The amalgamated free product of semifinite hyperfinite von Neumann algebras over atomic type I subalgebras

First consider an interpolated free group factor $A=L(F_{t})$, whose identity has trace $\lambda$. Then $\text{rdim}(A)=\lambda^{2}(t-1)$. Let $\gamma=\tau(p)$, then $pAp$ is isomorphic to $L(F_{1+(t-1)/(\gamma/\lambda)^{2}})$. Thus $\text{rdim}(pAp)=\gamma^{2}(1+(t-1)/(\gamma/\lambda)^{2}-1)=\lambda^{2}(t-1)$, as desired.

For a matrix algebra, we see that compression does not change the size of the minimal projection (note it is important that $p\in A$), and thus does not change the regulated dimension. For diffuse hyperfinite algebras the regulated dimension remains zero.

Since the regulated dimension is additive over direct sums, and since $p$ has full central support (thus all factor summands in $A$ are represented in $pAp$), the result follows. ∎

Let $\rho_{i}=\phi_{i}\circ\phi_{i-1}\circ\dots\circ\phi_{1}$. Choose a projection $p\in\mathcal{F}_{s_{1}}^{t_{1}}$ with finite non-zero trace $t^{\prime}$.

Then $\rho_{i}|_{p}$ considered as map from $p\mathcal{F}_{s_{1}}^{t_{1}}p\to\rho_{i}(p)\mathcal{F}_{s_{i+1}}^{t_{i+1}}\rho_{i}(p)$ is standard or an isomorphism. Furthermore we know $\rho_{i}(p)\mathcal{F}_{s_{i+1}}^{t_{i+1}}\rho_{i}(p)\cong\mathcal{F}_{s_{i+1}}^{t^{\prime}}$. Then we see each $\rho_{i-1}(p)\mathcal{F}_{s_{i}}^{t_{i}}\rho_{i-1}(p)$ is generated by $R_{i}$ and $S_{i}$, where $R_{i}$ is a copy of the hyperfinite II₁ factor, and $S_{i}=\{q_{j}X_{j}q_{j}\}_{j\in J_{i}}$ with $\sum_{j\in J_{i}}\tau(q_{j})^{2}=s_{i}$, where the $X_{j}$ are a semicircular system and $q_{j}\in R_{i}$, and so that $\phi_{i}(R_{i})=R_{i+1}$ and $\phi_{i}(S_{i})\subseteq S_{i+1}$. Thus $\rho_{i}(p)\mathcal{F}_{s_{i+1}}^{t_{i+1}}\rho_{i}(p)\cong\mathcal{F}_{s_{i+1}}^{t^{\prime}}$.

Then for each $i$, $\mathcal{F}_{s_{i}}^{t_{i}}$ is generated by $S_{i}$ and an amplification of $R_{i}$. The inductive limit is then generated by $S=\cup_{i=1}^{\infty}S_{i}$ and $R$, a hyperfinite II₁ (if $\lim_{i}t_{i}<\infty$) or II_∞ (if $\lim_{i}t_{i}=\infty$) factor. If we let $J=\cup_{i}J_{i}$ then we see $\sum_{j\in J}\tau(q_{j})^{2}=\lim_{i}s_{i}$ Thus the inductive limit is $\mathcal{F}_{s}^{t}$ as desired.

