Does the Markov property imply a finite Pimsner–Popa basis for connected inclusions of finite von Neumann algebras with diffuse centers?

Jones and Penneys (Quantum Topology, 2011) call an inclusion $M_0\subset (M_1,\mathrm{tr}_1)$ of finite von Neumann algebras Markov if the canonical faithful normal semifinite trace $\mathrm{Tr}_2$ on the Jones basic construction $M_2=\langle M_1,e_1\rangle$ is finite with $\mathrm{Tr}_2(1)^{-1}\mathrm{Tr}_2|_{M_1}=\mathrm{tr}_1$, and strongly Markov if moreover $M_1$ admits a finite Pimsner–Popa basis over $M_0$. Strongly Markov inclusions are exactly those to which their canonical planar $*$-algebra construction applies, the first step in their proof that every finite-depth subfactor planar algebra embeds in the bipartite graph planar algebra of its principal graph. Jolissaint (1990) proved that the Markov property implies the existence of a finite Pimsner–Popa basis for connected inclusions with atomic centers, and Jones and Penneys state that the case of connected inclusions with diffuse centers is unknown to them. Later work either stays within factors, multifactors (finite direct sums of factors), finite-dimensional algebras, regular or depth-2 subfactors, or center-preserving inclusions, or takes the finite basis as a hypothesis. The question — prove that every connected Markov inclusion with diffuse centers has a finite Pimsner–Popa basis, or exhibit one that does not — remains open.

Background

A von Neumann algebra is a unital $*$-algebra of bounded operators on a Hilbert space, closed in the weak operator topology; it is finite if it admits a faithful normal tracial state (a positive normalized trace functional). Its center $Z(M)$ is the subalgebra of elements commuting with all of $M$; $M$ is a factor when $Z(M)=\mathbb{C}$, and a II$_1$ factor is an infinite-dimensional finite factor. Subfactor theory studies inclusions $N\subset M$ of II$_1$ factors, their Jones index $[M:N]\in[1,\infty]$, and their standard invariants.

For a unital inclusion $M_0\subset (M_1,\mathrm{tr}_1)$ of finite von Neumann algebras with faithful normal tracial state $\mathrm{tr}_1$, the Jones projection $e_1$ is the orthogonal projection of $L^2(M_1,\mathrm{tr}_1)$ onto $L^2(M_0,\mathrm{tr}_0)$, where $\mathrm{tr}_0=\mathrm{tr}_1|_{M_0}$, and the basic construction is the von Neumann algebra $M_2=\langle M_1,e_1\rangle$ generated by $M_1$ and $e_1$; the trace-preserving conditional expectation $E_{M_0}:M_1\to M_0$ is determined by $e_1xe_1=E_{M_0}(x)e_1$. A Pimsner–Popa basis for $M_1$ over $M_0$ is a finite subset $B=\{b\}\subset M_1$ such that $1=\sum_{b\in B}be_1b^*$ in $M_2$; equivalently $x=\sum_b b\,E_{M_0}(b^*x)$ for all $x\in M_1$ (Proposition 2.2 of Jones–Penneys lists these conditions, together with $x=\sum_b E_{M_0}(xb)b^*$, as equivalent), i.e. $M_1$ is a finitely generated projective right $M_0$-module with $B$ as module basis. Popa showed that $M_2$ carries a canonical faithful normal semifinite trace $\mathrm{Tr}_2$ characterized by $\mathrm{Tr}_2(xe_1y)=\mathrm{tr}_1(xy)$. Following Jones–Penneys, the inclusion is called Markov if $\mathrm{Tr}_2$ is finite with $\mathrm{Tr}_2(1)^{-1}\mathrm{Tr}_2|_{M_1}=\mathrm{tr}_1$ — the exact analog, for inclusions with non-trivial centers, of the Markov trace condition of subfactor theory — and strongly Markov if moreover $M_1$ admits a finite Pimsner–Popa basis over $M_0$. When a basis exists, the Watatani index $\sum_b bb^*$ is independent of the basis and equals the scalar $\mathrm{Tr}_2(1)$; Watatani's index theory takes this finite quasi-basis condition as the very definition of finite index. Two classical classes are strongly Markov: finite-index inclusions of II$_1$ factors with their unique trace, for which Pimsner and Popa constructed orthonormal bases; and connected unital inclusions of finite-dimensional $C^*$-algebras with the Markov trace. Here an inclusion is connected when its two centers meet trivially, $Z(M_0)\cap Z(M_1)=\mathbb{C}$ (the meet in the lattice of von Neumann subalgebras of $M_1$ is one-dimensional), which for finite-dimensional algebras says the inclusion matrix is indecomposable; a center is atomic when the algebra is a finite direct sum of factors, and diffuse when it contains no minimal projections, e.g. $L^\infty$ of a non-atomic measure space.

