Some free entropy dimension inequalities for subfactors

Suppose $N \subset M$ is an inclusion of $II_1$-factors of finite index. If $N$ can be generated by a finite set of elements, then there exist finite generating sets $X$ for $N$ and $Y$ for $M$ such that $\delta_0(X) \geq \delta_0(Y)$, where $\delta_0$ denotes Voiculescu's microstates (modified) free entropy dimension. Moreover given $\epsilon >0$ one has $\delta_0(F) \geq \delta_0(G) \geq ([M:N]^{-2} -\epsilon) \cdot (\delta_0(F) -1) + 1 - \epsilon$ for certain generating sets $F$ for $N$ and $G$ for $M$.Comment: 11 page

Strongly 1-bounded von Neumann algebras

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $\alpha >0$, F is $\alpha$-bounded if the free packing $\alpha$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x in F with finite free entropy. It is shown that if M is strongly 1-bounded, then any finite set of selfadjoint generators G for M is 1-bounded and the microstates free entropy dimension of G is less than or equal to 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and the microstates free entropy dimension is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II_1-factors which have property Gamma, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of SL_n(Z), n >2. If M and N are strongly 1-bounded and their intersection is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II_1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.Comment: 15 pages, added references, minor correction

The free entropy dimension of hyperfinite von Neumann algebras

Suppose M is a hyperfinite von Neumann algebra with a tracial state $\phi$ and $\{a_1,...,a_n\}$ is a set of selfadjoint generators for M. We calculate $\delta_0(a_1,...,a_n)$, the modified free entropy dimension of $\{a_1,...,a_n\}$. Moreover we show that $\delta_0(a_1,...,a_n)$ depends only on M and $\phi$. Consequently $\delta_0(a_1,...,a_n)$ is independent of the choice of generators for M. In the course of the argument we show that if $\{b_1,...,b_n\}$ is a set of selfadjoint generators for a von Neumann algebra R with a tracial state and $\{b_1,...,b_n\}$ has finite dimensional approximants, then for any $b\in R$ $\delta_0(b_1,...,b_n)\geq \delta_0(b)$. Combined with a result by Voiculescu this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then $\delta_0(b_1,...,b_n)=1$.Comment: 34 pages, minor correction

The Rank Theorem and L2L^2-invariants in Free Entropy: Global Upper Bounds

Using an analogy with the rank theorem in differential geometry, it is shown that for a finite $n$-tuple $X$ in a tracial von Neumann algebra and any finite $m$-tuple $F$ of $*$-polynomials in $n$ noncommuting indeterminates, \begin{eqnarray*} \delta_0(X) & \leq & \text{Nullity}(D^sF(X)) + \delta_0(F(X):X) \end{eqnarray*} where $\delta_0$ is the (modified) microstates free entropy dimension and $D^sF(X)$ is a kind of derivative of $F$ evaluated at $X$. When $F(X) =0$ and $|D^sF(X)|$ has nonzero Fuglede-Kadison-L\"uck determinant, then $X$ is $\alpha$-bounded in the sense of \cite{j3} where $\alpha = \text{Nullity}(D^sF(X))$. Using Linnell's $L^2$ integral domain results in \cite{l} as well as Elek and Szab\'o's work on L\"uck's determinant conjecture for sofic groups in \cite{es} the following result is proven. Suppose $\Gamma$ is a sofic, left-orderable, discrete group with 2 generators and $\Gamma \neq \{0\}$. The following conditions are equivalent: (1) $\Gamma \not\simeq \mathbb F_2$. (2) $L(\Gamma) \not\simeq L(\mathbb F_2)$. (3) $L(\Gamma)$ is strongly $1$-bounded. (4) $\delta_0(X) = 1$ for any finite set of generators $X$ for $L(\Gamma)$. From Brodski\u{i} and Howie's results on local indicability (\cite{b}, \cite{h}), it follows that a sofic, torsion-free, one-relator group von Neumann algebra on two generators with nontrivial relator is strongly $1$-bounded. It also follows from the residual solvability of the positive one relator groups (\cite{baum}) that a one-relator group von Neumann algebra on two generators whose relator is a nontrivial, positive, non-proper word in the generators is strongly $1$-bounded.Comment: 67 page

A Free Entropy Dimension Lemma

Suppose M is a von Neumann algebra with normal, tracial state phi and {a_1,...,a_n} is a set of self-adjoint elements in M. We provide an alternative uniform packing description of delta_0(a_1,...,a_n), the modified free entropy dimension of {a_1,...,a_n}.Comment: 6 pages, minor correction

A propagation property of free entropy dimension

Let M be a tracial von Neumann algebra and A be a weakly dense unital C*-subalgebra of M. We say that a set X is a W*-generating set for M if the von Neumann algebra generated by X is M and that X is a C*-generating set for A if the unital C*-algebra generated by X is A. For any finite W*-generating set X for M we show that $\delta_0(X) \leq sup {\delta_0(Y): Y is a finite C*-generating set for A}$ where $\delta_0$ denotes the microstates free entropy dimension. It follows that if $sup {\delta_0(Y): Y is a finite C*-generating set for C*_{red}(\mathbb F_2)} < \infty$, then the free group factors are all nonisomorphic.Comment: 4 pages, minor correction

Fractal entropies and dimensions for microstate spaces

Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite dimensional algebra or where the set consists of a single selfadjoint. We show that the free Hausdorff dimension becomes additive for such sets in the presence of freeness.Comment: 25 pages, minor corrections, lifting of restrictive conditions for the computation of dimension of a single selfadjoint, additional lemma in section

All generating sets of all property T von Neumann algebras have free entropy dimension ≤1\leq 1

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of X, $\delta_0(X)$, is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then $\delta_0(X)=1$ and otherwise, \delta_0(X)=-\infinity. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(\mathbb Z)$ as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.Comment: 6 page

Graphs of functions and vanishing free entropy

Suppose X is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M. If z is a selfadjoint element in M and for some selfadjoint element y in the von Neumann algebra generated by X $\delta_0(y, z) < \delta_0(y) + \delta_0(z)$, then $\chi(X \cup \{z\}) = -\infty$ (here $\chi$ and $\delta_0$ denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X, then $\chi(X \cup \{z\}) = -\infty$. The statement and its proof are motivated by geometric-measure-theoretic results on graphs of functions. A similar statement for the nonmicrostates free entropy is obtained under the much stronger hypothesis that z lies in the algebra generated by X.Comment: 14 page

Dimension and Entropy Computations for L(Fr)L(F_r)

We show that certain generating sets of Dykema and Radulescu for $L(F_r)$ have free Hausdorff dimension r and nondegenerate free Hausdorff r-entropyComment: 9 page