2.1 Tracial von Neumann algebras and noncommutative laws

A tracial von Neumann algebra is a pair $(M,\tau )$ where M is a von Neumann algebra and $\tau \colon M\to {\mathbb C}$ is a faithful, normal, tracial state. We will primarily be interested in cases where M is diffuse, that is, it has no nonzero minimal projections. An interesting class of tracial von Neumann algebras are the group von Neumann algebras. Given a discrete group G the left regular representation $\lambda \colon G\to \mathcal {U}(\ell ^{2}(G))$ is given by

$$\begin{align*}(\lambda(g)\xi)(h)=\xi(g^{-1}h)\text{ for }\xi\in\ell^{2}(G),g,h\in G.\end{align*}$$

The group von Neumann algebra $L(G)$ is defined to be

$$\begin{align*}\overline{\operatorname{\mathrm{span}}\{\lambda(g):g\in G\}}^{SOT}. \end{align*}$$

The group von Neumann algebra can be turned into a tracial von Neumann algebra by defining $\tau \colon L(G)\to {\mathbb C}$ by $\tau (x)= \left \langle {x(\delta _{1}),\delta _{1}} \right \rangle $ . We may also view $\mathbb {M}_n({\mathbb C})$ as a tracial von Neumann algebra with the tracial state $\operatorname {\mathrm {tr}}_n$ given by

$$\begin{align*}\operatorname{\mathrm{tr}}_{n}(A) = \frac{1}{n}\sum_{i=1}^{n}A_{ii}. \end{align*}$$

The group von Neumann algebra is diffuse if and only if G is infinite. For a von Neumann algebra M, we use $M_{\operatorname {\mathrm {sa}}}$ for the self-adjoint elements of $M,$ and $\mathcal {U}(M)$ for the unitary elements of M.

Since tracial von Neumann algebras $(M,\tau )$ with M abelian correspond precisely to probability spaces, we may think of tracial von Neumann algebras as (a special case of) noncommutative probability spaces. Following this intuition, given a tracial von Neumann algebra $(M,\tau )$ and $1\leq p <\infty $ , we define the $\|\cdot \|_{p}$ on M by

$$\begin{align*}\|x\|_{p}=\tau(|x|^{p})^{1/p},\text{ where }|x|=(x^{*}x)^{1/2}. \end{align*}$$

It can be shown [Reference DixmierDix53] that this is indeed a norm on M. We use the notation $ \left \lVert {x} \right \rVert _\infty $ for the operator norm. Moreover, the definition of the norms can be extended to tuples by

$$\begin{align*}\left\lVert {(x_1,\dots,x_d)} \right\rVert _p = \begin{cases} \left( \sum_{j=1}^d \tau(|x_j|^p) \right)^{1/p}, & p \in [1,\infty), \\ \max_{j=1,\dots,d} \left\lVert {x_j} \right\rVert , & p = \infty. \end{cases} \end{align*}$$

If $(M,\tau )$ is viewed as a noncommutative probability space, then its elements maybe viewed as noncommutative random variables. In fact, a d-tuple $x = (x_1,\dots ,x_d) \in M_{\operatorname {\mathrm {sa}}}^d$ is the noncommutative analog of an ${\mathbb R}^d$ -valued random variable. Although one cannot define the law (or probability distribution) of x as a classical measure, we may define its noncommutative law as a certain linear function on a noncommutative polynomial algebra, just like the probability distribution d-tuple X of bounded classical random variables defines a map ${\mathbb C}[t_1,\dots ,t_d] \to {\mathbb C}$ sending p to $\mathbb {E}[p(X)]$

For $d \in {\mathbb N}$ , we let ${\mathbb C} \left \langle {t_1,\cdots ,t_d} \right \rangle $ be the algebra of noncommutative polynomials in d formal variables $t_1$ , …, $t_d$ , that is, the free ${\mathbb C}$ -algebra with d-generators. We give ${\mathbb C} \left \langle {t_{1},\cdots ,t_{d}} \right \rangle $ the unique $*$ -algebra structure which makes the $t_{j}$ self-adjoint. By universality, if A is any $*$ -algebra and $x=(x_{1},\cdots ,x_d)\in A^d$ is a self-adjoint tuple, then there is a unique $*$ -homomorphism ${\mathbb C} \left \langle {t_{1},\cdots ,t_d} \right \rangle \to A$ which sends $t_{j}$ to $x_{j}$ . For $p \in {\mathbb C} \left \langle {t_{1},\cdots ,t_d} \right \rangle $ , we use $p(x)$ for the image of p under this $*$ -homomorphism. Given a tracial von Neumann algebra $(M,\tau )$ and $x \in M^d_{\operatorname {\mathrm {sa}}}$ , we define the law of x, denoted $\ell _{x}$ , to be the linear functional $\ell _{x}\colon {\mathbb C} \left \langle {t_1,\cdots ,t_d} \right \rangle \to {\mathbb C}$ given by

$$\begin{align*}\ell_{x}(f)=\tau(f(x)). \end{align*}$$

The noncommutative laws can be characterized as follows.

Let $\Sigma _{d,R}$ be the set of all linear maps ${\mathbb C} \left \langle {t_1,\dots ,t_d} \right \rangle \to {\mathbb C}$ satisfying the equivalent conditions of Proposition 2.1. We equip $\Sigma _{d,R}$ with the weak $^*$ topology, that is, the topology of pointwise convergence on ${\mathbb C} \left \langle {t_1,\dots ,t_d} \right \rangle $ . It is easy to see that $\Sigma _{d,R}$ is compact and metrizable using Proposition 2.1 (ii).

2.2 Microstate spaces and $1$ -bounded entropy

Let $(M,\tau )$ be a diffuse tracial von Neumann algebra, and $x\in M_{\operatorname {\mathrm {sa}}}^d$ for some $d \in {\mathbb N}$ with $W^{*}(x)=M$ . Suppose that $ \left \lVert {x} \right \rVert _\infty \leq R$ . Following [Reference VoiculescuVoi94], for each open set $\mathcal {O}$ of $\Sigma _{d,R}$ and $N \in {\mathbb N}$ , we define

$$\begin{align*}\Gamma_R^{(n)}({\mathcal O}) = \{X \in \mathbb{M}_n({\mathbb C})_{\operatorname{\mathrm{sa}}}^d: \ell_X \in \mathcal{O} \}. \end{align*}$$

When $\mathcal {O}$ is a neighborhood of $\ell _x$ , we call $\Gamma _R^{(n)}(\mathcal {O})$ a microstate space for x.

Given $d,n\in {\mathbb N}$ , $p\in [1,\infty ]$ , $\varepsilon>0$ and $\Omega ,\Xi \subseteq {\mathbb M}_{n}({\mathbb C})^{d}$ then $\Xi $ is said to $(\varepsilon , \left \lVert {\cdot } \right \rVert _p)$ -cover $\Omega $ if, for every $A\in \Omega $ , there is a $B\in \Xi $ with $\|A-B\|_{p}<\varepsilon $ . We define the covering number of $\Omega \subseteq {\mathbb M}_{n}({\mathbb C})^{d}$ , denote $K_{\varepsilon }(\Omega ,\|\cdot \|_{p})$ to be the minimal cardinality of a set that $(\varepsilon ,\|\cdot \|_{p})$ -covers $\Omega $ . We will use covering numbers for different values of p in Section 3.

While these covering numbers work for many purposes, for unitarily invariant subsets of matrices, it is also natural to consider the orbital covering numbers modulo unitary conjugation. Given $n \in {\mathbb N}$ , $\varepsilon> 0$ and $\Omega ,\Xi \subseteq \mathbb {M}_{n}({\mathbb C})^d$ we say that $\Xi $ orbitally $(\varepsilon , \left \lVert {\cdot } \right \rVert _2)$ -covers $\Omega $ if for every $A\in \Omega $ , there is a $B \in \Xi $ and an $n \times n$ unitary matrix V so that

$$\begin{align*}\left\lVert {A-VBV^*} \right\rVert _2 < \varepsilon. \end{align*}$$

We define the orbital covering number $K_{\varepsilon }^{\text {orb}}(\Omega , \left \lVert {\cdot } \right \rVert _2)$ as the minimal cardinality of a set of $\Omega _0$ that orbitally $(\varepsilon , \left \lVert {\cdot } \right \rVert _2)$ -covers $\Omega $ . Since we will usually be concerned with $\|\cdot \|_{2}$ -norms we will frequently drop $\|\cdot \|_{2}$ from the notation and use $K_{\varepsilon }^{\text {orb}}(\Omega )$ instead of $K_{\varepsilon }^{\text {orb}}(\Omega ,\|\cdot \|_{2})$ . Let $R\in [0,\infty )$ be such that $ \left \lVert {x} \right \rVert _\infty < R$ .

