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Prime factors
We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.
In a series of papers, Murray and von Neumann (1–5) introduced and studied certain algebras of Hilbert space operators, also known as rings of operators. They are now known as “von Neumann algebras.”
A von Neumann algebra is a strongoperator closed selfadjoint subalgebra of the algebra of all bounded linear transformations on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity. They are the building blocks from which all the von Neumann algebras are built. The most elementary factors, type I_{n} factors, are isomorphic to the algebra M_{n} of all n × n complex matrices. One of the basic constructions with factors (producing other factors) is that of forming the tensor product. For factors of type I_{n}, M_{p} ⊗ M_{q} is isomorphic to M_{pq}. Of course, no such tensor decomposition of M_{p} is possible precisely when p is a prime. The theory of tensor decompositions of factors of type I_{n} is little more than the theory of factoring integers into their prime components.
Murray and von Neumann (1) classified factors by means of a relative dimension function. Finite factors are those for which this dimension function has a finite range. For finite factors, this dimension function gives rise to a (unique, when normalized) tracial state. In general, a von Neumann algebra admitting a faithful normal trace is said to be finite. Infinitedimensional finite factors are called factors of type II_{1}. They are “continuous” matrix algebras. Murray and von Neumann (2) spoke of “continuous dimensionality” in their factors of type II_{1}. In a parallel manner, when we study tensor products of factors of type II_{1} and their tensorproduct decompositions, we may speak of decomposition into “continuous primes.” Factors of type II_{1} tensored with one another (as von Neumann algebras) produce, again, factors of type II_{1}. Each factor of type II_{1} may be decomposed as the tensor product of M_{n} and a factor of type II_{1} for each n in ℕ. In a sense, the “discrete primes” (that is, 2, 3, 5, …) are not significant in the theory of decomposition into “continuous primes.” The first question of the theory of such decompositions has to be that of the existence of a continuous prime: Is there a factor of type II_{1} that is not (isomorphic to) the tensor product of two factors of type II_{1}? This problem and some related problems, concerning the basic structure of factors, have been asked and studied by many people (see, e.g., refs. 6 and 7, pp. 4.4.12 and 4.4.45). Popa (6) proves that there are prime factors of type II_{1} with a nonseparable predual. The separable case remained open. In this paper, we shall answer this question affirmatively.
We describe below, briefly, a basic construction of factors of type II_{1} by using regular representations of discrete groups. Our main result states that certain factors arising from free groups are prime. An outline of the proof follows the statement. We end with some open questions.
Main Result
There are two main classes of examples of von Neumann algebras introduced by Murray and von Neumann (1, 3). One is obtained from the “groupmeasure space construction”; the other is based on regular representations of a (discrete) group G (with unit e). The second class is the one needed in this paper. A brief description of that class follows.
The Hilbert space ℋ is l^{2}(G) (with its usual inner product). We assume that G is countable so that ℋ is separable. For each g in G, let L_{g} denote the left translation of functions in l^{2}(G) by g^{−1}. Then g → L_{g} is a faithful unitary representation of G on ℋ. Let ℒ_{G} be the von Neumann algebra generated by {L_{g}:g ∈ G}. Similarly, let R_{g} be the right translation by g on l^{2}(G) and ℛ_{G} be the von Neumann algebra generated by {R_{g}:g ∈ G}. Then the commutant ℒ′_{G} of ℒ_{G} is equal to ℛ_{G} and ℛ′_{G} = ℒ_{G}. The function u_{g} that is 1 at the group element g and 0 elsewhere is a cyclic trace vector for ℒ_{G} (and ℛ_{G}). In general, ℒ_{G} and ℛ_{G} are finite von Neumann algebras. They are factors (of type II_{1}) precisely when each conjugacy class in G (other than that of e) is infinite. In this case, we say that G is an infinite conjugacy class (i.c.c.) group.
Specific examples of such II_{1} factors result from choosing for G any of the free groups F_{n} on n generators (n ≥ 2), or the permutation group Π of integers ℤ (consisting of those permutations that leave fixed all but a finite subset of ℤ). Murray and von Neumann (3) prove that ℒ_{Fn} and ℒ_{Π} are not ∗ isomorphic to each other (a deep result). A factor is hyperfinite if it is the ultraweak closure of the ascending union of a family of finitedimensional selfadjoint subalgebras. In fact, ℒ_{Π} is the unique hyperfinite factor of type II_{1}; it is contained in any factor of type II_{1}; and the tensor product of the hyperfinite II_{1} factor with itself is ∗ isomorphic to itself. Now we state our main theorem.
