Hilbert transforms on graph products of finite von Neumann algebras

Abstract: We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interests on their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit $L_p$-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality and show, in particular, that it is equivalent to the graph product being generated by finite-dimensional von Neumann algebras with uniformly bounded dimensions. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide positive answers to a compactness problem posed by Ozawa in the setting of graph products of finite groups and right-angled Hecke von Neumann algebras.

• The paper establishes completely bounded Lp-boundedness for Hilbert transforms when the graph has uniformly bounded links and vertex algebras have uniformly bounded finite dimensions.
• It replaces the failed classical Cotlar identity with a generalized identity above the graph-dependent length threshold 3N, then combines it with Haagerup-type inequalities to obtain bounds for all 1<p<∞.
• The results apply to graph products of finite groups, right-angled Hecke von Neumann algebras, and compact quantum groups, while providing new positive cases of Ozawa’s compactness problem.

This paper develops the theory of Hilbert transforms on graph products of finite von Neumann algebras, establishing their $L_p$-boundedness for $1<p<\infty$ under a Haagerup-type inequality, and applies the results to Ozawa's compactness problem (2607.00194). The work extends the free-product theory of Mei and Ricard (Wand et al., 2016) to a setting that simultaneously generalizes free products and tensor products, where the key technical obstruction is that reduced operators no longer have uniquely determined initial syllables due to commutation relations.

Setting and definition of the Hilbert transform

Let $\Gamma=(V\Gamma,E\Gamma)$ be a simplicial graph and $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$ finite von Neumann algebras with normal faithful tracial states. The graph product $\mathcal{M}_\Gamma$ is again finite, since the graph product state is tracial by a result of Młotkowski. Reduced operators $a=a_1\cdots a_n$ (with $a_i\in\mathcal{M}_{s_i}^\circ$ and $s_1\cdots s_n$ a reduced word of the right-angled Coxeter group $W_\Gamma$) span a dense $*$-subalgebra, and the length $1<p<\infty$0 is well defined.

Because of braid relations, a reduced operator may admit several equivalent reduced expressions, so the initial syllable is not intrinsic. The authors therefore distinguish two families of $1<p<\infty$1-orthogonal projections: $1<p<\infty$2, projecting onto reduced operators whose every equivalent reduced expression begins with $1<p<\infty$3, and $1<p<\infty$4 for cliques $1<p<\infty$5, projecting onto operators whose maximal clique prefix $1<p<\infty$6 has type $1<p<\infty$7. The Hilbert transform is then

$1<p<\infty$8

with an adjoint-flipped variant $1<p<\infty$9. A second family $\Gamma=(V\Gamma,E\Gamma)$0, depending on the first $\Gamma=(V\Gamma,E\Gamma)$1 blocks, partitions operators of length at least $\Gamma=(V\Gamma,E\Gamma)$2 according to their length-$\Gamma=(V\Gamma,E\Gamma)$3 prefix, with a residual class $\Gamma=(V\Gamma,E\Gamma)$4. When $\Gamma=(V\Gamma,E\Gamma)$5 has no edges, $\Gamma=(V\Gamma,E\Gamma)$6 reduces to the Mei–Ricard free Hilbert transform.

Failure of the Cotlar identity and the generalized version

The Cotlar identity $\Gamma=(V\Gamma,E\Gamma)$7 underlying the classical proof of $\Gamma=(V\Gamma,E\Gamma)$8-boundedness fails in general for graph products. The paper gives an explicit counterexample: with reduced operators of types $\Gamma=(V\Gamma,E\Gamma)$9 and $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$0 over a path $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$1–$(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$2–$(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$3–$(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$4, the coefficient identity at the level of $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$5 requires $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$6, which fails for generic signs. The obstruction is precisely the non-uniqueness of reduced expressions.

The repair is a generalized Cotlar identity, valid only after projecting onto operators of length exceeding $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$7, where $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$8 bounds $(\mathcal{M}_s,\varphi_s)_{s\in V\Gamma}$9 over $\mathcal{M}_\Gamma$0 (condition (Fi)):

$\mathcal{M}_\Gamma$1

The proof rests on a factorization lemma for products of reduced operators (an operator-algebraic analogue of a result of Ciobanu–Holt–Rees): writing $\mathcal{M}_\Gamma$2 and $\mathcal{M}_\Gamma$3, the commutative core $\mathcal{M}_\Gamma$4 has length at most $\mathcal{M}_\Gamma$5 under (Fi), so whenever the product has length $\mathcal{M}_\Gamma$6, one of the outer factors has length $\mathcal{M}_\Gamma$7, forcing the maximal clique prefix to be determined by that factor alone and the identity to hold. For graph products of groups, a simpler direct proof via the group-theoretic lemma yields the identity for group elements $\mathcal{M}_\Gamma$8 with $\mathcal{M}_\Gamma$9. When $a=a_1\cdots a_n$0 has no edges ($a=a_1\cdots a_n$1), this recovers the Mei–Ricard Cotlar identity exactly.

Boundedness and the Haagerup-type inequality

The abstract bootstrap theorem is clean: any $a=a_1\cdots a_n$2-bounded $a=a_1\cdots a_n$3 satisfying the generalized Cotlar identity with respect to $a=a_1\cdots a_n$4 is bounded on $a=a_1\cdots a_n$5 for all $a=a_1\cdots a_n$6, provided $a=a_1\cdots a_n$7 extends boundedly from $a=a_1\cdots a_n$8 to $a=a_1\cdots a_n$9. The proof runs the Cotlar recursion from $a_i\in\mathcal{M}_{s_i}^\circ$0 to $a_i\in\mathcal{M}_{s_i}^\circ$1 and concludes by duality and interpolation, with an explicit bound $a_i\in\mathcal{M}_{s_i}^\circ$2 at each doubling step.