The $1$ -bounded entropy is defined in terms of the exponential growth rate of covering numbers of $\Gamma _R^{(n)}(\mathcal {O})$ (up to unitary conjugation) as $n \to \infty $ for neighborhoods $\mathcal {O}$ of $\ell _x$ . For a weak $^{*}$ -neighborhood $\mathcal {O}$ Of $\ell _{x}$ , we define

$$ \begin{align*} h_{R,\varepsilon}(\mathcal{O})& := \limsup_{n \to\infty}\frac{1}{n^2}\log K_{\varepsilon}^{\operatorname{\mathrm{orb}}}(\Gamma_{R}^{(n)}(\mathcal{O})), \\ h_{R,\varepsilon}(x) &:= \inf_{\mathcal{O} \ni \ell_x} h_{R,\varepsilon}(\mathcal{O}), \end{align*} $$

where the infimum is over all weak $^{*}$ -neighborhoods $\mathcal {O}$ of $\ell _{x}$ . We then define

$$\begin{align*}h_R(x) := \sup_{\varepsilon> 0} h_{R,\varepsilon}(x). \end{align*}$$

In [Reference HayesHay18], it is shown that $h_R(x)$ is independent of R provided that $ \left \lVert {x} \right \rVert _\infty \leq R$ , and hence we may unambiguously denote it by $h(x)$ . In fact, $h(x)$ only depends on the von Neumann algebra generated by x. Hence, for every finitely generated tracial von Neumann algebra M, we may define $h(M)$ as $h(x)$ for some generating tuple x. If M is not a factor, then the $1$ -bounded entropy depends upon the trace we choose on M. We will use $h(M,\tau )$ for a tracial von Neumann algebra $(M,\tau )$ if we wish to emphasize the dependence of the $1$ -bounded entropy on $\tau $ . However, we will typically suppress this from the notation and just use $h(M)$ unless there is a possibility of confusion. The definition of h can be extended even to the case where M is not finitely generated; see [Reference HayesHay18]. However, our results only require a direct appeal to the definition in the finitely generated setting. The importance of $1$ -bounded entropy to the study of strong $1$ -boundedness is encapsulated by the following result.

Because of this result, we will not use Jung’s original definition of strongly $1$ -bounded [Reference JungJun07b] and will instead show that algebras are strongly $1$ -bounded by showing that they have finite $1$ -bounded entropy.

2.3 Properties and applications of $1$ -bounded entropy

Since we will use general properties of $1$ -bounded entropy frequently, we list some of the main ones here. To state these we also need the notion of the $1$ -bounded entropy of N in the presence of M, denoted $h(N:M)$ , defined for an inclusion $N\leq M$ of tracial von Neumann algebras. The $1$ -bounded entropy in the presence is defined by modifying the definition of $1$ -bounded entropy above to only measure the size of the space of microstates for N which have an extension to microstates for M. We will not need the precise definition here, so we refer the reader to [Reference HayesHay18, Definition A.2] for details. We also use the microstates free entropy dimension $\delta _{0}(x)$ due to Voiculescu [Reference VoiculescuVoi96, Definition 6.1] for a tuple $x\in M^{k}_{\operatorname {\mathrm {sa}}}$ in a tracial von Neumann algebra $(M,\tau )$ . Here and throughout the paper, $\mathcal {R}$ refers to the hyperfinite $\textrm {II}_{1}$ -factor. For an inclusion $N\leq M$ , we let $\mathcal {N}_{M}^{wq}(N)$ be the set of $u\in \mathcal {U}(M)$ so that $uNu^{*}\cap N$ is diffuse. We assume all inclusions list below are trace preserving. We now list some general properties the $1$ -bounded entropy here.

1. 1. $h(M)=h(M:M)$ for every diffuse tracial von Neumann algebra $(M,\tau )$ ,

2. 2. $h(N:M)\geq 0$ if $N\leq M$ if diffuse and M embeds into an ultrapower of $\mathcal {R}$ , and $h(N:M)=-\infty $ if M does not embed into an ultrapower of $\mathcal {R}$ . (Exercise from the definitions.)

3. 3. $h(N_{1}:M_{1})\leq h(N_{2}:M_{2})$ if $N_{1}\leq N_{2}\leq M_{2}\leq M_{1}$ , if $N_{1}$ is diffuse. (Exercise from the definitions.)

4. 4. $h(N:M)\leq 0$ if $N\leq M$ and N is diffuse and hyperfinite. (Exercise from the definitions.)

5. 5. $h(M)=\infty $ if $M = \mathrm {W}^*(x_{1},\cdots ,x_{n})$ , where $x_{j}\in M_{sa}$ for all $1\leq j\leq n$ and $\delta _{0}(x_{1},\cdots ,x_{n})>1$ . For example, this applies if $M=L({\mathbb F}_{n})$ , for $n>1$ . (This follows from Theorem 2.2 and [Reference JungJun07b, Corollary 3.5]).)

6. 6. $h(N_{1}\vee N_{2}:M)\leq h(N_{1}:M)+h(N_{2}:M)$ if $N_{1},N_{2}\leq M$ and $N_{1}\cap N_{2}$ is diffuse. (See [Reference HayesHay18, Lemma A.12].)

7. 7. Suppose that $(N_{\alpha })_{\alpha }$ is an increasing chain of diffuse von Neumann subalgebras of a von Neumann algebra M. Then

   $$\begin{align*}h\left(\bigvee_{\alpha}N_{\alpha}:M\right)=\sup_{\alpha}h(N_{\alpha}:M).\end{align*}$$
   (See [Reference HayesHay18, Lemma A.10].)

8. 8. $h(N:M)=h(N:M^{\omega })$ if $N\leq M$ is diffuse, and $\omega $ is a free ultrafilter on an infinite set. (See [Reference HayesHay18, Proposition 4.5].)

9. 9. $h(\mathrm {W}^*(\mathcal {N}_{M}^{wq}(N)):M)=h(N:M)$ if $N\leq M$ is diffuse. Here, $\mathcal {N}_{M}^{wq}(N)=\{u\in \mathcal {U}(M):uN u^{*}=N\text { is diffuse}\}$ . (See [Reference HayesHay18, Theorem 3.8 and Proposition 3.2].)

10. 10. Let I be a countable set and $M=\bigoplus _{i\in I}M_{i}$ with $M_{i}$ diffuse for all i. Suppose that $\tau $ is a faithful trace on M and that $\lambda _{i}$ is the trace of the identity on $M_{i}$ . Endow $M_{i}$ with the trace $\tau _{i}=\frac {\tau |_{M_{i}}}{\lambda _{i}}$ . Then

    $$\begin{align*}h(M,\tau)\leq \sum_{i}\lambda_{i}^{2}h(M_{i},\tau_{i}).\end{align*}$$
    (See [Reference HayesHay18, Proposition A.13].)

The above axioms imply that all the von Neumann algebras in the following list have nonpositive $1$ -bounded entropy:

• • hyperfinite algebras,

• • factors with property gamma,

• • nonprime von Neumann algebras,

• • algebras with diffuse center,

• • algebras with diffuse, regular hyperfinite subalgebras (e.g., if the algebra has a Cartan subalgebra),

For proofs, see [Reference Hayes, Jekel, Nelson and SinclairHJNS21, Section 1.2]. The class of algebras with nonpositive $1$ -bounded entropy is also closed under direct sums (by item 10). Additionally, by item 9 if $N\leq M$ is regular (i.e., $W^{*}(\mathcal {N}_{M}(N))=M$ ), and $h(N:M)\leq 0$ , then $h(M)\leq 0$ . This applies if $h(N)\leq 0$ . In particular, if N is hyperfinite, or has gamma, or is not prime, or has diffuse center, and if N is regular in M, then $h(M)\leq 0$ .

2.4 Property (T)

Property (T) for groups, due to Kazhdan [Reference KaždanKaž67], is defined as follows. If $\pi : G \to \mathcal {U}(\mathcal {H})$ is a unitary representation of the group G on a Hilbert space, then a vector $\xi \in \mathcal {H}$ is said to be invariant if $\pi (g) \xi = \xi $ for all $g \in G$ . We say that the group G has property (T) if there exists a finite $F \subseteq G$ and $\delta> 0$ such that for every unitary representation $\pi $ on $\mathcal {H}$ , if there exists a nonzero $\xi \in \mathcal {H}$ such that $ \left \lVert {\pi (g) \xi - \xi } \right \rVert < \delta $ for all $g \in F$ , then $\mathcal {H}$ contains a nonzero invariant vector.

