Spectral dimension of $p$-adic integers

Connes’ formulation of noncommutative geometry centers around the concept of spectral triples [1]. These triples originate from fundamental properties of Dirac-type operators on manifolds. Importantly, a spectral triple for the algebra of continuous functions and Dirac operators on $L^{2}$ spinors has the remarkable ability to fully reconstruct a closed Riemannian manifold with a spin structure. This achievement is made possible through Connes’ reconstruction theorem [2] and the contributions of Lord et al [8].

Noncommutative concepts, described by spectral triples, provide a distinct perspective on the group of $p$-adic integers, redefining them as differentiable spaces. Operators associated with $p$-adic numbers hold significant number-theoretic implications, potentially giving rise to captivating spectral functions. Of note, Connes’ unpublished construction of a spectral triple for the algebra of continuous functions on the Cantor set is documented in [3]. Klimek et al [7] construct a spectral triple for the $C^{*}$-algebra of continuous functions on the space of $p$-adic integers. Their approach involves utilizing a rooted tree derived from a coarse-grained approximation of the space and applying the forward derivative on this tree. They not only validate the spectral triple’s compliance with the traits of a compact spectral metric space but also establish its equivalence to the traditional $p$-adic metric on the space of $p$-adic integers.

Motivated by Connes’ definition of dimension for spectral triples,Chakraborty and Pal [4] introduced the spectral dimension as an invariant for ergodic $C^{*}$-dynamical systems. They conjectured that for a homogeneous space of a classical compact Lie group, the spectral dimension matches its dimension as a differentiable manifold. The spectral dimensions of $SU(2)$, quaternion sphere $H^{n}$, and sphere $S^{n}$ computed in [4], [11] and [12] further bolster Chakraborty and Pal’s conjecture. Furthermore, in [4], spectral dimensions were computed for the noncommutative torus, $q$-deformation of $SU(l+1)$ $(l\geq 1)$, and the Cuntz algebra $A_{u}(Q)$. The spectral dimension of the $q$-deformation of $U(2)$ was computed in [6]. In this article, Section 3 computes the spectral dimension of the group of $p$-adic integers. Section 4 describes the $K$-groups of $\mathbb{Z}_{p}$.

In this section, we recall the notion of the spectral dimension of ahomogeneous space from the reference [4].

Since $D$ has compact resolvent, this is equivalent to asserting that $|D|^{-s}$ is trace-class on the complement of its kernel.

Every compact quantum group possesses a unique invariant Haar state $h$. The Haar state’s invariance is described as follows:

In the previous definition, for $j=2,3$, define $u_{1j}$ as $\phi_{1j}(u)$, where $\phi_{1j}:\mathcal{K}(\mathcal{H})\otimes C(G)\rightarrow\mathcal{K}(\mathcal{H})\otimes C(G)\otimes C(G)$ is given by $\phi_{12}(T\otimes a)=T\otimes a\otimes I$ and $\phi_{13}(T\otimes a)=T\otimes I\otimes a$ for all $T\in\mathcal{K}(\mathcal{H})$ and $a\in C(G)$. For a compact Hausdorff space $G$, $M\left(\mathcal{K}(\mathcal{H})\otimes C(G)\right)$ is isomorphic to $C_{b}^{str}\left(G,\mathcal{L}(\mathcal{H})\right)$, where $C_{b}^{str}\left(G,\mathcal{L}(\mathcal{H})\right)$ represents the set of bounded continuous functions from $G$ to $\mathcal{L}(\mathcal{H})$ equipped with the strict topology. The convergence of a net $\langle T_{i}\rangle_{i}$ to $T$ in $\mathcal{L}(\mathcal{H})$ in the strict sense is characterized by $\|T_{i}S-TS\|\to 0$ and $\|ST_{i}-ST\|\to 0$ for all compact operators $S\in\mathcal{K}(\mathcal{H})$.

We begin by recalling the group of $p$-adic integers from [13], discussing its defnition and key topological characteristics.

Evidently, $\mathbb{Z}$ (the set of integers) forms a dense subset within $\mathbb{Z}_{p}$. This implies that elements of $\mathbb{Z}_{p}$ can be regarded as formal power series $\sum_{n=0}^{\infty}x_{n}p^{n}$, where $0\leq x_{n}\leq p-1$. The topological group $(\mathbb{Z}_{p},+)$ is well-established as compact, totally disconnected, and Hausdorff. As a compact abelian group, $\mathbb{Z}_{p}$ possesses a unique probability Haar measure denoted as $\mu$.

Henceforth, we adopt the notation $\chi_{m,n}$ for all $(m,n)\in S$ to represent the character of $\mathbb{Z}_{p}$, where $\chi_{m,n}(1)=e^{\frac{2\pi im}{p^{n}}}$. We consider $C(\mathbb{Z}_{p})$ both as a compact quantum group and an unital $C^{*}$-algebra and compute the spectral dimension of the natural ergodic $C^{*}$-dynamical system $\left(C(\mathbb{Z}_{p}),C(\mathbb{Z}_{p}),\Delta\right)$ associated with $\mathbb{Z}_{p}$. The Haar state on $C(\mathbb{Z}_{p})$ serves as an invariant Haar state for the homogeneous space $C(\mathbb{Z}_{p})$. The relevant covariant representation of this system is given by the triple $\left(L^{2}(\mathbb{Z}_{p}),\pi,u\right)$, where $L^{2}(\mathbb{Z}_{p})$ is the GNS Hilbert space corresponding to the Haar state on $C(\mathbb{Z}_{p})$, $\pi$ is the representation of $C(\mathbb{Z}_{p})$ on $L^{2}(\mathbb{Z}_{p})$ through left multiplication, and $u$ is the right regular representation.