Main Theorem.
The free group factor ℒ_{Fn} associated with the free group on n (≥2) generators is prime, i.e., it is not isomorphic to the tensor product of any two factors of type II_{1}.
We prove this result with the aid of Voiculescu’s free probability theory (8–10) (especially, his recently introduced concept of free entropy) and some geometrical methods for estimating free entropies. We refer to ref. 10 for the basics of free probability theory.
Let ℳ be a von Neumann algebra with a normal faithful trace τ, X_{1}, … , X_{n} be selfadjoint elements in ℳ. As analogues of classical entropy and of Fisher’s information measure, Voiculescu (8) introduced free entropy χ(X_{1}, … , X_{n}). Roughly speaking, χ(X_{1}, … , X_{n}) is the limit of certain normalized measurement of all selfadjoint matrices that approximate X_{1}, … , X_{n} in joint distributions as the dimension of the matrices tends to infinity. We list some properties of free entropy in the following lemma.
Lemma 1.
Let X_{1}, … , X_{n}, n ≥ 1, be selfadjoint elements in ℳ (with trace τ), C be τ(X_{1}^{2} + ⋯ + X_{n}^{2})^{1/2} and R_{0} be max{∥X_{j}∥ : j = 1, … , n}. Then
(i) (ref. 8; p. 2.2) χ(X_{1}, … , X_{n}) ≤ n/2 log(2πeC^{2}n^{−1});
(ii) (ref. 8, p. 4.5) χ(X_{1}) = ∫∫ logs − tdμ_{1}(s)dμ_{1}(t) + 3/4 + 1/2 log 2π, where μ_{1} is the (measure on the spectrum of X_{1} corresponding to the) distribution of X_{1};
(iii) (ref. 8; p. 5.4) χ(X_{1}, … , X_{n}) = χ(X_{1}) + ⋯ + χ(X_{n}) when X_{1}, … , X_{n} are free random variables.
From the above lemma, we know that there are selfadjoint elements X_{1}, … , X_{n} in ℒ_{Fn} with finite free entropy such that X_{1}, … , X_{n} generate ℒ_{Fn} as a von Neumann algebra. In the following, we shall prove that if ℳ is not prime and X_{1}, … , X_{n}, n ≥ 2, generate ℳ as a von Neumann algebra, then χ(X_{1}, … , X_{n}) = −∞. Hence, ℒ_{Fn} is prime. In fact, we prove a slightly stronger result in the following lemma.
Lemma 2.
Let ℳ be a factor of type II_{1}, ℛ_{1} and ℛ_{2} be mutually commuting hyperfinite subfactors of ℳ. Let P_{1}, P_{2}, … and Q_{1}, Q_{2}, … be projections in ℳ with trace 1/2 that generate ℳ as a von Neumann algebra. Suppose that P_{1}, P_{2}, … commute with ℛ_{1}, Q_{1}, Q_{2}, … commute with ℛ_{2}. If X_{1}, … , X_{n} are selfadjoint elements in ℳ that generate ℳ as a von Neumann algebra, then χ(X_{1}, … , X_{n}) = −∞.
We give an outline of the proof here. The detailed argument will appear elsewhere.
From the assumptions in Lemma 2, we know that, for any positive ω, there are projections P_{1}, … , P_{p} and Q_{1}, … , Q_{q} in ℳ, p, q ∈ ℕ, and selfadjoint polynomials ϕ_{1}, … , ϕ_{n} in the noncommutative ∗ polynomial ring ℂ〈x_{1}, … , x_{p+q}〉 such that where ∥ ∥_{2} is the trace norm (∥X∥_{2}^{2} = τ(X*X), X ∈ ℳ).