It turns out that if G has property (T), then there exists a finite set $F \subseteq G$ and some $\gamma> 0$ such that for every unitary representation $\pi $ of G on a Hilbert space $\mathcal {H}$ , we have

$$\begin{align*}\left\lVert {\xi - P_{\operatorname{invariant}}\xi} \right\rVert \leq \frac{1}{\gamma} \sum_{g \in F} \left\lVert {\pi(g)\xi - \xi} \right\rVert , \end{align*}$$

where $P_{\operatorname {invariant}}: H \to H$ is the projection onto the subspace of invariant vectors. Such a set F is called a Kazhdan set and $\gamma $ is called the Kazhdan constant. For a proof, see [Reference Bekka, De la Harpe and ValetteBdlHV08, Definition 1.1.3, Proposition 1.1.9]. It is sometimes notationally convenient to list the elements of F as a tuple $(g_1,\dots ,g_{|F|})$ , which we call a Kazhdan tuple.

The von Neumann algebraic analogue of property (T), introduced by Connes and Jones [Reference Connes and JonesCJ85] for $\mathrm {II}_1$ factors and Popa [Reference PopaPop06a] for general tracial von Neumann algebras, uses Hilbert bimodules over a tracial von Neumann algebra M in place of group representations.

There is analogy between central vectors in a Hilbert M-M-bimodule and invariant vectors in a group representation motivated by the way in which representations of a discrete group G naturally give rise to bimodules over the group von Neumann algebra $L(G)$ . Let $\lambda $ and $\rho $ denote the left and right regular representations of G on $\ell ^2(G)$ . If $\pi $ is a representation of G on $\mathcal {H}$ , then $\pi \otimes \lambda $ is a left representation on $H \otimes \ell ^2(G)$ and $1 \otimes \rho $ is a right representation on $H \otimes \ell ^2(G)$ . These representations extend to normal left/right actions of $L(G)$ , making $H \otimes \ell ^2(G)$ into a Hilbert $L(G)$ - $L(G)$ -bimodule. If $\xi $ is an invariant unit vector in $\mathcal {H}$ , then $\xi \otimes \delta _e$ is a central and bitracial vector in $H \otimes \ell ^2(G)$ . Hence, property (T) for tracial von Neumann algebras is defined as follows.

It turns out that a group G has property (T) if and only if $L(G)$ has property (T) [Reference Connes and JonesCJ85, Theorem 2]. One can also formulate a von Neumann algebraic version of Kazhdan tuples as follows.

Definition 2.6. Let $(M,\tau )$ be a tracial von Neumann algebra. We say that $x = (x_1,\dots ,x_d) \in M_{\operatorname {\mathrm {sa}}}^d$ is a (self-adjoint) Kazhdan tuple if there exists $\gamma> 0$ such that for every Hilbert M-M-bimodule $\mathcal {H}$ and $\xi \in \mathcal {H}$ , we have

$$\begin{align*}\left\lVert {\xi - P_{\text{central}} \xi} \right\rVert \leq \frac{1}{\gamma} \left\lVert {x\xi - \xi x} \right\rVert _{H^{\oplus d}}, \end{align*}$$

where $x \xi = (x_1 \xi , \dots , x_d \xi )$ and $\xi x = (\xi x_1,\dots , \xi x_d)$ and where $P_{\text {central}}$ is the projection of H onto the subspace of central vectors. In this case, we call $\gamma $ the Kazhdan constant associated to the Kazhdan tuple x.

Unlike the group case, every tracial von Neumann algebra with property (T) need not admit a Kazhdan tuple (the finite set F in the definition of property (T) may depend a priori on $\varepsilon $ ). However, it is known for several natural classes of examples.

Proof. (i): Cutting M down by the minimal central projections in M, we see that M is a direct sum of $\textrm {II}_{1}$ -factors. Each such factor has property (T) by [Reference PopaPop06a, Proposition 4.7.2] and thus admits a Kazhdan tuple by [Reference Connes and JonesCJ85, Proposition 1]. Combining these Kazhdan tuples gives a Kazhdan tuple for M.

(ii): Choose a Kazhdan set $\{g_{1},\cdots ,g_{k}\}\subseteq G$ for G. Let

$$\begin{align*}x=(\operatorname{Re}(\pi(g_{1})),\operatorname{Re}(\pi(g_{2})),\dots,\operatorname{Re}(\pi(g_{k})),\operatorname{Im}(\pi(g_{1})),\dots,\operatorname{Im}(\pi(g_{k})))\in M_{\operatorname{\mathrm{sa}}}^{2k}.\end{align*}$$

Even though $\pi $ may not be an honest representation, note that if $\mathcal {H}$ is an M-M bimodule, then we do have a representation of G on $\mathcal {H}$ given by conjugating by $\pi (G)$ . Applying [Reference Bekka, De la Harpe and ValetteBdlHV08, Proposition 1.1.9] to this representation, we see that x is a Kazhdan tuple for $W^{*}(\pi (G))$ .

It is a standard exercise to show that a Kazhdan tuple for a group G must generate the group. A well-known result of Popa [Reference PopaPop86, Theorem 4.4.1] shows that the same is true for Kazhdan tuples associated to tracial von Neumann algebras. For the reader’s convenience, we recall the proof here.

2.5 Ultraproducts of matrix algebras

At several points, we use ultraproducts of matrix algebras, so we recall the relevant background as well as the connection between ultraproducts and microstate spaces.

Definition 2.9. Let J be an infinite set. A free ultrafilter is a unital homomorphism

$$\begin{align*}\omega\colon \ell^{\infty}(J)/c_{0}(J)\to {\mathbb C}.\end{align*}$$

For every $(a_{n})_{n\in J}\in \ell ^{\infty }(J)$ , we will use $\lim _{n\to \omega }a_{n}$ for $\omega ((a_{n})_{n\in J}+c_{0}(J))$ .

It follows from [Reference ConwayCon90, Proposition VIII.1.12] that $\lim _{n\to \omega }$ is a $*$ -homomorphism. It thus preserves inequalities and commutes with complex conjugation, by definition $\lim _{n\to \omega }$ preserves sums and products. By Gelfand–Naimark duality [Reference ConwayCon90, Theorem VIII.2.1], free ultrafilters exist in abundance (in fact, there are enough to separate points in $\ell ^{\infty }(J)/c_{0}(J)$ ). We use $\beta J\setminus J$ for the space of free ultrafilters on J.

Recall that if $\omega $ is a free ultrafilter on ${\mathbb N}$ , then the tracial ultraproduct of ${\mathbb M}_{n}({\mathbb C})$ with respect to $\omega $ is given by

$$\begin{align*}\prod_{n\to\omega}{\mathbb M}_{n}({\mathbb C})=\frac{\{(x_{k})_{n}\in \prod_{n}{\mathbb M}_{n}({\mathbb C}):\sup_{n}\|x_{n}\|_{\infty}<\infty\}}{\{(x_{n})_{n}\in \prod_{n}{\mathbb M}_{n}({\mathbb C}):\sup_{n}\|x_{n}\|_{\infty}<\infty,\lim_{n\to\omega}\|x_{n}\|_{2}=0\}}. \end{align*}$$

If $(x_{n})_{n}\in \prod _{n}{\mathbb M}_{n}({\mathbb C})$ and $\sup _{n}\|x_{n}\|_{\infty }<\infty $ , we let $[x_{n}]_{n}$ be the image of $(x_{n})_{n}$ under the natural quotient map

$$\begin{align*}\{(x_{n})_{n}\in \prod_{n}{\mathbb M}_{n}({\mathbb C}):\sup_{n}\|x_{n}\|_{\infty}<\infty\}\to \prod_{n\to\omega}{\mathbb M}_{n}({\mathbb C}). \end{align*}$$

It can be shown (see [Reference Brown and OzawaBO08, Lemma A.9]) that this is a tracial von Neumann algebra, with trace given by

$$\begin{align*}\tau_{\omega}((x_{n})_{n\to\omega})=\lim_{k\to\omega}\operatorname{\mathrm{tr}}_n(x_{n}).\end{align*}$$

We say that a tracial von Neumann algebra $(M,\tau )$ is Connes-embeddable if every von Neumann subalgebra with separable predual admits a trace-preserving embedding into a tracial ultraproduct of matrices.

