II1 Factors Research Articles

Exchange Relation Planar Algebras

An inclusion of II1 factors N⊂M of finite index has as an invariant, a double sequence of finite-dimensional algebras known as the standard invariant. Planar algebras were introduced by V. Jones as a geometric tool for computing standard invariants of existing subfactors as well as generating standard invariants for new subfactors. In this paper we define a class of planar algebras, termed exchange relation planar algebras, that provides a general framework for understanding several classes of known subfactor inclusions: the Fuss–Catalan algebras (i.e. those coming from the presence of intermediate subfactors) and all depth 2 subfactors. In addition, we present a new class of planar algebras (and thus a new class of subfactors) coming from automorphism subgroups of finite groups.

Multi-Sided Braid Type Subfactors

AbstractWe generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II1 factor R in [E1]—which arise by considering unitary braid representations with certain properties—to multisided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types A, B, C, D. We also compute the (first) relative commutants.

REMARKS ON THE SIMILARITY DEGREE OF AN OPERATOR ALGEBRA

The "similarity degree" of a unital operator algebra A was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type II1 factor with property Γ is at most 5, improving on a previous bound (≤ 44) due to E. Christensen.

Remarks on free products with respect to non-tracial states

Since the appearance of D. Voiculescu's free probability theory detailed study on free products of von Neumann algebras (based on his theory) has been carried by several authors, and free group factors are their typical examples. Indeed, his theory gives us very powerful tools to analyze free group factors. On the other hand, at the beginning of the 80's (i.e., before the appearance of D. Voiculescu's theory) S. Popa studied in detail free group factors and related type II1 factors and got several deep results ([P1], [P2]). He used there some techniques for type II1 factors related to his notion ``orthogonal pairs''. Then, at the mid 90's L. Ge generalized them to more general (but type II1) setting and solved some problems on maximal injective subalgebras ([G]). In this note, we will try to generalize the techniques to more general and not necessary type II1 setting. These generalizations are rather natural, but enable us to obtain some results on free products with respect to non-tracial states. In fact, as applications a factoriality and type classification result for free products and a result on (normal) conditional expectations from free products onto their subalgebras will be obtained. Based on the latter result we will discuss the recent question posed by M. Izumi, R. Longo and S. Popa ([ILP]): Let M L N be von Neumann algebras with a normal conditional expectation from M onto N. If N 0 \M ˆ C1, does there exist a normal conditional expectation from M onto L ?

A Galois Correspondence for II1 Factors and Quantum Groupoids

We establish a Galois correspondence for finite quantum groupoid actions on II1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows us to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain finite-dimensional C*-algebras related to it.

Free Composition of Paragroups

As a first step we define an extremality condition for paragroups, then we prove that any given extremal paragroup is equivalent to a Popa system constructed with the string algebras associated with the starting paragroup. As a corollary, we get that any extremal paragroup can be thought of as the paragroup coming from a (generally non-hyperfinite) extremal inclusion of II1 factors. Then we define a free composition for paragroups, using as a model the definition by V. Jones and D. Bisch of free composition of subfactors; and we prove that for any two given paragroups P1 and P2 there always exists a third paragroup P which realizes the free composition of P1 and P2 .

A Characterization of Depth 2 Subfactors of II1 Factors

We characterize finite index depth 2 inclusions of type II1 factors in terms of actions of weak Kac algebras and weak C*-Hopf algebras. If N⊂M⊂M1⊂M2⊂… is the Jones tower constructed from such an inclusion N⊂M, then B=M′∩M2 has a natural structure of a weak C*-Hopf algebra and there is a minimal action of B on M1 such that M is the fixed point subalgebra of M1 and M2 is isomorphic to the crossed product of M1 and B. This extends the well-known results for irreducible depth 2 inclusions.

Free entropy with respect to a completely positive map

Given an element X in a tracial von Neumann algebra ( M , τ), a unital subalgebra B ⊂ M , and a completely positive map η: B → B , we define the free Fisher information Φ*( X : B , η) of X relative to B with respect to η. The definition is a generalization of Voiculescu's definition of Φ*( X : B ), which corresponds to η = τ. We show that many facts about Φ*( X : B ) generalize to Φ*( X : B , η). As an application, we show that if B is commutative and X is singular with respect to B , then Φ*( X : B ) is infinite. Φ*( X : B , η) is minimized when X is a B -valued semicircular variable with covariance η. Therefore, Φ*(· : B , η) is an appropriate quantity to study B -valued semicircular systems with arbitrary covariance. Associating to a pair of completely positive maps μ, η: B → B the quantity Φ*( X μ : B , η), where X μ is a B -valued semicircular variable with covariance μ, allows us to measure "absolute continuity" of μ and η. To a completely positive map μ we can naturally associate the number Φ*( X μ : B , id). If μ is the conditional expectation onto a subfactor A ⊂ B of a II 1 factor B , we show that Φ*( X E : B , id) is equal to the Jones' index [ B ⊂ A ].

