Gevrey Regularity and Compact Quantum Metric Spaces for $L^p$-Group Algebras

Abstract: We introduce the beta-Gevrey lp-rapid decay property (GRD){beta,p}, for 0 < beta <= 1 and 1 <= p < infinity, for countable discrete groups. This property is a subexponential analogue of classical rapid decay, in which polynomial control is replaced by logarithmic subexponential control of order o(R^beta). We establish basic results for (GRD){beta,p}. We then apply this framework to compact quantum metric structures on reduced Lp-group algebras. We introduce strongly dense-core beta-Gevrey regular lp-spectral triples and give two classes of examples. For countable discrete groups satisfying (GRD)_{beta,p}, we prove, using Rieffel's criterion, that the corresponding Gevrey seminorms induce metrics on the Banach-algebra state space which metrize the weak-* topology. This yields compact quantum metric space structures in settings beyond classical rapid decay, including groups of intermediate growth such as the first Grigorchuk group.

• The paper introduces a novel β-Gevrey ℓ^p-rapid decay property that extends classic RD conditions to subexponential frameworks.
• It constructs strongly dense-core β-Gevrey regular L^p-spectral triples where the decay controls ensure compatibility with quantum metric structures.
• The approach successfully metrizes the weak-* topology on state spaces for groups with intermediate and subexponential growth.

Gevrey Regularity and Compact Quantum Metric Spaces for $L^p$-Group Algebras: An Expert Analysis

Introduction and Problem Formulation

This paper addresses the development and characterization of compact quantum metric spaces (CQMS) utilizing reduced $L^p$-group algebras, extending the reach of noncommutative geometry and quantum metric theory to settings governed by subexponential, Gevrey-type regularity. The central innovation is the introduction of the $\beta$-Gevrey $\ell^p$-rapid decay property $(GRD)_{\beta,p}$ for discrete groups, which interpolates between polynomial (RD) and subexponential decay paradigms. Theoretical development proceeds by constructing strongly dense-core $\beta$-Gevrey regular $L^p$-spectral triples and demonstrating that Gevrey-type estimates suffice to metrize the weak-$*$ topology on state spaces of reduced $L^p$-group algebras—broadening the class of groups and algebras admitting a CQMS structure, with applications to intermediate-growth groups.

The $\beta$-Gevrey $L^p$0-Rapid Decay Property

The $L^p$1-Gevrey decay property, denoted $L^p$2 for $L^p$3 and $L^p$4, requires for a countable discrete group $L^p$5 with proper length function $L^p$6 the existence of subexponential control: $L^p$7 for all $L^p$8 supported in balls $L^p$9. This constitutes a subexponential logarithmic version of RD, where the classical polynomial control is replaced by subexponential envelopes modeled on Gevrey sequences—directed by the parameter $\beta$0, which interpolates between analytic ($\beta$1) and ultradifferentiable ($\beta$2) settings. Notably, when $\beta$3 has polynomial growth, bounded doubling, or subexponential growth, it satisfies $\beta$4 for all relevant $\beta$5. The property is stable under direct products and passage to subgroups, with the sharpest growth restrictions appearing for amenable groups.

The main technical implications are as follows:

• Classical $\beta$6 implies $\beta$7 for all $\beta$8.
• Subexponential group growth rates, such as those for the Grigorchuk group, yield $\beta$9 with $\ell^p$0 determined by the upper bounds on $\ell^p$1.
• Amenable groups with $\ell^p$2 necessarily have volume functions $\ell^p$3.

Strongly Dense-Core $\ell^p$4-Gevrey Regular $\ell^p$5-Spectral Triples

Spectral triples in the $\ell^p$6-Banach algebra context are constructed using closed (usually unbounded) operators $\ell^p$7 with compact resolvent and a norm-dense subalgebra over which all higher commutators $\ell^p$8, $\ell^p$9 are defined. Gevrey regularity is imposed via factorial-type bounds on the iterates of the derivation: $(GRD)_{\beta,p}$0 for $(GRD)_{\beta,p}$1 in a norm-dense subalgebra and any $(GRD)_{\beta,p}$2. The existence of such a structure places the algebra in the context of strongly dense-core $(GRD)_{\beta,p}$3-Gevrey regular $(GRD)_{\beta,p}$4-spectral triples.

Concrete settings include:

• $(GRD)_{\beta,p}$5, where the Gevrey growth of derivatives guarantees that trigonometric polynomials are contained in all $(GRD)_{\beta,p}$6 algebras.
• Reduced group algebras: $(GRD)_{\beta,p}$7, with $(GRD)_{\beta,p}$8 the multiplication operator by the length function. Here, the iterated commutators yield explicit powers of $(GRD)_{\beta,p}$9, and the Gevrey regularity holds on the algebra generated by group elements.

Compact Quantum Metrics and Weak-$\beta$0 Metrization

The Gevrey seminorm $\beta$1 is used in the Rieffel-type construction to define a Monge–Kantorovich metric on the Banach-algebraic state space: $\beta$2 Applying Rieffel's criterion for CQMSes requires showing that the image of the unit ball of the seminorm in the quotient $\beta$3 is totally bounded for the quotient norm. The proof is nontrivial, leveraging Gevrey tail estimates to produce strong control over the decay of matrix coefficients, even in the absence of RD, in turn preventing the escape of mass phenomena that are problematic in weak or exponential growth settings.

The main result is that for every countable discrete group $\beta$4 with $\beta$5, and for every $\beta$6, the metric $\beta$7 metrizes the weak-$\beta$8 topology on $\beta$9, thus providing a CQMS structure.

For polynomial growth groups and those of intermediate/subexponential growth (Grigorchuk group: cusp $L^p$0), the construction covers new territory not accessible by RD-based methods. In the bounded doubling case, all $L^p$1 (i.e., all Gevrey regularities) are available.

Implications and Future Directions

This framework conclusively extends quantum metric geometry to $L^p$2-operator algebras associated with groups of intermediate and subexponential growth, establishing a new class of CQMSes whose geometry is governed by Gevrey scales. Beyond proving the existence of strongly $L^p$3-Gevrey $L^p$4-quantum compact metric spaces, the machinery developed here suggests various directions:

• $L^p$5-theory and Noncommutative Analysis: The presence of Gevrey regularity opens pathways to the analysis of spectral invariants, local index formulas, and potential analytic torsion in these Banach-algebraic contexts.
• Quantum Gromov–Hausdorff Convergence: The parameter $L^p$6 can be interpreted as a regularity scale, which may provide refined structures for convergence of quantum metric spaces under deformation.
• C*- and $L^p$7-algebra Interactions: The generalization aligns with ongoing efforts to transfer $L^p$8-algebraic results to Banach settings, thus extending the toolbox for non-Hilbertian noncommutative geometry.

Conclusion

The paper provides a rigorous and general analytic foundation for the construction of compact quantum metric spaces from $L^p$9-group algebras, substituting Gevrey-type subexponential decay estimates for RD, and verifying that the associated spectral triples yield metrics compatible with the weak-$*$0 topology. The approach enables a uniform treatment of CQMS structures for groups beyond the RD case, with detailed tail estimates and functional calculi rooted in Gevrey analysis. These results significantly extend the analytical reach of quantum metrics in noncommutative geometry and suggest substantial theoretical and practical avenues for future work.