X, q]∥_{2} < δ, and τ(q) = β = m/n is rational and close to α. We may moreover find a family of matrix units e_{ij} ∈ B ∨ A, 1 ≤ i, j ≤ n, fixed by the modular group, and so that

e_{ij}^{∗} = e_{ji}, e_{ij}e_{kl} = δ_{jk}e_{il}
τ(e_{ii}) = \frac{1}{n}, q = ∑_{i=1}^{m} e_{ii}.

Denote by C the algebra generated in B ∨ A by {e_{ij}}_{1≤i,j≤n}. Note that C ≅ M_{n×n}, the algebra of n × n matrices. The restriction of ϕ ∗ τ to C is the usual matrix trace. Then

Φ_{ϕ}^{∗}(X : B) = Φ_{ϕ∗τ}^{∗} (X : B ∨ A) ≥ Φ_{ϕ∗τ}^{∗} (X : C).

Write X_{ij} = e_{1i}Xe_{j1}. Then the inequality ∥[X, q]∥_{2} < δ implies that

δ > ∥xq − qx∥_{2}
= ∥qxq + (1 − q)xq − qxq − qx(1 − q)∥_{2}
= ∥(1 − q)xq − qx(1 − q)∥_{2}
= \sqrt{2} · ∥qx(1 − q)∥_{2},

since (1 − q)xq and qx(1 − q) are orthogonal. Hence

∥qX(1 − q)∥_{2} < δ/\sqrt{2.}

It follows that

∑_{1≤i≤m, m<j≤n} ϕ(X_{ij}^{∗}X_{ij}) + ∑_{m<i≤n, 1≤i≤n} ϕ(X_{ij}^{∗}X_{ij}) < δ^{2}.

Denote by ϕ′ the state n(ϕ ∗ τ)(e_{11} · e_{11}) on e_{11}W^{∗}(X, C)e_{11}. Then

Φ_{ϕ′}^{∗}({X_{ij}}) ≥ ∑_{i,j} Φ_{ϕ′}^{∗}({X_{ij}})

> 2m(n − m)\frac{1}{n(δ^{2}/2m(n − m))}
= \frac{(2m(n − m))^{2}}{nδ^{2}}
= n^{3}4\frac{β^{2}(1 − β)^{2}}{δ^{2}} .

Arguing exactly as in [4, Proposition 4.1], we get that

Φ_{ϕ∗τ}^{∗} (X : C) = \frac{1}{n^{3}} Φ_{ϕ′}^{∗}(\{X_{ij}\}) > 4\frac{β^{2}(1 − β)^{2}}{δ^{2}} .

Since β was a rational number, arbitrarily close to α, we get the desired estimate for Φ_{ϕ}^{∗}(X : B).

Theorem 4.4. Assume that M is a von Neumann algebra with a faithful normal trace τ, and X_{i} are a family of self-adjoint elements in M, ∥X_{i}∥ = 1. Assume that B_{i} form an increasing sequence of subalgebras of M, so that M = \overline{∪B_{i}}^{w}. Assume further that for some normal faithful state ϕ on M and for each j,

lim inf_{i} Φ_{ϕ}^{∗}(X_{i} : B_{j}) < +∞.

Then M is a factor.

Proof. In view of Theorem 4.1, we may assume that ϕ is a trace, τ. Assume that M is not a factor. Then there exists a central projection p ∈ M of some trace α = τ(p), α(1 − α) ≠ 0. Moreover, [p, X_{i}] = 0 for all i. Since B_{i} increase to all of M, given δ > 0, there is a large enough j and a projection q ∈ B_{j}, so that ∥q − p∥_{2} < δ/2. Then for any k,

∥[q, X_{k}]∥_{2} = ∥qX_{k} − X_{k}q∥_{2}
= ∥(q − p)X_{k} − X_{k}(q − p) + pX_{k} − X_{k}p∥_{2}
≤ ∥(q − p)X_{k}∥_{2} + ∥X_{k}(q − p)∥_{2} + 0
≤ 2∥(q − p)∥_{2}∥X_{k}∥
< 2(δ/2) = δ.