∎

Let $D=\bigoplus_{k\in I_{D}}\overset{p_{k}^{D}}{M_{n_{k}^{D}}}$ (where we allow $n_{k}=\infty$, in which case we mean $B(\mathcal{H})$). For each $k\in I_{D}$, let $\{e_{i,j}^{(k)}\}$ be the standard basis for $M_{n_{k}^{D}}$. Let $e=\sum_{k\in I_{D}}e_{1,1}^{(k)}$ (with the sum converging in the SOT). Note that this is a projection in $D$ with central support $I$, and $eDe$ is abelian. Let $A^{\prime}=eAe,B^{\prime}=eBe,$ and $D^{\prime}=eDe$. Let $\mathcal{M}=A*_{D}B$ and let $\mathcal{M}^{\prime}=vN(A^{\prime}\cup B^{\prime})$, and since $A^{\prime}\subseteq A$, $B^{\prime}\subseteq B$ we see that $A^{\prime}$ and $B^{\prime}$ are free with amalgamation over $D^{\prime}$, and thus $\mathcal{M}^{\prime}=A^{\prime}*_{D^{\prime}}B^{\prime}$. Let $V=\{e_{1,j}^{(k)}|k\in I_{D},1<j\leq n_{k}^{D}\}$. Note then that $A=vN(A^{\prime}\cup V)$, $B=vN(B^{\prime}\cup V)$, and $D=vN(D^{\prime}\cup V)$. Then $\mathcal{M}=vN(\mathcal{M}^{\prime}\cup V)$. Thus $\mathcal{M}^{\prime}=e\mathcal{M}e$. By our assumption we know $A^{\prime}$ and $B^{\prime}$ are in $\mathcal{R}$ and $D^{\prime}$ is an abelian multimatrix algebra. Thus by assumption $\mathcal{M}^{\prime}$ is of the correct form, which implies $\mathcal{M}$ is of the same form with the same number of factor summands of each kind (where the kinds are: diffuse hyperfinite, type I atomic, and $\mathcal{F}_{s}^{t}$ as this step may change interpolated free group factors to $\mathcal{F}_{t}^{\infty}$ or matrix algebras to $B(\mathcal{H})$). Now if $\text{rdim}(A^{\prime}*_{D^{\prime}}B^{\prime})=\text{rdim}(A^{\prime})+\text{rdim}(B^{\prime})-\text{rdim}(D^{\prime})$, then since the central support of $e$ is the identity $\text{rdim}(A*_{D}B)=\text{rdim}(A)+\text{rdim}(B)-\text{rdim}(D)$.

∎

First we assume $N$ is finite a finite set and begin by dividing it into type I and type II parts and approximating them separately. If $A$ is type I and finite, and thus of the form $B\oplus(C\otimes L^{\infty}([0,1]))$, where $B$ and $C$ are multimatrix algebras. We can choose a representation so that every $p_{n}$, $n\in N$, is composed of diagonal matrices in $B$ and $C$ and characteristic functions in $L^{\infty}([0,1])$. Let $C=\oplus_{k\in K}C_{k}$ where each $C_{k}$ is a matrix algebra. Then for each $k$ in $K$ we can choose a partition $P_{k}$ of $[0,1]$ into a countable number of measurable subsets so that for every $S\in P_{k}$ and $e_{j,j}^{(k)}\in C_{k}$, $e_{j,j}^{(k)}\otimes\chi_{S}\leq p_{n}$ for some $n\in N$. Then we set $A_{1}=B\oplus\bigoplus_{k\in K}\left(C_{k}\otimes L^{\infty}(P_{k})\right)$, and note $D\subseteq A_{1}$. We can then define each $A_{j}$ by further refining $P_{k}$ to get the desired sequence.

Next we do the finite type II case (with finite $N$). To make this easier to extend to the infinite $N$ semifinite case, we will not assume normalised trace, and consider it as a cut-down of a larger algebra. We consider it as the cutdown of an algebra $A^{\prime}=L^{\infty}(X,\mu)\otimes R$, with centre-valued trace $\tau_{Z}$ so that $\tau_{Z}(A^{\prime})$ is the constant function on $X$ of value $2^{K}$ for some $K$. If we let $p_{n^{\prime}}=I_{A^{\prime}}-I_{A}$, we can rephrase this as finding an approximating sequence for $A^{\prime}$ with subalgebra $D^{\prime}$ spanned by ${p_{n}}_{n\in N\cup\{n^{\prime}\}}$, and then cut down the result by $I_{A}$. So without loss of generality we will assume our algebra is $L^{\infty}(X,\mu)\otimes R$ (essentially we are back to working in the normalised trace case).

As before we pick subalgebras $M_{2^{k+K}}$ of $R$ so that $\cup_{k=-K}^{\infty}M_{2^{k+K}}$ is dense in $R$, and denote the standard basis elements of $M_{2^{k+K}}$ by $e_{i,j}^{(k)}$ and the trace of $e_{i,i}^{(k)}$ is $2^{-k}$, and use the inclusion $M_{2^{k+K}}\to M_{2^{k+1+K}}$ where $e_{i,j}^{(k)}\to e_{2i-1,2j-1}^{(k+1)}+e_{2i,2j}^{(k+1)}$.