Suppose that $d \in {\mathbb N}$ , and $R>0$ . If $(\ell _{n})_{n}$ is a sequence in $\Sigma _{d,R}$ , and $\ell \in \Sigma _{d,R}$ , we say that $\lim _{n\to \omega }\ell _{n}=\ell $ if $\lim _{n\to \omega }\ell _{n}(P)=\ell (P)$ for all $P\in {\mathbb C} \left \langle {t_{1},\cdots ,t_{d}} \right \rangle $ . Suppose $(M,\tau )$ is a tracial von Neumann algebra and $x\in M_{\operatorname {\mathrm {sa}}}^{d}$ . If $X^{(n)}\in {\mathbb M}_{n}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{d}$ satisfies $\lim _{n\to \omega }\ell _{X^{(n)}}=\ell _{x}$ , then there is a unique trace-preserving embedding $\pi \colon M\to \prod _{n\to \omega }{\mathbb M}_{n}({\mathbb C})$ which satisfies

$$\begin{align*}\pi(x_{j})=[X_{j}^{(n)}]_{n}\text{ for }j=1,\cdots,d; \end{align*}$$

(see, for instance, [Reference Gangbo, Jekel, Nam and ShlyakhtenkoGJNS22, Lemma 5.10] for further detail). Because of this, we may heuristically think of the microstate spaces $\Gamma ^{(n)}_{R}(\mathcal {O})$ as $\mathcal {O}\to \ell _{x},n\to \infty $ as parameterizing the space of embeddings of M into an ultraproduct of matrices.

See [Reference JungJun07a] and [Reference Atkinson and KunnawalkamAKE21, Sections 1.2, 1.3] for further discussion of the connections between ultraproducts and microstate spaces. These concepts also relate to random matrix theory; see, for instance, [Reference VoiculescuVoi94, Reference VoiculescuVoi98, Reference Dykema, Jung and ShlyakhtenkoDJS05, Reference Brown, Dykema and JungBDJ08, Reference Hayes, Jekel, Nelson and SinclairHJNS21].

4.1 Direct sums and strong $1$ -boundedness

In order to prove Propositions 1.2 and 1.3, we must understand the behavior of h under direct sums of tracial von Neumann algebras. Let J be a countable index set and $(\lambda _j)_{j \in J} \in (0,\infty )^J$ such that $\sum _{j \in J} \lambda _j = 1$ , and for each $j \in J$ , let $(M_j,\tau _j)$ be a tracial von Neumann algebra. Then

$$\begin{align*}\bigoplus_{j \in J} \lambda_j(M_j,\tau_j) \end{align*}$$

is defined as the tracial von Neumann algebra $(M,\tau )$ , where

$$\begin{align*}M = \{(x_j)_{j \in J}: \sup_j \left\lVert {x_j} \right\rVert < \infty\} \end{align*}$$

and

$$\begin{align*}\tau((x_j)_{j \in J}) = \sum_{j \in J} \lambda_j \tau_j(x_j). \end{align*}$$

It is shown in [Reference HayesHay18, Proposition A.13(i)] that

(4.1) $$ \begin{align} h(M,\tau) \leq \sum_{j \in J} \lambda_j^2 h(M_j,\tau_j). \end{align} $$

Our goal in this section is to give a corresponding lower bound for $h(M,\tau )$ (see Lemma 4.2). As with many results in free entropy theory, we run into the issue that we do not know whether using a $\liminf $ instead of $\limsup $ in the definition of h would yield the same result. Thus, we will need to use the ‘ $\liminf $ version’ of $1$ -bounded entropy $\underline {h}$ in our formula. It is also convenient for the proof to use the description of $1$ -bounded entropy in terms of relative microstates rather than unitary orbits. We therefore recall the following definitions.

Definition 4.1. Let $(M,\tau )$ be a tracial von Neumann algebra, $l\in {\mathbb N}$ and $y\in M_{\operatorname {\mathrm {sa}}}^{l}$ . A microstates sequence for y is a sequence $(Y^{(n)})\in \prod _{n}{\mathbb M}_{n}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{l}$ with

$$\begin{align*}\sup_{n}\|Y^{(n)}\|_{\infty}<\infty,\end{align*}$$

$$\begin{align*}\ell_{Y^{(n)}}\to_{n\to\infty}\ell_{y}.\end{align*}$$

Suppose that $d\in {\mathbb N}$ , that $x\in M_{\operatorname {\mathrm {sa}}}^{d}$ and that $Y^{(n)}$ is a microstates sequence for y as above. Fix

$$\begin{align*}R>\max(\|x\|_{\infty},\sup_{n}\|Y^{(n)}\|_{\infty}).\end{align*}$$

For $n\in {\mathbb N}$ and $\mathcal {O}$ a weak $^{*}$ -neighborhood of $\ell _{x,y}$ , we let

$$\begin{align*}\Gamma^{(n)}_{R}(\mathcal{O}|Y^{(n)})=\{B\in {\mathbb M}_{n}({\mathbb C})_{\operatorname{\mathrm{sa}}}^{r}:(B,Y^{(n)})\in \Gamma^{(n)}(\mathcal{O})\}.\end{align*}$$

We let $K_{\varepsilon }(\Omega ,\|\cdot \|_{2})$ be the minimal cardinality of a subset of ${\mathbb M}_{n}({\mathbb C})^{d}$ which $(\varepsilon , \left \lVert {\cdot } \right \rVert _2)$ -covers $\Omega $ . Let $(M,\tau )$ be a tracial von Neumann algebra, $d,s,l\in {\mathbb N}$ and let $x\in M_{\operatorname {\mathrm {sa}}}^{d},y\in M_{\operatorname {\mathrm {sa}}}^{s},a\in M_{\operatorname {\mathrm {sa}}}^{l}$ . Suppose that $(A^{(n)})_{n}$ is a microstates sequence for a. Fix $R\in (0,\infty )$ with $R>\max (\|x\|_{\infty },\|y\|_{\infty },\sup _{n}\|A^{(n)}\|_{\infty })$ . For $n\in {\mathbb N}$ , let $\pi _{r}\colon M_{n}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{d+s}\to M_{n}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{d}$ be the projection onto the first d-coordinates. For a weak $^{*}$ -neighborhood $\mathcal {O}$ of $\ell _{x,y,a}$ , and $n\in {\mathbb N}$ , we set

$$\begin{align*}\Gamma^{(n)}_{R}(x|A^{(n)}:\mathcal{O})=\pi_{r}(\Gamma^{(N)}_{R}(\mathcal{O}|A^{(k)})).\end{align*}$$

We will heuristically refer to the collection of spaces $(\Gamma ^{(n)}_{R}(x|A^{(n)}:\mathcal {O}))_{n,\mathcal {O}}$ as the microstates spaces for x relative to a in the presence of y. Roughly speaking, the usual microstates spaces correspond to embeddings of $W^{*}(x,a)$ into an ultraproduct of matrices. As in Section 2.5, relative microstates spaces in the presence correspond to embeddings of $W^{*}(x)$ into ultraproducts of matrices which both restrict to the embedding of $W^{*}(a)$ given by $(A^{(n)})_{n}$ , and have an extension to $W^{*}(x,y,a)$ .

We now define

$$ \begin{align*} \underline{h}_{R,\varepsilon}(x|(A^{(n)})_{n}:\mathcal{O})&:=\liminf_{n\to\infty}\frac{1}{n^{2}}\log K_{\varepsilon}(\Gamma_{R}^{(n)}(x|A^{(n)};\mathcal{O}),\|\cdot\|_{2}), \\ \underline{h}_{R,\varepsilon}(x|(A^{(n)})_{n}:y) &:= \inf_{\mathcal{O}} \underline{h}_{R,\varepsilon}(x|(A^{(n)})_{n}:\mathcal{O}), \end{align*} $$

where the infimum is over all weak $^{*}$ -neighborhoods $\mathcal {O}$ of $\ell _{x,y,a}$ . If $W^{*}(a)$ is diffuse and hyperfinite, and if $W^{*}(a)\subseteq W^{*}(x)$ , we then define

$$\begin{align*}\underline{h}(x:y) := \sup_{\varepsilon> 0} \underline{h}_{R,\varepsilon}(x|(A^{(n)})_{n}:y). \end{align*}$$

