*-Regular Dilation in Operator Theory

• *-Regular dilation is defined for contractive operator representations that are lifted to isometries or unitaries while ensuring the complete positivity of associated Toeplitz kernels.
• It is equivalent to having a minimal Nica-covariant isometric dilation, satisfying Brehmer-type and functional operator inequalities in settings like graph products and right LCM semigroups.
• Its applications span noncommutative function theory, operator algebras, and spectral dilation of annular operators, unifying classical results by Brehmer, Frazho–Bunce–Popescu, and Popescu.

A *-regular dilation is a central concept in operator theory and semigroup representation, characterizing when a tuple of (typically contractive) operators can be “lifted” or dilated to a tuple of isometries or unitaries while preserving certain algebraic and positivity properties. Developed extensively in the context of right-angled Artin monoids, quasi-lattice ordered semigroups, and more generally right LCM semigroups, *-regular dilation unifies classical results from Brehmer’s theory for $\mathbb{N}^k$, Frazho–Bunce–Popescu theory for row contractions, and recent advances on graph products and $q$-commuting tuples.

1. Algebraic Definition and Toeplitz Kernels

Let $\Gamma$ be a simple graph with vertex set $\Lambda$, and let $P_\Gamma = \ast_{i \in \Lambda} \mathbb{N}$ denote its right-angled Artin monoid (graph product of $\mathbb{N}$) with generators $\{e_i: i \in \Lambda\}$ subject only to $e_i e_j = e_j e_i$ if $(i, j) \in E(\Gamma)$ (Li, 2016). A contractive representation

$T : P_\Gamma \to \mathcal{B}(\mathcal{H})$

is specified by contractions $T_i = T(e_i)$, respecting the commutation relations of $\Gamma$.

One constructs the associated Toeplitz kernel

$$K(p, q) = \begin{cases} T(v) T(u)^* & \text{if } p^{-1}q = u^{-1}v,\; u, v \text{ no common initial generator, } u, v \text{ commute} \ 0 & \text{otherwise} \end{cases}$$

$\ast$-regularity of $T$ is the condition that for every finite subset of $P_\Gamma$, the matrix $[K(p_i, p_j)]_{i, j}$ is positive semidefinite, i.e., $K$ is a completely positive definite (CPD) kernel.

This definition generalizes to right LCM semigroups $P$: $T: P \to \mathcal{B}(\mathcal{H})$ has a *-regular dilation if the kernel

$$K(p, q) = \begin{cases} T(p^{-1} r) T(q^{-1} r)^* & \text{if } pP \cap qP = rP \ 0 & \text{otherwise} \end{cases}$$

is CPD (Li, 2017, Li et al., 24 Dec 2025).

2. Equivalence with Nica-Covariant Isometric Dilations

The foundational theorem is that for graph products of $\mathbb{N}$ or general right LCM semigroups, the following are equivalent for a contractive representation $T$ (Li, 2016, Li, 2017):

• $T$ is $\ast$-regular (its kernel $K$ is CPD).
• $T$ admits a minimal isometric Nica-covariant dilation $V$.
• $T$ satisfies Brehmer-type operator inequalities:

$$\sum_{\substack{U \subseteq W\ U\text{ is a clique}}} (-1)^{|U|} \, T(e_U) T(e_U)^* \geq 0$$

for all finite $W \subseteq \Lambda$ (with $e_U = \prod_{i \in U} e_i$).

Nica-covariance for $V$ requires

$$V(p) V(q)^* = \begin{cases} V(p \wedge q) V(p \wedge q)^* & \text{if } p \wedge q \text{ exists} \ 0 & \text{otherwise} \end{cases}$$

where $\wedge$ denotes the quasi-lattice meet.

The *-regular dilation thus acts as both a positivity and factorization condition, hinging on the combinatorics of the underlying monoid or semigroup structure.

3. Unified Classical Results: Brehmer, Frazho–Bunce–Popescu, Popescu

The *-regular criterion unifies:

• Brehmer's condition: For $\Gamma$ complete ($P_\Gamma \cong \mathbb{N}^k$), the positivity inequalities specialize to

$\sum_{U \subseteq W} (-1)^{|U|} T_U T_U^* \geq 0$

recovering regular dilation on $\mathbb{N}^k$.