Let $f_{n}\in L^{\infty}(X,\mu)$ be the centre-valued trace of $p_{n}$ in $A$, for all $n\in N$. Then let $S_{n,k}=f^{-1}_{n}\left(\cup_{\ell=0}^{\infty})\left[\frac{1}{2^{k}},\frac{2}{2^{k}}\right)\right)$ for $k\geq-K$. Then note $f_{n}=\sum_{k=-K}^{\infty}\frac{1}{2^{k}}\chi_{S_{n,k}}$.

Now define $P_{0,k}$ to be a countable partition of $X$ so that for every $n\in N$, each $S_{n,k}$ with non-zero measure is the union of sets in $P_{0,k}$, and so that $P_{0,k}$ refines $P_{0,k-1}$ (note since $N$ is finite and $k$ is bounded below, this can be accomplished with a finite set of intersections). For each $S\in P_{0,k}$, let $s_{S,k}$ be the number of $n\in N$ such that $S\subseteq S_{n,k}$. Inductively define $r_{S,k}=s_{S,k}+2r_{S^{\prime},k-1}$ where $S\subseteq S^{\prime}\in P_{0,k-1}$, starting with $P_{0,-K}=\{X\}$ and $r_{X,-K}=0$.

Then define $A_{0}$ to be the multimatrix algebra spanned by the basis

Now then we can partition the diagonal basis elements, $e_{i,i}^{(k)}\otimes\chi_{S}$, from $B_{0}$ into sets $\{F_{n}\}_{n\in N}$ by placing one element of the form $e_{i,i}^{(k)}\otimes\chi_{S}$ into each $F_{n}$ where $S\subseteq S_{n,k}$, which works because there are exactly $s_{S,k}$ such elements.

This means for any $x\in X$ (except possibly on a set of measure zero an $n\in N$, we see that:

Thus we have a set of orthogonal projections $\{p_{n}^{\prime}\}_{n\in N}$ with centre-valued trace equal to $\{f_{n}\}_{n\in N}$. These are then equivalent to the $p_{n}$ in $D$, and thus without loss of generality we can identify them with $D$.

To define $B_{m}$, we first define $P_{m,k}$ to be a refinement of $P_{m-1,k}$ such for any $S\in P_{m,k}$ either $0<\mu(S)\leq 2^{-m}$ or $S$ is a single atom. We define $r_{S,k}$ and $s_{S,k}$ in the same way (note for $S\in P_{m,k}$ if there exists an $S^{\prime}\in P_{m^{\prime},k}$ with $S\subseteq S^{\prime}$ and $m>m^{\prime}$ then $r_{S,k}=r_{S^{\prime},k}$ and $s_{S,k}=s_{S^{\prime},k}$ so this is well defined, and we do not need to add an index $m$).

Then let $A_{m}$ be the multimatrix algebra spanned by

Recalling the inclusion of $M_{2^{m-1}}$ into $M_{2^{m}}$ and that each $P_{m,k}$ is a refinement of $P_{m-1,k}$, we see $A_{m-1}\subseteq A_{m}$.

We have established that for any $x\in X$ (except possibly on a set of measure zero) as $m$ goes to infinity the sum of $\{\tau(e_{i,i}^{(k)})=2^{-k}|e_{i,i}^{(k)}\otimes\chi_{S}\in B_{0},x\in S,k\leq m\}$ goes to $2^{K}$. Since for $A_{m}$ there is a set $S$ containing $x$ with measure less than $\max\{\mu(\{x\}),2^{-m}\}$ and matrix subalgebra $M_{m^{\prime}}\otimes\chi_{S}\subseteq M_{2^{m}}\otimes\chi_{S}$, whose trace is $\mu(S)$ times that sum. Thus this is dense in $\cup_{m}M_{2^{m}}\otimes L^{\infty}(X,\mu)$ and thus dense in $A$

If $N$ is only countable (and thus $A$ may be only semifinite), then let $q_{n}$ be the sum of the first $n$ $p_{j}$ in $D$. Then use the finite case to construct an approximating chain $A_{i,1}$ of $q_{1}Aq_{1}$ which contains $q_{1}Dq_{1}$. We claim we can use the finite case again to let $A_{i,n}$ be an approximating chain of $q_{n}Aq_{n}$ containing $q_{n}Dq_{n}$ and so that $A_{i,n}\subseteq A_{i,n+1}$ for all $i$ and for $n\geq 1$.