The remaining hypothesis is characterized in three equivalent ways: $a_i\in\mathcal{M}_{s_i}^\circ$3 is ultracontractive; $a_i\in\mathcal{M}_{s_i}^\circ$4 is a graph product of finite-dimensional algebras with $a_i\in\mathcal{M}_{s_i}^\circ$5; and $a_i\in\mathcal{M}_{s_i}^\circ$6 satisfies a Haagerup-type inequality, i.e., each homogeneous projection $a_i\in\mathcal{M}_{s_i}^\circ$7 is bounded from $a_i\in\mathcal{M}_{s_i}^\circ$8 to $a_i\in\mathcal{M}_{s_i}^\circ$9 with polynomial dependence on $s_1\cdots s_n$0. The implication (ii)$s_1\cdots s_n$1(iii) is proved via a length-additivity lemma ($s_1\cdots s_n$2 forces $s_1\cdots s_n$3) and a Cauchy–Schwarz argument yielding complete boundedness of $s_1\cdots s_n$4 with polynomial cb-norm. This equivalence is a strengthening of Caspers–Klisse–Larsen's Khintchine-based proof of the Haagerup inequality for graph product Hecke algebras, and it avoids Khintchine inequalities entirely. Consequently:

• Theorem B: under (Fi) and the uniform finite-dimensionality condition, $s_1\cdots s_n$5 and $s_1\cdots s_n$6 are bounded on $s_1\cdots s_n$7 for all $s_1\cdots s_n$8, with completely bounded versions also holding.
• Theorem C: the block-dependent transforms $s_1\cdots s_n$9 and $W_\Gamma$0 satisfy the generalized Cotlar identity modulo $W_\Gamma$1 and are $W_\Gamma$2-bounded.

A corollary gives Khintchine-type estimates for the families $W_\Gamma$3 in the mixed column-row norm of Pisier's noncommutative vector-valued $W_\Gamma$4-spaces.

Classes of examples

The framework covers three notable families. Graph products of finite groups $W_\Gamma$5 with $W_\Gamma$6: here the Haagerup-type inequality coincides with property (RD) of $W_\Gamma$7 with respect to the block length, and the paper's equivalence shows (RD) for the block length holds exactly when the vertex groups are finite with uniformly bounded orders. Right-angled Hecke von Neumann algebras $W_\Gamma$8 for arbitrary multiparameters $W_\Gamma$9: the paper proves the polynomial estimate $*$0 for graphs satisfying (Fi), extending the finite-graph result of Caspers–Klisse–Larsen and giving a Khintchine-free proof. Graph products of compact quantum groups of Kac type: under (RD) assumptions on the vertex duals and clique duals, the three formulations (matrix-coefficient estimates, Fourier-side (RD), and the $*$1-estimate) are shown equivalent, again yielding $*$2-bounded Hilbert transforms.

Application to Ozawa's compactness problem

Ozawa asked whether, for $*$3, the commutator of $*$4 with the projection $*$5 onto words ending in a fixed $*$6 maps the unit ball to an $*$7-compact set. The paper answers this affirmatively in two new settings: for graph products of finite groups satisfying the uniform cardinality bound, and for right-angled Hecke von Neumann algebras. The key commutator estimate is that $*$8 and $*$9 have finite rank, vanishing on all $1<p<\infty$00 with $1<p<\infty$01 (respectively $1<p<\infty$02); finiteness of the low-length part uses finiteness of the vertex groups. Combined with Hölder's inequality and Theorem C, this yields that $1<p<\infty$03 maps the unit ball of $1<p<\infty$04 (or $1<p<\infty$05) into a compact subset of $1<p<\infty$06 for every $1<p<\infty$07 in the $1<p<\infty$08-space, $1<p<\infty$09. Since graph products of finite groups over non-affine irreducible Coxeter systems are ICC, and certain right-angled Hecke von Neumann algebras are $1<p<\infty$10 factors by Garncarek's factoriality criterion, this produces new instances of Ozawa's phenomenon beyond free group factors.

Limitations and open questions

Several hypotheses constrain the scope of the results. The graph must satisfy (Fi), i.e., uniformly bounded vertex degrees; the paper does not treat graphs with unbounded links. The boundedness theorem requires finite-dimensionality of the vertex algebras with uniformly bounded dimensions, which the equivalence shows is not merely technical but necessary for the ultracontractivity route—though the authors note the free-product case ($1<p<\infty$11) requires no such assumption. The theory is developed for tracial (finite) von Neumann algebras; the authors state that the arguments should extend to type III with suitable modifications, but this is carried out only in remark form. Finally, the generalized Cotlar identity holds only above a length threshold $1<p<\infty$12 (or $1<p<\infty$13), and the paper leaves open the question of whether an unconditional identity, or a threshold-free argument via other means, is possible in the presence of commutation relations.

Conclusion

The paper establishes that Hilbert transforms on graph products of finite von Neumann algebras are $1<p<\infty$14-bounded for $1<p<\infty$15, provided the graph has uniformly bounded links and the vertex algebras are finite-dimensional of uniformly bounded dimension. The two structural innovations—a generalized Cotlar identity valid above a graph-determined length threshold, and the equivalence between ultracontractivity of the low-length projection and a Haagerup-type inequality—are of independent interest and give a new, Khintchine-free route to rapid decay for graph products. The resulting positive answers to Ozawa's compactness problem for graph products of finite groups and right-angled Hecke von Neumann algebras extend the known examples from free group factors to a substantially broader class of $1<p<\infty$16 factors.