[Paper Review] Free group $C^*$-algebras associated with $\ell_p$

Rui Okayasu|arXiv (Cornell University)|Mar 5, 2012

Advanced Operator Algebra Research6 references3 citations

TL;DR

This paper generalizes Haagerup's characterization of positive definite functions on free groups to the $C^*$-algebra associated with $\ell_p$ for $p \geq 2$, showing that these algebras are mutually non-isomorphic and each admits a unique tracial state. The characterization hinges on the decay rate of positive definite functions in $\ell_p$-norm, extending known results for the reduced group $C^*$-algebra ($p=2$) and establishing isomorphisms with $D_p^\pm$-completions.

ABSTRACT

For every $p\geq 2$, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to the positive linear functionals on the free group $C^*$-algebra associated with the ideal $\ell_p$. This is a generalization of Haagerup's characterization for the case of the reduced free group $C^*$-algebra. As a consequence, the associated $C^*$-algebras are mutually non-isomorphic, and they have a unique tracial state.

Motivation & Objective

• To extend Haagerup's characterization of positive definite functions on free groups to $C^*$-algebras associated with $\ell_p$ for $p \geq 2$, generalizing the known result for the reduced group $C^*$-algebra ($p=2$).
• To establish that the $C^*$-algebras $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \geq 2$ are mutually non-isomorphic.
• To prove that each $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \in [2, \infty)$ admits a unique tracial state.
• To show that the $C^*$-algebras associated with the ideals $D_p^\pm(\Gamma)$ coincide with those associated with $\ell_p(\Gamma)$.

Proposed method

• Introduces a characterization of positive definite functions on a free group $\mathbb{F}_d$ that extend to positive linear functionals on $C^*_{\ell_p}(\mathbb{F}_d)$, using decay conditions in $\ell_p$-norm.
• Applies Hölder's inequality and estimates on the growth of conjugacy classes in free groups to bound the $\ell_p$-norm of positive definite functions.
• Uses the structure of word length and conjugacy classes in $\mathbb{F}_d$ to derive lower bounds on the number of elements in $W_{k+2n} \cap K$ for nontrivial conjugacy classes $K$, showing exponential growth.
• Establishes isomorphisms between $C^*_{\ell_p}(\mathbb{F}_d)$ and $C^*_{D_p^\pm}(\mathbb{F}_d)$ by approximating $\ell_p$-functions via $\ell_q$-functions with $q$ approaching $p$ from above or below.
• Employs weak-* topology convergence of approximating functionals $\varphi\varphi_\beta$ to $\varphi$ as $\beta \to 1$, ensuring extension to $C^*_{\ell_p}$-functionals.
• Leverages known results from Brown and Guentner on $C^*$-completions via ideals in $\ell_\infty(\Gamma)$, particularly the characterization of amenability and the Haagerup property.

Experimental results

Research questions

• RQ1 Which positive definite functions on a free group $\mathbb{F}_d$ extend to positive linear functionals on the $C^*$-algebra $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \geq 2$?
• RQ2 Are the $C^*$-algebras $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \in [2, \infty]$ mutually non-isomorphic?
• RQ3 Does each $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \in [2, \infty)$ admit a unique tracial state?
• RQ4 How do the $C^*$-algebras associated with the ideals $D_p^\pm(\Gamma) = \bigcap_{\varepsilon>0} \ell_{p+\varepsilon}(\Gamma)$ and $\bigcup_{\varepsilon>0} \ell_{p-\varepsilon}(\Gamma)$ relate to those associated with $\ell_p(\Gamma)$?

Key findings

• For every $p \geq 2$, the $C^*$-algebra $C^*_{\ell_p}(\mathbb{F}_d)$ is isomorphic to $C^*_{D_p^+}(\mathbb{F}_d)$, where $D_p^+(\Gamma) = \bigcap_{\varepsilon>0} \ell_{p+\varepsilon}(\Gamma)$, showing that the completion is stable under small perturbations of the exponent above $p$.
• For every $p > 2$, the $C^*$-algebra $C^*_{\ell_p}(\mathbb{F}_d)$ is isomorphic to $C^*_{D_p^-}(\mathbb{F}_d)$, where $D_p^-(\Gamma) = \bigcup_{\varepsilon>0} \ell_{p-\varepsilon}(\Gamma)$, indicating stability under small decreases in the exponent below $p$.
• The $C^*$-algebras $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \in [2, \infty]$ are mutually non-isomorphic, as shown by the distinct decay conditions required for positive definite functions to extend to $C^*_{\ell_p}$-functionals.
• Each $C^*_{\ell_p}(\mathbb{F}_d)$ for $p \in [2, \infty)$ has a unique tracial state, derived from the fact that any tracial state would correspond to a positive definite function constant on conjugacy classes, which cannot lie in $\ell_p$ unless zero, contradicting normalization.
• The full group $C^*$-algebra $C^*(\mathbb{F}_d)$ is isomorphic to $C^*_{D_\infty^-}(\mathbb{F}_d)$, where $D_\infty^-(\Gamma) = \bigcup_{\varepsilon>0} \ell_{\infty - \varepsilon}(\Gamma)$, extending the known isomorphism $C^*_{\lambda}(\mathbb{F}_d) = C^*_{D_2^+}(\mathbb{F}_d)$.
• The characterization of extendable positive definite functions on $\mathbb{F}_d$ to $C^*_{\ell_p}(\mathbb{F}_d)$ is a strict generalization of Haagerup's result for $p=2$, with the decay rate $|\varphi(s)| \lesssim \alpha^{|s|}$ for $\alpha < (2d-1)^{-1/p}$ being necessary and sufficient for $\varphi \in \ell_p$.