• Row contraction case: For $\Gamma$ edgeless ($P_\Gamma$ is the free semigroup), only singleton cliques contribute, and the condition becomes

$I - \sum_{i=1}^k T_i T_i^* \geq 0$

matching Frazho–Bunce–Popescu's dilation theory for row contractions.

• Popescu's polyball property (P): For complete $k$-partite graph, weights $r^{|U|}$ are introduced in the clique sum, underlying Popescu’s dilation results for polydomains (Li, 2016).

4. Generalizations: Right LCM Semigroups, $q$-commuting Tuples

*-regular dilation extends to contractive representations on right LCM semigroups $P$ (Li, 2017, Li et al., 24 Dec 2025): The necessary and sufficient criterion for dilation is

$$Z_F = \sum_{U \subseteq F} (-1)^{|U|} T_U T_U^* \geq 0$$

with $T_U T_U^* = T_r T_r^*$ if $\cap p_i P = rP$; otherwise $0$.

For $q$-commuting tuples $(T_1, \dots, T_k)$, where $T_i T_j = q_{ij} T_j T_i$ with $|q_{ij}| = 1$, the *-regular dilation matches all mixed moments modulo the $q$-twist, and the positivity criterion (the $q$-Brehmer sum) is

$$S(u) = \sum_{v \subseteq u} (-1)^{|v|} T(e(v))^* T(e(v)) \geq 0$$

where $e(v)$ is the multi-index for the subset $v$ (Pal et al., 2024, Tomar, 13 Aug 2025). Minimal *-regular dilation is equivalent to existence of a tuple of doubly $q$-commuting isometries or unitaries matching the original tuple’s algebraic structure.

5. Functional Models and Spectral Dilation: Annular Operators

For tuples of commuting normal $\mathbb{A}_r$-contractions (operators with spectrum in a closed annulus $\overline{\mathbb{A}_r}$), a *-regular dilation produces boundary unitaries $(U_1, \dots, U_m)$ with joint spectrum on the distinguished boundary, constructed via solution of a Dirichlet problem on the polyannulus and measure-theoretic dilation (Naimark extension) (Pal et al., 2023). This construction generalizes Sz.-Nagy’s dilation for the disk to the annular setting, and applies to both normal and doubly commuting subnormal tuples.

6. Kernel Positivity and the Brehmer Criterion

The operator inequalities central to *-regularity are encapsulated in the Brehmer-type kernel positivity. Verifying *-regularity at the level of finite sets (cliques, blocks, generators) often reduces the problem to positive-definiteness of alternating sums of operator block matrices. If the underlying semigroup has the descending chain condition, these checks can often be restricted to minimal generators, rendering the criteria combinatorial (Li, 2017).

For $q$-commuting tuples, phase factors must be incorporated in all kernel and moment computations, but structural results and equivalences parallel the commutative case.

Setting	Positivity Criterion	Dilation Type
$\mathbb{N}^k$	Brehmer sum over subsets	Commuting isometries
Free semigroup $\mathbb{F}_k^+$	$I - \sum T_i T_i^*$	Row isometries, orthogonal
Graph product $P_\Gamma$	Sum over cliques in $\Gamma$	Nica-covariant isometries
$q$-commuting tuple	$q$-Brehmer sum (twisted moments)	$q$-commuting unitaries

7. Applications and Structural Implications

*-regular dilation provides a structural tool for decomposing complex multi-operator systems into well-behaved model tuples (isometries, unitaries, Nica-covariant dilations), enabling explicit analysis of their operator-theoretic properties. In particular, for representations of graph products, right LCM semigroups, and $q$-commuting families, it yields necessary and sufficient conditions for dilation and informs the classification of operator tuples in terms of their commutation and positivity behavior. Applications include noncommutative function theory, operator algebras, and dilation of spectral sets in harmonic analysis.

A plausible implication is that further generalizations are possible to broader classes of semigroups and modules, with kernel positivity serving as the universal test for dilation existence. The integration of measure-theoretic and combinatorial conditions remains a unifying theme across most dilation frameworks.