The motivation comes from planar algebras. A planar algebra, in the sense of Jones, is a sequence of vector spaces ('$n$-box spaces') equipped with multilinear operations indexed by isotopy classes of shaded planar tangles; the subfactor planar algebra of a finite-index subfactor is its standard invariant, realized on the tower of higher relative commutants obtained by iterating the basic construction. A subfactor has finite depth when only finitely many simple bimodules occur in the tensor powers of the generating bimodule; equivalently its principal graph — the bipartite graph recording induction and restriction of simple bimodules across the subfactor — is finite. To every finite bipartite graph with a base vertex, Jones associated a bipartite graph planar algebra, spanned by loops on the graph. The embedding theorem of Jones and Penneys states that every finite-depth subfactor planar algebra $Q_\bullet$ is a planar subalgebra of the bipartite graph planar algebra of its principal graph. Its proof has three steps: (1) construct the canonical planar $*$-algebra of a strongly Markov inclusion from its basic-construction tower — the Pimsner–Popa basis is the essential ingredient, so the construction is available exactly for strongly Markov inclusions; (2) identify this canonical planar algebra with the bipartite graph planar algebra in the finite-dimensional case; (3) the shift trick — for finite depth one can choose $r\ge 0$ so that $Q_{2r,+}\subset Q_{2r+1,+}\subset (Q_{2r+2,+},e_{2r+1})$ is a basic construction, and then $M_0=Q_{2r,+}\subset M_1=Q_{2r+1,+}$ is a strongly Markov finite-dimensional inclusion whose canonical planar algebra contains $Q_\bullet$ via adding $2r$ or $2r+1$ strands. Step (3) is possible if and only if $Q_\bullet$ has finite depth, and this is precisely where the proof uses finite depth. (While the paper was being written, Morrison and Walker found a different proof, available so far only as an unpublished preprint, valid for every subfactor planar algebra without the finite-depth hypothesis; it works directly from the planar algebra rather than from towers of algebras.)

Because the canonical planar $*$-algebra of step (1) exists only under the basis hypothesis, the exact relationship between the Markov property and the existence of a basis determines the scope of the construction. Jolissaint proved that for connected inclusions with atomic centers the Markov property does imply a finite basis, so 'Markov' and 'strongly Markov' coincide in that case. For connected inclusions whose centers are diffuse, Jones and Penneys state that the implication is unknown to them (Remark 2.9 of the published version; Question 2.14 of the arXiv version), and no later resolution is known to date.

Problem Statement

Let $M_0\subset (M_1,\mathrm{tr}_1)$ be a unital inclusion of finite von Neumann algebras, with $\mathrm{tr}_1$ a faithful normal tracial state, such that the inclusion is connected ($Z(M_0)\cap Z(M_1)=\mathbb{C}$) and both centers are diffuse. Let $M_2=\langle M_1,e_1\rangle$ be the Jones basic construction, where $e_1$ is the orthogonal projection of $L^2(M_1,\mathrm{tr}_1)$ onto $L^2(M_0,\mathrm{tr}_0)$ with $\mathrm{tr}_0=\mathrm{tr}_1|_{M_0}$, and let $E_{M_0}:M_1\to M_0$ be the trace-preserving conditional expectation. Let $\mathrm{Tr}_2$ be the canonical faithful normal semifinite trace on $M_2$ determined by $\mathrm{Tr}_2(xe_1y)=\mathrm{tr}_1(xy)$. Suppose the inclusion is Markov:

$$\mathrm{Tr}_2 \text{ is finite and } \mathrm{Tr}_2(1)^{-1}\,\mathrm{Tr}_2\big|_{M_1}=\mathrm{tr}_1.$$

Does it follow that $M_1$ admits a finite Pimsner–Popa basis over $M_0$, i.e. a finite subset $B=\{b\}\subset M_1$ with

$$1_{M_2}=\sum_{b\in B} b\,e_1\,b^*,$$

equivalently $x=\sum_{b\in B} b\,E_{M_0}(b^*x)$ for all $x\in M_1$ (Proposition 2.2 of the source lists this, and also $x=\sum_{b\in B} E_{M_0}(xb)\,b^*$, as equivalent to the displayed identity)?

This is the question left open by Jones and Penneys (Remark 2.9 of the published version of 'The embedding theorem for finite depth subfactor planar algebras'; Question 2.14 of the arXiv version), which generalizes Jolissaint's affirmative answer for connected inclusions with atomic centers. No additional hypotheses — atomicity, finite dimensionality, containment of one center in the other, or irreducibility — may be added, since the question concerns exactly the diffuse connected case left open by the source.

References

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