From the definition of free entropy (8), we shall estimate certain measurement of all (finitedimensional) selfadjoint matrices that approximate X_{1}, … , X_{n} in joint distributions. For technical reasons, we use the notion of “modified” free entropy (9). More precisely, we estimate the free entropy of X_{1}, … , X_{n} in the presence of P_{1}, … , P_{p} and Q_{1}, … , Q_{q}. When selfadjoint elements A_{1}, … , A_{n} in M_{k} approximate X_{1}, … , X_{n} in joint distributions, there are projections E_{1}, … , E_{p} and F_{1}, … , F_{q} in M_{k} as well, corresponding to elements in certain Grassmann manifolds, that approximate projections P_{1}, … , P_{p} and Q_{1}, … , Q_{q} (in ℳ) in joint distributions. At the same time, A_{j} are close (in tracenorm) to ϕ_{j}(E_{1}, … , E_{p}, F_{1}, … , F_{q}).
From this observation, we are able to reduce the estimate of the free entropy of X_{1}, … , X_{n} to the volume estimate of the image of the cartesian product of the Grassmann manifolds under maps given by (noncommutative) polynomials ϕ_{1}, … , ϕ_{n}. Let k be the degree of the matrices that approximate X_{j}’s, D be an upper bound of the first derivatives of ϕ_{j}’s in the domain of the cartesian product of the Grassmann manifolds and d be the dimension of the manifolds. By using Szarek’s results (11) on nets in unitary groups and Grassmann manifolds, we have that where C is a universal constant, a = max{∥X_{j}∥_{2} + 1 : 1 ≤ j ≤ n}, and Γ(·) is the classical Γfunction.
The assumptions that P_{1}, P_{2}, … commute with ℛ_{1} and Q_{1}, Q_{2}, … commute with ℛ_{2} give restrictions on the dimensions of the Grassmann manifolds. Hence, we can choose the dimension d so that d(p + q)/k^{2} ≤ ω and d(p + q)/k^{2}log(CD ) ≤ log(2C). From Stirling’s formula for the Γfunction, we have Choosing ω arbitrarily small, we have χ(X_{1}, … , X_{n}) = −∞.
Using classical von Neumann algebra techniques, one can show that if ℳ is the tensor product of two factors of type II_{1} (i.e., not prime), then ℳ satisfies all the hypotheses of Lemma 2.
Open Problems
Some questions about decompositions into continuous primes, analogous to simple facts about (discrete)primefactor decomposition, and about continuous primes, themselves, come instantly to mind:
1. Are there infinitely many (nonisomorphic) prime factors of type II_{1}?
2. Is ℒ_{F2}⊗̄ ℒ_{F2} ∗ isomorphic to ℒ_{F2}⊗̄ ℒ_{F2}⊗̄ ℒ_{F2} (“uniqueness” of prime decomposition)?
3. With ℳ a factor of type II_{1}, let p(ℳ) be the set of integers n for which there are prime factors ℳ_{1}, … , M_{n}, n ∈ ℕ, of type II_{1} such that ℳ ≅ ℳ_{1}⊗̄ ⋯ ⊗̄ ℳ_{n}. If there is no such n, let p(ℳ) be {∞}. Does p(ℳ) contain only one number?
Finally, we propose the project of classifying all von Neumann subalgebras of free group factors as an analogue of Connes’s classification of von Neumann subalgebras of the hyperfinite II_{1} factor (12), and ask a question suggested by Lemma 2. Is the relative commutant of a nonatomic injective (or abelian) von Neumann subalgebra of ℒ_{Fn} in ℒ_{Fn} always injective?
Acknowledgments
This work was supported by the National Science Foundation.
Footnotes

↵ email: liming{at}math.mit.edu.

Richard V. Kadison, University of Pennsylvania, Philadelphia, PA
 Received August 6, 1996.
 Accepted August 19, 1996.
 Copyright © 1996, The National Academy of Sciences of the USA
References
 ↵
 Murray F J,
 von Neumann J
 ↵
 Murray F J,
 von Neumann J
 ↵
 Murray F J,
 von Neumann J
 ↵
 von Neumann J
 ↵
 Popa S
 ↵
 Sakai S
 ↵
 Voiculescu D
 ↵
 Voiculescu D
 ↵
Voiculescu, D., Dykema, K. & Nica, A. (1992) Free Random Variables, Centre de Recherches Mathématiques Université de Montréal Monograph Series (Am. Math. Soc., Providence, RI), Vol. 1.
 ↵
 Lin BL
 Szarek S J
 ↵
 Connes A