By standard methods, it can be shown that $\underline {h}(x:y)$ does not depend upon the choice of R, and this justifies dropping it from the notation. If $W^{*}(a)$ is diffuse and hyperfinite, then the above quantity is independent of a and the choice of $(A^{(n)})_{n}$ by the same arguments in [Reference HayesHay18, Lemma A.5, Corollary A.6]. In [Reference HayesHay18], this is only shown when a is a single element which generates a diffuse, abelian von Neumann algebra, but the same argument works for any tuple generating a diffuse, hyperfinite von Neumann algebra. By the same arguments as in [Reference HayesHay18, Theorem A.9], one can show that if $d',s'\in {\mathbb N}$ and $x'\in M_{\operatorname {\mathrm {sa}}}^{d'},y\in M_{\operatorname {\mathrm {sa}}}^{s'}$ satisfies $W^{*}(x')=W^{*}(x)$ , $W^{*}(x,y)=W^{*}(x',y')$ and if $W^{*}(x),W^{*}(y)$ are diffuse, then $\underline {h}(x:y)=\underline {h}(x':y')$ . Thus, if $M_{1}\leq M_{2}\leq M$ and $M_{1},M_{2}$ are finitely generated and diffuse, we can set

$$\begin{align*}\underline{h}(M_{1}:M_{2})=\underline{h}(x:y),\end{align*}$$

where $x_{j},j=1,2$ are finite self-adjoint tuples in M with $W^{*}(x_{1})=M_{1}$ , $W^{*}(x_{1},x_{2})$ . We then set

$$\begin{align*}\underline{h}(M_{1}:M)=\inf_{Q}\underline{h}(M_{1}:Q),\end{align*}$$

where the infimum is over all finitely generated subalgebras Q of M which contain $M_{1}$ . Finally, for $N\leq M$ , we let

$$\begin{align*}\underline{h}(N:M)=\sup_{P}\underline{h}(P:M),\end{align*}$$

where the supremum is over all finitely generated, diffuse subalgebras P of N. We call $\underline {h}(N:M)$ the lower $1$ -bounded entropy of N in the presence of M, and we set $\underline {h}(M)=\underline {h}(M:M)$ . The above definition using relative microstates can intuitively be thought of describing the $1$ -bounded entropy as a measurement of how many embeddings N has into an ultraproduct of matrices which both restrict to a given embedding of $W^{*}(a)$ and have an extension to M. The definition of $1$ -bounded entropy using unitary conjugation orbits can be thought of as a measurement of how many embeddings M has into an ultraproduct of matrices modulo unitary conjugation. It is a fact that when $N = M$ , these measurements are the same [Reference HayesHay18, Lemma A.5].

We remark that the properties of h given in Section 2 are satisfied, mutatis mutandis, for $\underline {h}$ with the exception that the analogue of item 6 is that

$$\begin{align*}\underline{h}(N_{1}\vee N_{2}:M)\leq \underline{h}(N_{1}:M)+h(N_{2}:M)\text{ if }N_{1}\cap N_{2}\text{ is diffuse}. \end{align*}$$

The lower bound for the $1$ -bounded entropy of direct sums is as follows.

Lemma 4.2. Suppose that J is a countable index set and that $((M_{j},\tau _{j}))_{j\in J}$ are diffuse tracial von Neumann algebras. Let $(\lambda _{j})_{j\in J}$ be positive numbers with $\sum _{j}\lambda _{j}=1$ , and set $(M,\tau )=\bigoplus _{j}\lambda _{j}(M_{j},\tau _{j})$ . Suppose that $N_{j}\leq M_{j},j\in J$ are diffuse, and set $N=\bigoplus _{j}N_{j}$ . Then

$$\begin{align*}\underline{h}(N:M)\geq \sum_{j}\lambda_{j}^{2}\underline{h}(N_{j}:M_{j}),\end{align*}$$

$$\begin{align*}h(N:M)\geq \lambda_{i}^{2}h(N_{i}:M_{i})+\sum_{j\ne i}\lambda_{j}^{2}\underline{h}(N_{j}:M_{j})\text{ for all }i\in J. \end{align*}$$

Proof. If there is a $j \in J$ so that $M_{j}$ is not Connes-embeddable, then both sides are $-\infty $ , and there is nothing to prove. So we will assume without further comment throughout the proof that all algebras involved are Connes-embeddable. The proofs for h and $\underline {h}$ are nearly identical, so we will only give the proof for $\underline {h}$ . First, consider the case where $J = \{1,2\}$ . Fix tracial von Neumann algebras $(M_{j},\tau _{j}),j=1,2$ , and diffuse $N_{j}\leq M_{j}$ . Without loss of generality, we may assume that $N_{j},M_{j}$ are finitely generated, so let $r_{j},s_{j}\in {\mathbb N},j=1,2$ , and let $x_{j}\in (N_{j})_{\operatorname {\mathrm {sa}}}^{r_{j}}$ , $y_{j}\in (M_{j})_{\operatorname {\mathrm {sa}}}^{s_{j}}$ be generating sets. Let $z_{j},j=1,2$ be the identity of $M_{j}$ , regarded as a projection in $M_{1}\oplus M_{2}$ . Let $a_{j}\in (N_{j})_{\operatorname {\mathrm {sa}}},j=1,2$ be an element with diffuse spectrum. We will use microstates relative to $b=(a_{1}\oplus 0,0\oplus a_{1},z_{1},z_{2})$ to compute the entropy. So let $C^{(n)}\in {\mathbb M}_{n}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{4}$ be a microstates sequence for b. To ease notation set $A_{j}^{(n)}=C^{(n)}_{j},j=1,2$ and $Z^{(n)}_{j}=C^{(n)}_{j+2},j=1,2$ . We may, and will, assume that

• • $Z^{(n)}_{1},Z^{(n)}_{2}$ are diagonal projections with $Z^{(n)}_{1}+Z^{(n)}_{2}=1$ for all n,

• • $n\operatorname {\mathrm {tr}}_{n}(Z^{(n)}_{1})=\lfloor {\tau (z_{1})n\rfloor }$ for all n,

• • $Z^{(n)}_{j}A^{(n)}_{j}Z^{(n)}_{j}=A^{(n)}_{j}$ for all n and all $j=1,2,$

• • $A^{(n)}_{j}$ are diagonal for all n and all $j=1,2$ .

Set $b_{j}=(a_{j},z_{j}),j=1,2,$ and let $B^{(n)}_{j}=(A^{(n)}_{j},\operatorname {\mathrm {id}}_{\ell _{n,j}})$ where we regard $A^{(n)}_{j}$ as an element of $M_{\ell _{n,j}}$ , where $\ell _{n,j}=n\tau (Z^{(n)}_{j})$ Set $x=(x_{1}\oplus 0,0\oplus x_{2})$ , and $y=(y_{1}\oplus 0,0\oplus y_{2})$ . Finally, fix

$$\begin{align*}R>\max(\|x\|_{\infty},\|y\|_{\infty},\sup_{n}\|B^{(n)}\|_{\infty}).\end{align*}$$

Given a neighborhood $\mathcal {O}$ of $\ell _{x,y,b}$ , we may find neighborhoods $\mathcal {O}_{j},j=1,2$ of $\ell _{x_{j},y_{j},b_{j}}$ so that

$$\begin{align*}\Gamma^{(n)}_{R}(x|B^{(n)}:\mathcal{O}) &\supseteq \Gamma^{(\lfloor{\tau(z_{1})n\rfloor})}_{R}(x_{1}|B^{(\lfloor{\tau(z_{1})n\rfloor}))}_{1}:\mathcal{O}_{1})\oplus \Gamma^{(\lceil{\tau(z_{2})n\rceil})}_{R} \\ &\quad \times (x_{2}|B^{(\lceil{\tau(z_{2})n\rceil})}_{2};\mathcal{O}_{2})\text{ for all sufficiently large }n.\end{align*}$$

Given $\varepsilon>0$ , let $\Omega _{n,j}\subseteq \Gamma ^{(\ell _{n,j})}(x_{j}|B^{(\ell _{n,j})}_{j};\mathcal {O}_{j})$ be $\varepsilon $ -separated with respect to $\|\cdot \|_{2}$ (meaning that distinct points in $\Omega _{n,j}$ are distance at least $\varepsilon $ from each other). Then for all sufficiently large n, we have that $\Omega _{n,1}\oplus \Omega _{n,2}$ is an $\varepsilon $ -separated subset of $\Gamma ^{(n)}(x|B^{(n)};\mathcal {O})$ and thus has cardinality at most $K_{\varepsilon /2}(\Gamma ^{(n)}_{R}(x|B^{(n)};\mathcal {O}),\|\cdot \|_{2})$ .