On the Fixed-point Algebra under a Minimal Free Product-type Action of the Quantum Group SUq(2)

In this note, a minimal action of SUq(2) is investigated as a natural continuation of our previous work. The type (in the sense of Murray-von Neumann and Connes) of the fixed-point algebra is determined. As a simple application, an example of a pair of a type II1 factor and its irreducible subfactor with an arbitrary index greater than four is constructed.

Cartan subalgebras of finite von Neumann algebras

Sorin Popa [Po 1] [Po 2] has proved several results on the existence and properties of hyperfinite subfactors R of a type II1 factor M with the relative commutant R0 \M of R in M equal to C1. These theorems have been used in various cohomology calculations [CES, CS, PS, CPSS], as averaging over an amenable subgroup that generates the hyperfinite subfactor is a major step in showing that the continuous and completely bounded Hochschild cohomology groups are equal. It has seemed reasonable that Popa's results could be extended from factors to general type II1 von Neumann algebras by direct integral theory [KR, Chapter 4]. However, we do not know of such an attempt. Direct integral theory can be used directly to prove cohomology is zero and deduce results like [CPSS, Theorems 5.4 and 5.5] however these theorems on the continuous Hochschild cohomology for a von Neumann algebra with Cartan subalgebras are deduced from the theorems in this paper. This paper provides direct proofs of Popa's main two results in [Po 1] by modifying his proofs using an interpolation type result for projections in a maximal abelian selfadjoint sub-algebra (masa) of the type II1 algebra. This introduction contains a more detailed description of how our results extend Popa's, the basic definitions, and a brief reference to their use in the calculation of Hochschild cohomology groups in von Neumann algebras. Though averaging plays an important role in calculating the Hochschild cohomology of von Neumann algebras for all types I ; II1; II1; III of von Neumann algebras (see [Ri]), the results proved here are only used in the type II1 situation. The reason is that the type I 's are already trivially hyperfinite, and the type II1 and III von Neumann algebras may be handled by their stability under tensoring with B H†. This tensor factor B H† of M in the II1 and III cases gives a suitable hyperfinite algebra over which to average. Popa [Po 1] restricts his attention to type II II1 and II1† factors. MATH. SCAND. 85 (1999), 105^120

On subdiagonal algebras for subfactors

On subdiagonal algebras for subfactors

Outer Conjugacy Classes of Trace Scaling Automorphisms of Stable UHF Algebras

Connes [C] classified automorphisms of the II1 approximately finite-dimensional (AFD) factor R0;1 up to outer conjugacy. An automorphism of R0;1 determines a modulus or scaling factor on a trace ˆ . It follows from this classification and [CT] that automorphisms which do not leave a trace invariant are classified up to conjugacy by their modulus; for each 2 0; 1† there is up to conjugacy a unique automorphism of R0;1 which scales the trace by . Consequently, for such there is a unique factor of type III the Powers factor R whose associated type II1 factor is R0;1; necessarily, R ˆ R0;1 Z. A key idea for model building in Connes's work on the classification of automorphisms of AFD factors [C] and in subsequent work of Jones [J], Ocneanu [O], and Kawahigashi, Sutherland and Takesaki [KST] on amenable group actions is a non-commutative Rohlin lemma for an aperiodic automorphism of a finite von Neumann algebra. Recall that an automorphism of a von Neumann algebra M is said to be properly outer if the restriction to Me is outer for each non-zero invariant projection e, and aperiodic if every non-zero power is properly outer. Connes showed that an automorphism of M is properly outer if and only if for each non-zero projection e in M, and > 0, there exists a non-zero projection f in M such that kf f †k 0, there exist projections F1; ;Fn in M such that Pn jˆ1 Fj ˆ 1 and Fj† y Fj‡1 2 , j ˆ 1; . . . ; n, where Fn‡1 ˆ F1 and kxk2 ˆ x x† 1 2. MATH. SCAND. 83 (1998), 74^86