For the type I part, the above construction naturally allows us to do this. For the type II case we have to make some minor modifications to make this work.

For the type II construction we only note that in going from $q_{n}Aq_{n}$ to $q_{n+1}Aq_{n+1}$ we can consider it now as a cutdown of some $L^{\infty}(X^{\prime},\mu^{\prime})\otimes R\otimes M_{2^{K^{\prime}}}$ where $X^{\prime}$ contains $X$, and $\mu^{\prime}$ restricts to $\mu$ on $X$. We can then write this as $(L^{\infty}(X,\mu)\otimes R\otimes M_{2^{K^{\prime}}})\oplus(L^{\infty}(X^{\prime}\backslash X,\mu^{\prime})\otimes R\otimes M_{2^{K^{\prime}}})$, and approximate the second direct summand separately. The tensor with $M_{2^{K^{\prime}}}$ only changes the value $K$, which does not affect the $B_{m}$ for $m>-K$. We are also adding another projection $p_{n+1}$ (or two if we need a new $p_{(n+1)^{\prime}}$) , which creates a new partition $S_{n,K}$, but this merely further refines $P_{m,k}$, meaning with this construction each $B_{m}^{\prime}$ will contain our original $B_{m}$, thus completing the proving the claim.

Then define our approximating chain to be $A_{i}=A_{i,i}\oplus(1-q_{i})D(1-q_{i})$, completing the proof.

∎

The proof is identical to Lemma 3.5 in [6], following directly from examining either the Bratteli diagrams or the minimal projections. The only difference is that since we have a countably infinite number of matrix algebras, each of which can be split into a countably infinite number of matrix algebras, we may need a countably infinite number of steps. ∎

By Lemma 3.8 assume without loss of generality that $D$ is abelian. If $D$ is finite dimensional we can apply Theorem 2.9 (since if $A$ and $B$ were not finite, then the trace on the finite dimensional $D$ would not be semifinite), so we may also assume $D=\bigoplus\limits_{k=1}^{\infty}\overset{p_{k}^{D}}{\underset{t_{k}^{D}}{\mathbb{C}}}$.

Use Lemma 3.9 to define the sequences $A_{i}$ and $B_{j}$ to be chains of subalgebras of $A$ in $B$, respectively, containing $D$. Let $q_{k}^{D}=\sum_{k^{\prime}=1}^{k}p_{k^{\prime}}^{D}$. Then let

and let $\mathcal{N}(i,j,k)$ be

Note that in the construction from Lemma 3.9, if $A^{(a)}$ is the atomic part of $A$, then each $A_{i}$ contains $q_{k}^{D}A^{(a)}q_{k}^{D}$ for some $k$ (and the same is true for $B$).

For fixed $k$, the sequence $\mathcal{M}(i,j,k)$, is an approximating sequence for the amalgamated free product of hyperfinite von Neumann algebras over $q_{k}^{D}D$ (which is finite dimensional), as in Theorem 2.9. Thus we can choose $i_{k}$ and $j_{k}$ sufficiently large to be the start of the sequence in the proof of that theorem. In particular, for any minimal projection $p\in A_{i_{k}}$ (resp $B_{j_{k}}$) from the diffuse part of $A$ (resp. $B$), if $p\leq p_{k^{\prime}}^{D}$ with $k^{\prime}\leq k$ then $\tau(p)<t_{k^{\prime}}^{D}-\tau(p^{\prime})$ for any minimal projection $p^{\prime}\in B$ (resp $A$) with $p^{\prime}\leq p_{k^{\prime}}^{D}$, unless $p_{k^{\prime}}^{D}$ is minimal in $B$ (resp. $A$), and also if the number of interpolated free group factors in $\mathcal{M}(i,j,k)$ is stable for all $i\geq i_{k}$ and $j\geq j_{k}$. Furthermore we can assume that if $\text{rdim}(A)$ (resp $B$) is finite then $\text{rdim}(q_{k}A_{i_{k}}q_{k})>\text{rdim}(q_{k}Aq_{k})-1/k$. We chose $i_{k}$ and $j_{k}$ also sufficiently large so that $A_{i}$ contains $q_{k}^{D}A^{(a)}q_{k}^{D}$ and $B_{j}$ contains $q_{k}^{D}B^{(a)}q_{k}^{D}$ for all $i,j\geq i_{k},j_{k}$.