Thus, for all $\varepsilon>0$ and for all sufficiently large n,

$$\begin{align*} & K_{\varepsilon/2}(\Gamma^{(n)}_{R}(x|B^{(n)}:\mathcal{O}),\|\cdot\|_{2}) \\ &\quad \geq K_{\varepsilon}(\Gamma^{(\lfloor{\tau(z_{1})n\rfloor})}_{R}(x_{1}|B^{(n)}_{1}:\mathcal{O}_{1}),\|\cdot\|_{2})K_{\varepsilon}( \Gamma^{(\lceil{\tau(z_{2})n\rceil})}_{R}(x_{2}|B^{(n)}_{2}:\mathcal{O}_{2}),\|\cdot\|_{2}).\end{align*}$$

Since $\{\lfloor {\tau (z_{1})n\rfloor }:n\in {\mathbb N}\}$ and $\{\lceil {\tau (z_{2})n\rceil }:n\in {\mathbb N}\}$ have finite complement in ${\mathbb N}$ , and since limit infimums are super additive, the above inequality shows that for all $\varepsilon>0$ we have

$$\begin{align*}\underline{h}_{R,\varepsilon/2}(x|(B^{(n)})_{n}:\mathcal{O})\geq \tau(z_{1})^{2}\underline{h}_{R,\varepsilon}(x_{1}|B^{(n)}_{1}:y_{1})+\tau(z_{2})^{2}\underline{h}_{R,\varepsilon}(x_{2}|B^{(n)}_{2}:y_{2}).\end{align*}$$

Letting $\mathcal {O}$ decrease to $\ell _{x,y,b}$ and letting $\varepsilon \to 0$ completes the proof of the case where $J = \{1,2\}$ .

The case where J is finite follows by induction. Now, suppose that J is infinite and assume without loss of generality that $J = {\mathbb N}$ . Let $(N_{j}\leq M_{j})_{j\in J}$ be as in the statement of the lemma. For each $j\in J$ , fix a diffuse abelian $A_{j}\leq N_{j}$ . For $r\in {\mathbb N}$ , let

$$\begin{align*}N_{\leq r}=\bigoplus_{j=1}^{r}N_{j}\oplus \bigoplus_{j=r+1}^{\infty}A_{j}.\end{align*}$$

Since $\bigoplus _{j=r+1}^{\infty }A_{j}$ has $1$ -bounded entropy zero, we have by the case of finite J that

$$\begin{align*}\underline{h}(N_{\leq r}:M)\geq \sum_{j=1}^{r}\lambda_{j}^{2}\underline{h}(N_{j}:M_{j}).\end{align*}$$

Since $N_{\leq r}$ are an increasing sequence of subalgebras of N and $\bigvee N_{\leq r}=N$ , we have that

$$\begin{align*}\underline{h}(N:M)=\sup_{r}\underline{h}(N_{\leq r}:M)\geq \sum_{j=1}^{\infty}\lambda_{j}^{2}\underline{h}(N_{j}:M_{j}). \end{align*}$$

4.2 Amplification and strong $1$ -boundedness

This section will show that $h(M^t) = t^{-2} h(M)$ for $\mathrm {II}_1$ factor M (Proposition 1.3) and then conclude the proof of Proposition 1.2. To prove Proposition 1.3 for $t \in (0,1)$ , we consider a projection $p \in M$ with trace t and analyze the $1$ -bounded entropy of certain subalgebras of M related to $pMp$ . At a key point, we use the fact that if N is a von Neumann subalgebra of M, then for $h(N:M)$ to equal $h(M)$ , it is sufficient that every sequence of matricial microstates for N extends to a sequence of matricial microstates for M. We state this property more precisely as follows.

The following lemma formulates the microstates extension property in terms of microstates spaces. The proof is an exercise in understanding the definitions and is left to the reader.

Lemma 4.4. Let $(M,\tau )$ be a tracial von Neumann algebra, and let $N\leq M$ . Then the inclusion $N\leq M$ has the microstates extension property if and only if for every $d,s\in {\mathbb N}$ , every $x\in N_{\operatorname {\mathrm {sa}}}^{d},y\in M_{\operatorname {\mathrm {sa}}}^{s}$ , every $R>\max (\|x\|_{\infty },\|y\|_{\infty })$ and every neighborhood $\mathcal {V}$ of $\ell _{x,y}$ there is a neighborhood $\mathcal {O}$ of $\ell _{x}$ so that

$$\begin{align*}\Gamma^{(n)}_{R}(\mathcal{O})\subseteq \pi_{d}(\Gamma^{(n)}_{R}(\mathcal{V}))\text{ for all sufficiently large }n,\end{align*}$$

where $\pi _{d}\colon {\mathbb M}_{k}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{d+s}\to {\mathbb M}_{k}({\mathbb C})_{\operatorname {\mathrm {sa}}}^{d}$ is the projection onto the first d coordinates.

The following lemma gives some examples and also explains the relevance to $1$ -bounded entropy.

We remark that (i) makes sense from our intuitive description of $1$ -bounded entropy. The quantity $h(N:Q)$ (resp. $h(N:M)$ ) is supposed to be a measurement of ‘how many’ embeddings there are of N into an ultraproduct of matrices which have an extension to Q (resp. M). If $Q\leq M$ has the microstates extension property, then every embedding of Q extends to M and thus the quantities $h(N:Q)$ , $h(N:M)$ should be the same.

Now, we are ready to prove Proposition 1.3, and in fact we also give the analogous result for $\underline {h}$ . We remark that the point $h(M^t) \leq t^{-2} h(M)$ for $t \in (0,1)$ was already shown in [Reference HayesHay18, Propostion A.13(ii)].

Proposition 4.6. Suppose that M is a $\textrm {II}_{1}$ -factor and $\tau $ is its canonical trace. For $t\in (0,\infty )$ , let $M^{t}$ be the $t^{th}$ amplification of M. Then

$$\begin{align*}h(M^{t}) = \frac{1}{t^{2}}h(M) \quad \text{and} \quad \underline{h}(M^{t}) = \frac{1}{t^{2}}\underline{h}(M). \end{align*}$$

Proof. We first handle the case when $t \in (0,1)$ . Fix a hyperfinite $\textrm {II}_{1}$ -subfactor R of M. Let $p\in R$ be a projection with $\tau (p)=t$ . Observe that $M^{t}\cong pMp$ . Let $N=pMp+(1-p)R(1-p)$ Since R is a factor, we may find partial isometries $v_{1},\cdots ,v_{n}\in R$ so that $v_{j}^{*}v_{j}\leq p$ and $\sum _{j}v_{j}v_{j}^{*}=1$ . Thus,

$$\begin{align*}N\vee R\supseteq W^{*}(v_{1},v_{2},\cdots,v_{n})\vee (pMp+(1-p))=M.\end{align*}$$

So

(4.2) $$ \begin{align} N\vee R=M. \end{align} $$

Step 1. We show that $\underline {h}(M^{t})\geq \frac {1}{t^{2}}\underline {h}(M)$ . By Equation (4.2) and the fact that $N\cap R=pRp+(1-p)R(1-p)$ is diffuse, we have

$$\begin{align*}\underline{h}(M) &=\underline{h}(N\vee R)\leq \underline{h}(N)+h(R)\\ & =\underline{h}(N)\leq t^{2}\underline{h}(pMp)+(1-t)^{2}h((1-p)R(1-p))=t^{2}\underline{h}(pMp)=t^{2}\underline{h}(M^{t}).\end{align*}$$

Here, we use the analogues of [Reference HayesHay18, Lemma A.12 and Proposition A.13 (i)] for $\underline{\mathrm{h}}$ .

Step 2. We prove that $\underline {h}(M^{t})\leq \frac {1}{t^{2}}\underline {h}(M)$ . We start with the following claim.

Claim. $N\leq M$ has the microstates extension property

Suppose that $\omega $ is a free ultrafilter on the natural numbers, and let $\mathcal {M}=\prod _{k\to \omega }{\mathbb M}_{k}({\mathbb C})$ be the tracial ultraproduct of ${\mathbb M}_{k}({\mathbb C})$ . Let $\Theta \colon N\to \mathcal {M}$ be a trace-preserving embedding. Since $N\cap R$ is hyperfinite, we know that $N\cap R\leq R$ has the microstates extension property by Lemma 4.5 (ii). So there exists a trace-preserving embedding $\Psi \colon R\to \mathcal {M}$ with $\Psi \big |_{N\cap R}=\Theta \big |_{N\cap R}$ . Let $v_{1},\cdots ,v_{n}$ be as before the proof of Step 1. Define $\widetilde {\Theta }\colon M\to \mathcal {M}$ by

$$\begin{align*}\widetilde{\Theta}(x)=\sum_{i,j}\Psi(v_{i})\Theta(v_{i}^{*}xv_{j})\Psi(v_{j})^{*}.\end{align*}$$