The noncommutative Hilbert transform approach to free entropy

Free entropy is the entropy quantity with the right behavior with respect to free independence. We began the study of free entropy with the one-variable case [7], taking as definition, based on random matrix heuristics, the negative of the logarithmic energy of the distribution. We then found in [8] a general definition of χ(X1, . . . , Xn), the free entropy of an n-tuple of noncommutative random variables in a tracial W ∗-probability space, using matricial microstates. The parallel to classical entropy suggests a number of natural properties free entropy should satisfy and several of these have been established ([7, 8, 10, 4, 6]). This new machinery has had striking applications to the solution of some old problems on von Neumann algebras ([9, 2, 3, 1]). On the other hand, from the point of view of this parallel to classical entropy, the theory is still incomplete, which keeps certain further applications to von Neumann algebras out of reach. The main reason for the difficulties is that we know very little about matrix-approximants to elements in type II1 von Neumann algebras. Even the general existence question for such approximants, which would be a first step in this direction, is unsolved and coincides with Alain Connes’ well known problem about embedding type II1 factors into the ultraproduct of the hyperfinite II1 factor. Under these circumstances we began in [11] to look for another approach to free entropy which avoids matricial microstates. This should not be viewed only as an alternative to microstates, since some of the results actually can be used within the microstates approach. This note summarizes the lecture we intended to give at the Gdansk Quantum Probability meeting about our current work towards a “microstates-free” approach to free entropy [11].

Singular extensions of the trace and the relative Dixmier property in the type 𝐼𝐼₁ factors

If N ⊂ M N \subset M is an inclusion of type I I 1 \mathrm {II}_1 factors with N ′ ∩ M = C I , N’\cap M = \mathbf {C} I, we study the connection between the existence of singular states on M M which extend the trace on N N and the Dixmier approximation property in M M with unitaries in N . N. We also prove the existence of singular conditional expectations from certain free product factors onto irreducible hyperfinite subfactors.

Entropy for extensions of Bernoulli shifts

AbstractWe give a condition for automorphisms α and β on finite von Neumann algebras which induces the tensor product formula for entropies: H(α ⊗ β) = H(α) + H(β). As an application, the Bernoulli shift (1/n, 1/n, …, 1/n) has extensions to ergodic outer automorphisms {αk; k = 1,2, …} on the hyperfinite II1 factor R with the entropies H(αk) = (1/2)kn log n.

Factors generated by direct sums of II1 factors

Factors generated by direct sums of II1 factors

Singular Traces on Semifinite von Neumann Algebras

Singular traces are constructed on a general semifinite von Neumann algebra, thus generalizing the result of Dixmier (C.R. Acad. Sci. Paris262, 1966.) Moreover our technique produces singular traces on type II1 factors. Such traces, though vanishing on all bounded operators, are non trivial on the *-algebra of affiliated unbounded operators. On a semifinite factor, we show that all traces are given by a dilation invariant functional on the cone of positive decreasing functions on [0, ∞), and we prove that the existence of a singular trace which is non trivial on a given operator is equivalent to an eccentricity condition on the singular values function, a result which generalizes the theorem given in (S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, J. Funct. Anal., to appear.) for B(H).

ENTROPY FOR CANONICAL SHIFTS AND STRONG AMENABILITY

When N⊂M is an inclusion of type II1 factors with finite index, let Γ be the canonical shift on the von Neumann algebra R generated by the derived tower of N⊂M, H(R|Γ(R)) the relative entropy, and H(Γ) the dynamical entropy. We characterize the equality cases of the inequalities [Formula: see text] in connection with the subexponential growth condition and the strong amenability condition. Also the topological entropy of Γ and the convergence of entropy increments on the derived tower are discussed.

An example of an irreducible subfactor of the hyperfinite II1 factor with rational, noninteger index.

An example of an irreducible subfactor of the hyperfinite II1 factor with rational, noninteger index.

On a Family of Almost Commuting Endomorphisms

If gi is a central sequence of unitaries in a II1 factor, we show that under certain circumstances limn → ∞ Ad(∏ni = 1gi) is an automorphism. Examples come naturally from solutions of the Yang-Baxter equation with a spectral parameter, and the study of endomorphisms of the Cuntz algebra.