Spectral Invariance and Gevrey Regularity for Groups with strongly subexponential growth

Abstract: We study spectral invariance and Gevrey regularity for convolution operators with kernels in suitable weighted function spaces on locally compact groups equipped with a locally bounded length function $\ell$. The main analytic scale is given by the subexponential weights. For groups whose volume growth is bounded above by $e^{R^γ}$ for some $0<γ<1$, we establish spectral comparison result for compactly supported functions. For compactly supported Hermitian functions, we prove spectral radius invariance across the symmetric $q$-pseudofunction $$-algebra, the weighted and unweighted group algebras, and the full and reduced group $C^$-algebras. For unimodular groups satisfying strong subexponential growth of exponent at most $β$, we construct a Gevrey-Beurling operator algebra inside the unitized $q$-pseudofunction algebra. We prove that this algebra is inverse-closed and that its inclusion induces an isomorphism in topological $K$-theory. The inverse-closedness theorem may be viewed as a quantitative Gevrey-type noncommutative Wiener lemma. As an application, we show that whenever a convolution operators with kernels in the corresponding weighted Gevrey-Beurling space is invertible in the unitized $q$-pseudofunction algebra, then its inverse belongs to the same Gevrey-Beurling operator algebra and satisfies explicit Gevrey seminorm estimates. We also develop a relative theory for pairs of finitely generated groups using Schreier graph lengths and quasi-regular representations. This provides a subexponential analogue of rapid decay for group pairs, when subgroup is normal, it reduces to the usual theory on the quotient. The framework can apply to intermediate-growth examples, including the Grigorchuk group, and is stable under products with polynomial growth groups and under compact extensions.

• The paper extends spectral invariance from polynomial to subexponential weighted algebras on groups with strong subexponential growth.
• It introduces Gevrey regularity for convolution operators with factorial-type norm controls, ensuring inverse-closedness of operator algebras.
• The results yield K-theoretic equivalences and extend to group pairs, providing robust tools for noncommutative geometry and index theory.

Spectral Invariance and Gevrey Regularity on Groups of Strongly Subexponential Growth

Introduction

This paper addresses spectral invariance and a novel form of regularity—Gevrey regularity—for convolution operators on locally compact groups exhibiting strong subexponential growth. The primary analytic tool is provided by a family of subexponential weights $\omega_s(x)=\exp(s \ell(x)^\beta)$ for $0<\beta < 1$, constructed via a locally bounded length function $\ell$ on the group. The authors systematically analyze how these weights lead to a precise control of function space and operator algebra regularity, yielding significant results in noncommutative harmonic analysis and $C^*$-algebra theory, with extensions to the context of group pairs and intermediate growth groups.

Spectral Invariance with Subexponential Weights

A core achievement is the extension of spectral invariance results, classically known for polynomial weights and groups of polynomial growth, to algebras with subexponential weights on groups whose growth is upper-bounded by $e^{R^\gamma}$ for some $0 < \gamma < 1$. The authors prove that for such groups, the spectrum of a compactly supported function $f \in C_c(G)$ is invariant under passage among weighted $L^1(G, \omega_s)$, unweighted $L^1(G)$, the full and reduced group $C^*$-algebras, and, in the Hermitian case, in the symmetric $0<\beta < 1$0-pseudofunction algebra $0<\beta < 1$1. This is formalized in their Theorem~A, and makes explicit that, under strong subexponential growth, $0<\beta < 1$2 inherits quasi-symmetry.

Notably, these results show the equivalence of spectral properties between various Banach $0<\beta < 1$3-algebras associated with $0<\beta < 1$4, provided the functions are compactly supported. Further, the proofs leverage a refined version of the Barnes-Hulanicki theorem, and rely on the geometric control offered by the subexponential weights.

Gevrey Regularity and Its Operator-Algebraic Realization

The paper develops a class of Gevrey-type spaces of functions and their operator-algebraic analogues, adapted to the group setting and parameterized by the subexponential weight exponent $0<\beta < 1$5. Function $0<\beta < 1$6 belongs to $0<\beta < 1$7 if it lies in every $0<\beta < 1$8. The authors show that the convolution operator associated to such a function exhibits controlled regularity with respect to the commutator derivation $0<\beta < 1$9, satisfying factorial-like bounds dictated by the Gevrey order corresponding to $\ell$0.