Observe that $v_{i}^{*}Mv_{j}\subseteq pMp\subseteq N$ for all $i,j$ , so the formula above makes sense. For all $1\leq i,j,k,l\leq n$ and all $x,y\in M$ , we have

$$ \begin{align*} \Psi(v_{i})\Theta(v_{i}^{*}xv_{j})\Psi(v_{j})^{*}\Psi(v_{k})\Theta(v_{k}^{*}yv_{l})\Psi(v_{l})^{*}&=\Psi(v_{i})\Theta(v_{i}^{*}xv_{j})\Psi(v_{j}^{*}v_{k})\Theta(v_{k}^{*}yv_{l})\Psi(v_{l})^{*}\\ &=\delta_{j=k}\Psi(v_{i})\Theta(v_{i}^{*}xv_{j})\Theta(v_{j}^{*}v_{j})\Theta(v_{j}^{*}yv_{l})\Psi(v_{l})^{*}\\ &=\delta_{j=k}\Psi(v_{i})\Theta(v_{i}^{*}xv_{j}v_{j}^{*}v_{j}v_{j}^{*}yv_{l})\Psi(v_{l})^{*}\\ &=\delta_{j=k}\Psi(v_{i})\Theta(v_{i}^{*}xv_{j}v_{j}^{*}yv_{l})\Psi(v_{l})^{*}. \end{align*} $$

From here, it is direct to show that $\widetilde {\Theta }(xy)=\widetilde {\Theta }(x)\widetilde {\Theta }(y)$ for all $x,y\in M$ . It is also direct to show that $\widetilde {\Theta }$ preserves adjoints. Finally, for all $x\in M$ :

$$\begin{align*}\tau_{\omega}(\widetilde{\Theta}(x))=\sum_{i,j}\tau_{\omega}(\Theta(v_{i}^{*}xv_{j})\Psi(v_{j}^{*}v_{i}))=\sum_{i}\tau_{\omega}(\Theta(v_{i}^{*}xv_{i}v_{i}^{*}v_{i}))=\sum_{i}\tau(xv_{i}v_{i}^{*})=\tau(x).\end{align*}$$

This proves the claim.

By Equation (4.2) and the fact that $N\cap R$ is diffuse, we have that

$$\begin{align*}\underline{h}(M) = \underline{h}(N\vee R:M) = \underline{h}(N:M) = \underline{h}(N), \end{align*}$$

the last step following by the claim and Lemma 4.5 (i). By Lemma 4.2,

$$\begin{align*}\underline{h}(M)=\underline{h}(N)\geq t^{2}\underline{h}(pMp). \end{align*}$$

This completes the proof of Step 2 and hence also the proof for $\underline {h}$ when $t\in (0,1)$ . The case $t=1$ is trivial, whereas if $t>1$ , then $\frac {1}{t}\in (0,1)$ and so

$$\begin{align*}\underline{h}(M)=\underline{h}((M^{t})^{\frac{1}{t}})=t^{2}\underline{h}(M^{t}), \end{align*}$$

For the case of h, it was already shown in [Reference HayesHay18, Proposition A.13(ii)] that $h(M^{t})\leq \frac {1}{t^{2}}h(M)$ for $t \in (0,1)$ , and the proof of the opposite inequality proceeds in a similar manner to Step 2 above.

With Lemma 4.2 on direct sums and Proposition 4.6 on amplifications in hand, we are ready to finish the proof of Proposition 1.2 showing that strong $1$ -boundedness of all tracial von Neumann algebras with property (T) is equivalent to $h(M) \leq 0$ for all $\mathrm {II}_1$ factors with property (T).

Proof of Proposition 1.2

(iii) $\implies $ (ii). Assume that every $\mathrm {II}_1$ property (T) factor has nonpositive $1$ -bounded entropy. Let M be a tracial von Neumann algebra with property (T), and we will show that $h(M) \leq 0$ . Decomposing the center of M into diffuse and atomic parts, we see that there is a countable index set J (potentially empty) so that

$$\begin{align*}M=M_{0}\oplus \bigoplus_{j\in J}M_{j},\end{align*}$$

where $M_{0}$ either has diffuse center or is $\{0\}$ , and each $M_{j}$ is a factor. Since $M_{0}$ either has diffuse center or is $\{0\}$ , we know $h(M_{0})\leq 0$ . Since M has property (T), each $M_{j}$ has property (T) by [Reference PopaPop06a, Proposition 4.7.2]. Thus, by Equation (4.1), we have $h(M) \leq 0$ .

(ii) $\implies $ (i). If every tracial von Neumann algebra with property (T) satisfies $h(M) \leq 0$ , then it is strongly $1$ -bounded since strong $1$ -boundedness is equivalent to $h(M) < \infty $ by [Reference HayesHay18, Proposition A.16].

(i) $\implies $ (iii). Proceeding by contraposition, assume that N is a property (T) factor with $h(N)> 0$ , and we will show that there is a tracial von Neumann algebra $(M,\tau )$ with property (T) such that $h(M,\tau ) = \infty $ . Let $(N^{4^{-k}},\tau _k)$ be the compression of N, and let

$$\begin{align*}(M,\tau) = \bigoplus_{k \in {\mathbb N}} 2^{-k} (N^{4^{-k}},\tau_k). \end{align*}$$

By [Reference PopaPop06a, Proposition 4.7.1], we know that M has property (T). By Lemma 4.2 and Proposition 4.6, for each $j \in {\mathbb N}$ , we have

$$\begin{align*}h(M,\tau) \geq 4^{-j} h(N^{4^{-j}}) + \sum_{k \neq j} 4^{-k} \underline{h}(N^{4^{-j}}) = 4^j h(N) + \sum_{k \neq j} 4^k \underline{h}(N). \end{align*}$$

Since $h(N)>0$ , we know that N is Connes-embeddable, and thus $\underline {h}(N)\geq 0$ . So

$$\begin{align*}h(M)\geq 4^j h(N) \text{ for all }j\in {\mathbb N}. \end{align*}$$

Letting $j\to \infty $ we see that $h(M)=\infty $ , that is, M is not strongly $1$ -bounded.

Finally, we show that for each finite von Neumann algebra M with property (T), there exists a faithful normal tracial state $\tau $ such that $(M,\tau )$ is strongly $1$ -bounded. As in (iii) $\implies $ (ii), write $M = \bigoplus _{j=0}^{\infty } M_j$ such that $M_0$ is zero or has diffuse center, and $M_j$ is a factor for $j \geq 1$ . Let $\tau _j$ be the unique tracial state on $M_j$ . Since $h(M_0) \leq 0$ and since $h(M_j) < \infty $ for $j \geq 1$ by Theorem 1.1, we may choose nonnegative constants $(\lambda _j)_{j=0}^{\infty }$ such that $\sum _{j=0}^{\infty } \lambda _j = 1$ , $\sum _{j \in {\mathbb N}_0} \lambda _j^2 h(M_j) < \infty $ , and $\lambda _{j}>0$ if and only if $M_{j}\ne 0$ . Let $\tau $ be the faithful normal tracial state on M given by $\bigoplus _{j=0}^{\infty } \lambda _j \tau _j$ . It follows by Equation (4.1) that $h(M,\tau ) < \infty $ .

4.3 Direct sums and free entropy dimension

In this section, we show that Theorem 1.1 implies Jung and Shlyakhtenko’s result that a property (T) tracial von Neumann algebra has microstates free entropy dimension at most $1$ with respect to every finite generating tuple [Reference Jung and ShlyakhtenkoJS07]. In fact, at the end of the section, we also sketch how to generalize the argument to show that $\delta _{0}(x)\leq 1$ for any infinite generating tuple x. First, we recall the definition of Voiculescu’s microstates free entropy dimension.

Definition 4.7. Let $(M,\tau )$ be a tracial von Neumann algebra, $x\in M^{d}_{\operatorname {\mathrm {sa}}}$ . Fix $R>\|x\|_{\infty }$ . For $\varepsilon>0$ , and a weak $^{*}$ -neighborhood $\mathcal {O}$ of $\ell _{x}$ , we set

$$\begin{align*}\delta_{R,\varepsilon}(\mathcal{O}) &=\limsup_{k\to\infty}\frac{\log K_{\varepsilon}(\Gamma^{(k)}_{R}(\mathcal{O}),\|\cdot\|_{2})}{k^{2}|\log(\varepsilon)|},\\ \delta_{R,\varepsilon}(x) &=\inf_{\mathcal{O}}\delta_{\varepsilon}(\mathcal{O}),\end{align*}$$

where the infimum is over all weak $^{*}$ -neighborhoods $\mathcal {O}$ of $\ell _{x}$ . We then set

$$\begin{align*}\delta_{0}(x)=\limsup_{\varepsilon\to 0}\delta_{R,\varepsilon}(x).\end{align*}$$

We call $\delta _{0}(x)$ the microstates free entropy dimension of x.

By standard methods, $\delta _{0}(x)$ does not depend upon the choice of R and this justifies dropping it from the notation. This is not the original definition in [Reference VoiculescuVoi96]; however, by [Reference JungJun03a, Corollary 2.4] they are the same. The following lemma, based on previous work of Jung, describes the behavior free entropy dimension under direct sums.