$\mathbb{Z}_{p}$ can alternatively be identified as the inverse (or projective) limit of the system $\left\{\left\langle\mathbb{Z}/{p^{n}\mathbb{Z}}\right\rangle_{n\in\mathbb{N}},\left\langle\Phi_{n}\right\rangle_{n=2}^{\infty}\right\}$, where for each $n\in\mathbb{N}$ with $n\geq 2$, the transition map $\Phi_{n}:\mathbb{Z}/{p^{n}\mathbb{Z}}\rightarrow\mathbb{Z}/{p^{n-1}\mathbb{Z}}$ is defined as $\Phi_{n}\left(x\mod p^{n}\right):=x\mod p^{n-1}$ for all $x\in\mathbb{Z}$. This identification is precisely given by the isomorphism $\phi:\mathbb{Z}_{p}\rightarrow\varprojlim\mathbb{Z}/{p^{n}\mathbb{Z}}$ defined by $\phi(x)=\left\langle\phi_{n}(x)\right\rangle_{n\in\mathbb{N}}$ where for each $n\in\mathbb{N}$, $\phi_{n}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}/{p^{n}\mathbb{Z}}$ is the projection map defined by $\phi\left(\sum_{k=0}^{\infty}x_{k}p^{k}\right)$ $=\sum_{k=0}^{n-1}x_{k}p^{k}+p^{n}\mathbb{Z}$. Consequently, $C(\mathbb{Z}_{p})$ can be seen as a $C^{*}$-algebraic inductive limit of the induced system $\left\{C\left(\mathbb{Z}/{p^{n}\mathbb{Z}}\right),\left\langle\Psi_{n}\right\rangle_{n=2}^{\infty}\right\}$, where $\Psi_{n}:C\left(\mathbb{Z}/{p^{n-1}\mathbb{Z}}\right)\rightarrow C\left(\mathbb{Z}/{p^{n}\mathbb{Z}}\right)$ is the induced transition map defined by $\left(\Psi_{n}f\right)(x\mod p^{n}):=f(x\mod p^{n-1})$ for all $f\in C\left(\mathbb{Z}/{p^{n-1}\mathbb{Z}}\right)$ and $x\in\mathbb{Z}$. The induced isomorphism $\psi:\varinjlim C\left(\mathbb{Z}/{p^{n}\mathbb{Z}}\right)\rightarrow C(\mathbb{Z}_{p})$ is given by $\psi\left(\langle f_{n}\rangle_{n\in\mathbb{N}}\right)(x)=\lim_{n\rightarrow\infty}f_{n}\phi_{n}(x)$ for all $\langle f_{n}\rangle_{n\in\mathbb{N}}\in\varinjlim C\left(\mathbb{Z}/{p^{n}\mathbb{Z}}\right)$ and $x\in\mathbb{Z}_{p}$. As a result, this identification $C(\mathbb{Z}_{p})=\varinjlim C\left(\mathbb{Z}/{p^{n}\mathbb{Z}}\right)$ implies $K_{i}\left(C(\mathbb{Z}_{p})\right)=\varinjlim K_{i}\left(C\left(\mathbb{Z}/{p^{n}\mathbb{Z}}\right)\right)$ for $i=1,2$. Since $K_{1}\left(C(\mathbb{Z}/{p^{n}\mathbb{Z}})\right)=0$ for all $n\in\mathbb{N}$, it follows that $K_{1}\left(C(\mathbb{Z}_{p})\right)=0$.

For every natural number $r$, $\mathbb{Z}_{p}$ can be expressed as the disjoint union of $p^{r}$ balls $x+p^{r}\mathbb{Z}_{p}$ for $x\in\{0,1,\ldots,p^{r}-1\}$. Consequently, each ball $x+p^{r}\mathbb{Z}_{p}$ is both open and closed. For a subset $B$ of a set $X$, we denote the characteristic function of $B$ on $X$ as $\mathbbm{1}_{B}$.

We recall a well-known continuous mapping from $\mathbb{Z}_{p}$ to the interval $[0,1]$, known as the Monna map [9]. This mapping, denoted as $T:\mathbb{Z}_{p}\rightarrow[0,1]$, is defined by $T\left(\sum_{k=0}^{\infty}x_{k}p^{k}\right)=\sum_{k=0}^{\infty}\frac{x_{k}}{p^{k+1}}$, where $0\leq x_{k}\leq p-1$. The Monna map $T$ is continuous with respect to the $p$-adic metric on $\mathbb{Z}_{p}$ and the standard Euclidean metric on $[0,1]$. Furthermore, it preserves measures, with $T$ being measure-preserving concerning the probability Haar measure on $\mathbb{Z}_{p}$ and the Lebesgue measure $\lambda$ on $[0,1]$. Additionally, we note that the set $E=\{x\in[0,1]:x\text{ has multiple base }p\text{ representations}\}$ is countable and thus has measure zero. Consequently, when we restrict the map $T$ to $\mathbb{Z}_{p}\setminus T^{-1}[E]$, it becomes a bijection. As a result, the induced map $\widetilde{T}:L^{2}([0,1])\rightarrow L^{2}(\mathbb{Z}_{p})$ defined as $\widetilde{T}(f):=f\circ T$, where $f\in L^{2}([0,1])$, is unitary.