On the operator side, they construct a Gevrey-Beurling operator algebra $\ell$1, defined as the intersection of the unitized $\ell$2-pseudofunction algebra and the Gevrey class in $\ell$3. The main theorem here (Theorem~B) establishes that this algebra is inverse-closed: If a convolution operator with kernel in $\ell$4 is invertible in the unitized pseudofunction algebra, then its inverse remains in $\ell$5, and explicit norm estimates are provided for Gevrey seminorms of the inverse. This is achieved through a sophisticated adaptation of norm-controlled inversion for Dales-Davie and Gevrey algebras.

K-Theory Isomorphism and Stability Under Constructions

Leveraging the spectral invariance and holomorphic functional calculus properties, the paper shows (Corollary~K) that the inclusion of the Gevrey-Beurling operator algebra into the unitized pseudofunction algebra induces an isomorphism in topological $\ell$6-theory. Thus, $\ell$7 provides a robust, $\ell$8-theoretically equivalent smooth subalgebra, suitable for applications in index theory and noncommutative geometry.

Furthermore, the framework is shown to be stable under group-theoretic constructions: The subexponential growth properties persist under direct products, compact extensions, and passage to quotients, as illustrated for the Grigorchuk group and products with polynomial growth groups.

Relative Gevrey Regularity for Group Pairs

Significantly, the authors introduce a relative Gevrey regularity theory for pairs $\ell$9, where $C^*$0 is a finitely generated subgroup of $C^*$1. By employing the Schreier graph length $C^*$2 and quasi-regular representations, they develop subexponential-weighted spaces and corresponding operator algebras. For these, an analogue of their previous inverse-closedness result holds (Theorem~D). If $C^*$3 is normal, the theory collapses to the quotient group, but for non-normal $C^*$4, the regularity only controls growth in directions transverse to $C^*$5, as detailed by concrete examples.

Numerical Strengths and Contradictory Claims

• The spectral invariance theorems explicitly equate the spectrum and spectral radius in disparate algebras for compactly supported elements, under strong subexponential growth.
• The Gevrey-type operator algebras are shown to be inverse-closed with explicit factorial norm estimates, generalizing the classical Wiener lemma to this noncommutative, geometrically refined context.
• The framework applies to intermediate-growth groups like the Grigorchuk group, which fall outside the reach of rapid decay (polynomial-weight) smooth algebras.

Theoretical and Practical Implications

These results have extensive implications for noncommutative geometry, index theory, and the analysis of $C^*$6-algebras of groups beyond polynomial growth. The inverse-closed Gevrey-Beurling algebras become natural domains for cyclic cocycles and higher index invariants, with $C^*$7-theory unchanged from the ambient $C^*$8- or pseudofunction algebra. This provides new analytic tools for groups such as those of intermediate growth, relevant for both rigidity and deformation problems in $C^*$9-theory and operator algebras. The findings also illuminate key distinctions between rapid decay properties and more refined subexponential smoothness conditions.

Developing relative Gevrey regularity for group pairs further expands this analytic machinery, making it compatible with modern treatments of groupoids, Fell bundles, and higher index theory for singular spaces.

Future Directions

The concluding section poses several open questions, including:

• Intrinsic versions of the main spectral invariance and inverse-closedness results for the non-unitized algebra, and for all convolution operators (not only those associated to functions in $e^{R^\gamma}$0).
• Development of a full cyclic and local cyclic homology theory for these Gevrey-Beurling algebras, especially their extension properties, and the characterization of smooth cyclic cocycles.
• Extensions to étale groupoids and Fell bundles, as suggested by recent advances in the field.

Conclusion

The paper develops a rigorous, explicit theory of subexponential-weighted Gevrey regularity for convolution algebras on groups of strong subexponential growth. It proves robust spectral invariance, constructs inverse-closed smooth subalgebras with full $e^{R^\gamma}$1-theoretic equivalence, and establishes powerful results in a relative context. This analytic and $e^{R^\gamma}$2-algebraic framework greatly expands the landscape of possible analytic and geometric applications in harmonic analysis, noncommutative geometry, and operator algebras—especially for groups beyond the scope of the rapid decay paradigm.