Now applying Lemma 4.3, we deduce that for any i > j,

Φ_{τ}^{∗}(X_{i} : B_{j}) > 4\frac{α^{2}(1 − α)^{2}}{δ^{2}} .

Hence lim inf_{i} Φ_{τ}^{∗}(X_{i} : B_{j}) > 4α^{2}(1 − α)^{2}/δ^{2}, which is a contradiction, since δ was arbitrary.

The hypothesis of the theorem is satisfied for some von Neumann algebras. For example, let M = L(𝔽_{∞}) generated by an infinite semicircular family X_{i}, i = 1, 2, 3, . . .. Then if B_{i} = W^{∗}(X_{j} : j < i), the assumptions of the theorem are satisfied. In fact if Y_{i} is any family of elements of a tracial von Neumann algebra, so that ∥Y_{i}∥ = 1, and X_{i} are a free semicircular family, then letting Z_{j}(ϵ) = Y_{i} + ϵX_{i}, M_{ϵ} = W^{∗}(Z_{1}(ϵ), Z_{2}(ϵ), …) and B_{j} = W^{∗}(Z_{1}(ϵ), …, Z_{j}(ϵ)), we see that M_{ϵ} is a factor. In other words, generators of an arbitrary tracial von Neumann algebra can be perturbed (in a certain representation of this algebra) by an arbitrarily small amount ϵ in uniform norm, to produce a II_{1} factor. Another way of putting it is to note that the free Brownian motion ϵ ↦ Z_{j}(ϵ) started at {Y_{1}, Y_{2}, … } generates a factor at any time ϵ > 0.

4.3. Factoriality in the non-tracial case. In a similar way, we get the following:

Theorem 4.5. Assume that M is a von Neumann algebra with a faithful normal state ϕ, and X_{i} are a family of self-adjoint elements in M, ∥X_{i}∥ = 1. Assume that B_{i} form an increasing sequence of subalgebras of M, σ_{t}^{ϕ}(B_{i}) = B_{i} for all t and i, and assume that M^{ϕ} = \overline{∪(B_{i} ∩ M^{ϕ})}^{w}. Let R_{i} be the operator of right multiplication by X_{i} densely defined on L^{2}(M, ϕ). Assume that sup_{i} ∥R_{i}∥ = C < +∞. Assume further that for each j,

lim inf_{i} Φ_{ϕ}^{∗}(X_{i} : B_{j}) < +∞.

Then M is a factor.

Proof. Assume that M is not a factor. Then there exists a central projection p ∈ M, α = ϕ(p), α(1−α) ≠ 0. Moreover, [p, X_{i}] = 0 for all i. Since automatically p ∈ M^{ϕ} and B_{i} ∩ M^{ϕ} increase to all of M^{ϕ}, given δ > 0, there is a large enough j and a projection q ∈ B_{j} ∩ M^{ϕ}, so that ∥q − p∥_{2} < δ/(1 + C). Then for any k,

∥[q, X_{k}]∥_{2} = ∥qX_{k} − X_{k}q∥_{2}
= ∥(q − p)X_{k} − X_{k}(q − p) + pX_{k} − X_{k}p∥_{2}
≤ ∥(q − p)X_{k}∥_{2} + ∥X_{k}(q − p)∥_{2} + 0
≤ ∥R_{k}∥∥(q − p)∥_{2} + ∥(q − p)∥_{2}∥X_{k}∥
< (1 + C)(δ/(1 + C)) = δ.

Note that the assumption on the norms of R_{i} is satisfied if each X_{i} is analytic for σ_{t}^{ϕ} and satisfies sup_{k} ∥σ_{i}^{ϕ}(X_{k})∥ = C < +∞.

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Department of Mathematics, UCLA, Los Angeles, CA 90095

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