Lemma 4.8. Let $(M,\tau )$ be a tracial von Neumann algebra, $d\in {\mathbb N}$ and $x\in M^{d}_{\operatorname {\mathrm {sa}}}$ . Let J be a countable set and $(z_{j})_{j\in J}$ be central projections in M with $\sum _{j\in J}z_{j}=1$ . For $j\in J$ , let $x_{z_{j}}$ be $xz_{j}$ regarded as an element of $Mz_{j}$ . Then

$$\begin{align*}\delta_{0}(x)-1\leq \sum_{j\in J}\tau(z_{j})^{2}(\delta_{0}(x_{z_{j}})-1). \end{align*}$$

Proof. Fix $R>\|x\|_{\infty }$ . We first handle the case where J is finite. By induction, to handle the case of finite J it suffices to handle the case where $J=\{1,2\}$ . In this case, we use z for $z_{1}$ . Let $P_{n}\in {\mathbb M}_{n}({\mathbb C})_{\operatorname {\mathrm {sa}}}$ be microstates for z such that each $P_{n}$ is an orthogonal projection. By [Reference JungJun06, Lemma 3.2 and Corollary 4.3], we have

$$\begin{align*}\delta_{0}(x)=\delta_{0}(x,z)=\delta_{0}(z)+\limsup_{\varepsilon\to 0}\frac{h_{R,\varepsilon}(x|(P_{k}))}{\log(1/\varepsilon)}=2\tau(z)(1-\tau(z))+\limsup_{\varepsilon\to 0}\frac{h_{R,\varepsilon}(x|(P_{k}))}{\log(1/\varepsilon)},\end{align*}$$

where in the last step we use [Reference JungJun03b, Corollary 5.8]. It follows from the proof of [Reference HayesHay18, Proposition A.13(i)] that

$$\begin{align*}h_{R,6\varepsilon}(x|(P_{k})_{k})\leq \tau(z)^{2}h_{R,\varepsilon}(x_{z})+(1-\tau(z))^{2}h_{R,\varepsilon}(x_{1-z}), \text{ for all sufficiently small }\varepsilon.\end{align*}$$

Dividing by $\log (1/\varepsilon )$ and letting $\varepsilon \to 0$ , we obtain that

$$\begin{align*}\delta_{0}(x)\leq 2 \tau(z)(1-\tau(z))+\tau(z)^{2}\delta_{0}(x_{z})+(1-\tau(z))^{2}\delta_{0}(x_{1-z}),\end{align*}$$

and by direct computation this is equivalent to the desired inequality.

We now handle the case of infinite J. We may, and will, assume that $J={\mathbb N}$ . For $n\in {\mathbb N}$ , let $z_{\leq n}=\sum _{j=1}^{n}z_{j}$ . Then, by the case of finite J:

$$\begin{align*}\delta_{0}(x)-1\leq (1-\tau(z_{\leq n}))^{2}(\delta_{0}(x_{1-z_{\leq n}})-1)+\sum_{j=1}^{n}\tau(z_{j})^{2}\delta_{0}(x_{z_{j}}).\end{align*}$$

We have that $\delta _{0}(x_{1-z_{\leq n}})\leq d$ , and thus

$$\begin{align*}\delta_{0}(x)-1\leq (1-\tau(z_{\leq n}))^{2}(d-1)+\sum_{j=1}^{n}\tau(z_{j})^{2}\delta_{0}(x_{z_{j}}).\end{align*}$$

The proof is thus completed by letting $n\to \infty .$

Since being strongly $1$ -bounded implies microstates free entropy dimension at most $1$ with respect to any set of generators, the preceding lemma automatically implies the following.

We now recover the results of Jung–Shlyakhtenko.

We remark that if one works carefully with free entropy dimension in the presence in Lemma 4.8, then a proof of the Corollary 4.9 can be given for infinite tuples as well. Using this, one can show that if M is a property (T) algebra, then $\delta _{0}(x)\leq 1$ for every tuple which generates M, even if x is infinite. We will not give the full proof here, but sketch the details for the interested reader.

Given a tracial von Neumann algebra $(M,\tau )$ and finite tuples $x\in M_{\operatorname {\mathrm {sa}}}^{d},y\in M_{\operatorname {\mathrm {sa}}}^{s}$ with $d,s\in {\mathbb N}$ , an $R>\max (\|x\|_{\infty },\|y\|_{\infty })$ , a weak $^{*}$ -neighborhood $\mathcal {O}$ of $\ell _{x,y}$ and $\varepsilon>0$ , we set

$$\begin{align*}\delta_{R,\varepsilon}(x:\mathcal{O}) &=\limsup_{n\to\infty}\frac{K_{\varepsilon}(\Gamma_{R}^{(n)}(x:\mathcal{O}),\|\cdot\|_{2})}{\log(1/\varepsilon)},\\ \delta_{R,\varepsilon}(x:y) &=\inf_{\mathcal{O}}\delta_{R,\varepsilon}(x:\mathcal{O}),\end{align*}$$

where the infimum is over all weak $^{*}$ -neighborhoods $\mathcal {O}$ of $\ell _{x,y}$ . We then define the free microstates free entropy dimension of x in the presence of y by

$$\begin{align*}\delta_{0}(x:y)=\limsup_{\varepsilon\to 0}\delta_{R,\varepsilon}(x:y),\end{align*}$$

by standard methods we can show that this is independent of R, and this justifies dropping R from the notation. One can show that if $a\in M_{\operatorname {\mathrm {sa}}}^{t}$ for some $t\in {\mathbb N}$ and $W^{*}(x,y)=W^{*}(x,a)$ , then $\delta _{0}(x:y)=\delta _{0}(x:a)$ . So we set

$$\begin{align*}\delta_{0}(x:W^{*}(x,y))=\delta_{0}(x:y),\end{align*}$$

and this does not depend upon the choice of y.

Now, suppose that $x=(x_{j})_{j\in J}\in M_{\operatorname {\mathrm {sa}}}^{J}$ for some set J. For a finite $F\subseteq J$ , let $x_{F}\in M_{\operatorname {\mathrm {sa}}}^{F}$ be given by $x_{F}=(x_{j})_{j\in F}$ . We then set

$$\begin{align*}\delta_{0}(x_{F}:M) & =\inf_{Q}\delta_{0}(x_{F}:Q), \\ \delta_{0}(x:M) &=\sup_{F}\delta_{0}(x_{F}:M),\end{align*}$$

where the infimum is over all finitely generated $Q\leq M$ with $x_{F}\in Q^{F}_{\operatorname {\mathrm {sa}}}$ , and the supremum is over all finite subsets F of J. We set $\delta _{0}(x)=\delta _{0}(x:W^{*}(x))$ .

To generalize Corollary 4.9 to infinite tuples, one first proves a modification of Lemma 4.8. Namely, if x is a self-adjoint tuple in M and $(z_{j})_{j\in J}$ are projections in $Z(M)\cap W^{*}(x)$ , then

(4.3) $$ \begin{align} \delta_{0}(x:M)-1\leq \sum_{j}\tau(z_{j})^{2}(\delta_{0}(x_{z_{j}}:Mz_{j})-1). \end{align} $$

To prove Equation (4.3), one first handles the case x is a finite tuple, and $J=\{1,2\}$ . The proof of Equation (4.3) in this case is a minor modification of Lemma 4.8. Namely, one modifies the proof of [Reference JungJun06, Lemma 3.2 and Corollary 4.3] to show that if $z\in Z(M)\cap W^{*}(x)$ is a central projection, then

$$\begin{align*}\delta_{0}(x,z:M)=2\tau(z)(1-\tau(z))+\limsup_{\varepsilon\to 0}\frac{h_{R,\varepsilon}(x|(P_{k})_{k}:M)}{\log(1/\varepsilon)} .\end{align*}$$

One then modifies the proof of [Reference JungJun03b, Corollary 5.8] to say that $\delta _{0}(x:M)=\delta _{0}(x,z:M)$ . After changing the results in [Reference JungJun06, Reference JungJun03b] to work for free entropy dimension in the presence, the proof of Equation (4.3) in the case that x is a finite tuple and $J=\{1,2\}$ proceeds exactly as in Lemma 4.8. The proof of the general case of Equation (4.3) from this special case also follows precisely as in Lemma 4.8.

The inequality (4.3) automatically shows that if M is a direct sum of strongly $1$ -bounded algebras and if x is any self-adjoint tuple in M, then $\delta _{0}(x)\leq 1$ , and from this one deduces a version of Corollary 4.10 where x is any self-adjoint tuple (finite or not).