Topological Free Entropy Dimension of in Unital C^*-algebras

TL;DR

This paper computes the topological free entropy dimension for individual self-adjoint elements and families of generators in unital C*-algebras, providing new insights into their structural properties.

Contribution

It introduces explicit calculations of topological free entropy dimensions for specific elements and classes of unital C*-algebras, expanding understanding of their entropy characteristics.

Findings

Computed topological free entropy dimension of one self-adjoint element.

Calculated topological orbit dimension of one self-adjoint element.

Determined entropy dimensions for generators of irrational rotation and free group tensor product C*-algebras.

Abstract

The notion of topological free entropy dimension of $n-$tuples of elements in a unital C$^*$ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological orbit dimension of one self-adjoint element in a unital C$^*$ algebra. Moreover, we calculate the values of topological free entropy dimensions of families of generators of some unital C$^*$ algebras (for example: irrational rotation C$^*$ algebras or minimal tensor product of two reduced C$^*$ algebras of free groups).

Equations

$$\Vert (A_1,\dots ,A_n)\Vert =\max \{\Vert A_1\Vert ,\dots ,\Vert A_n\Vert \}$$

$$\Vert (A_1,\dots ,A_n){\Vert }_2=\sqrt{{\tau }_k(A_1^{\ast }{A}_1)+\dots +{\tau }_k(A_n^{\ast }{A}_n)}$$

$$\Vert (A_1,\dots ,A_n)-(B_1,\dots ,B_n)\Vert <\omega .$$

$$\Vert (A_1,\dots ,A_n)-(B_1,\dots ,B_n){\Vert }_2<\omega .$$

$${\Gamma }_R^{(top)}(x_1,\dots ,x_n,y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r)$$

$$(A_1,\dots ,A_n,B_1,\dots ,B_m)\in ({\mathcal{M}}_k^{s.a}(\mathbb{C}){)}^{n+m}$$

$$\max \{\Vert A_1\Vert ,\dots ,\Vert A_n\Vert ,\Vert B_1\Vert ,\dots ,\Vert B_m\Vert \}\le R$$

$$|\Vert P_j(A_1,\dots ,A_n,B_1,\dots ,B_m)\Vert -\Vert P_j(x_1,\dots ,x_n,y_1,\dots ,y_m)\Vert |\le ϵ,\forall 1\le j\le r.$$

$$P_j(x_1,\dots ,x_n,y_1,\dots ,y_m)={\alpha }_0\cdot I_{\mathcal{A}}+\sum _{s=1}^N\sum _{1\le i_1,\dots ,i_s\le n+m}{\alpha }_{i_1\cdots i_s}{z}_{i_1}\cdots z_{i_s}$$

$$P_j(A_1,\dots ,A_n,B_1,\dots ,B_m)={\alpha }_0\cdot I_k+\sum _{s=1}^N\sum _{1\le i_1,\dots ,i_s\le n+m}{\alpha }_{i_1\cdots i_s}{Z}_{i_1}\cdots Z_{i_s}$$

$$\{X_1,\dots ,X_n,Y_1,\dots ,Y_m\}\subset \{P_1,\dots ,P_r\}$$

$${\Gamma }_R^{(top)}(x_1,\dots ,x_n,y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r)={\Gamma }_{top}(x_1,\dots ,x_n,y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r)$$

$${\Gamma }_R^{(top)}(x_1,\dots ,x_n:y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r)$$

$$(A_1,\dots ,A_n,B_1,\dots ,B_m)\to (A_1,\dots ,A_n).$$

$${\nu }_{\infty }({\Gamma }_R^{(top)}(x_1,\dots ,x_n:y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r),\omega )$$

$${\delta }_{top}(x_1,\dots ,$$

$$=\underset{R>0}{\sup }\underset{ϵ>0,r\in \mathbb{N}}{\inf }\underset{k\to \infty }{\mathrm{limsup}}\frac{\log ({\nu }_{\infty }({\Gamma }_R^{(top)}(x_1,\dots ,x_n:y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r),\omega ))}{-k^2\log \omega }.$$

$${\delta }_{top}(x_1,\dots ,x_n:y_1,\dots ,y_m)=\underset{\omega \to 0^{+}}{\mathrm{limsup}}{\delta }_{top}(x_1,\dots ,x_n:y_1,\dots ,y_m;\omega ).$$

$${\delta }_{top}(x_1,\dots ,$$

$$=\underset{\omega \to 0^{+}}{\mathrm{limsup}}\underset{ϵ>0,r\in \mathbb{N}}{\inf }\underset{k\to \infty }{\mathrm{limsup}}\frac{\log ({\nu }_{\infty }({\Gamma }_M^{(top)}(x_1,\dots ,x_n:y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r),\omega ))}{-k^2\log \omega }.$$

$$\prod _{m=1}^{\infty }{\mathcal{M}}_{k_m}(\mathbb{C})=\{(Y_m{)}_{m=1}^{\infty }|\forall m\ge 1,Y_m\in {\mathcal{M}}_{k_m}(\mathbb{C})\text{ and }\underset{m\ge 1}{\sup }\Vert Y_m\Vert <\infty \}.$$

$${\mathcal{I}}_{\infty }$$

$${\mathcal{I}}_2$$

$${\Gamma }_R^{(top)}(x_1,\dots ,x_n:y_1,\dots ,y_m;k,ϵ,P_1,\dots ,P_r)$$

$${\delta }_{top}(x_1,\dots ,x_n:y_1,\dots ,y_m)$$

$$\Vert U^{\ast }AU-\left(\begin{array}{rlrl}{\lambda }_1{I}_{k_1}& 0& \cdots & 0\\ 0& {\lambda }_2{I}_{k_2}& \cdots & 0\\ \cdots & \cdots & \ddots & \cdots \\ 0& 0& \cdots & {\lambda }_n{I}_{k_n}\end{array}\right)\Vert \le 2\omega ,$$

$$\Vert U^{\ast }{A}_{m}U-\left(\begin{array}{rlrl}{\lambda }_1{I}_{s_1}& 0& \cdots & 0\\ 0& {\lambda }_2{I}_{s_2}& \cdots & 0\\ \cdots & \cdots & \ddots & \cdots \\ 0& 0& \cdots & {\lambda }_n{I}_{s_n}\end{array}\right)\Vert >2\omega ,$$

$$|{\lambda }_i-{\lambda }_j|\ge \theta \text{ for all 1≤i<j≤m.}$$

$$t=\frac{k-n+m}{m}.$$

$$B=diag({\lambda }_{m+1},\dots ,{\lambda }_n)$$

$$A=diag({\lambda }_1{I}_t,{\lambda }_2{I}_t,\dots ,{\lambda }_m{I}_t,B)$$

$$\Omega (A)=\{U^{\ast }AU|U\text{ is in }\mathcal{U}(k)\}.$$

Full text

Topological Free Entropy Dimension

in Unital C∗-algebras

Don Hadwin     and     Junhao Shen111The second author is supported by an NSF grant.

Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824

Email: don@math.unh.edu     and     jog2@cisunix.unh.edu

**Abstract: ** The notion of topological free entropy dimension of $n$-tuple of elements in a unital C∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C∗ algebras, including irrational rotation C∗ algebra, UHF algebra, and minimal tensor product of two reduced C∗ algebras of free groups.

Keywords: Topological free entropy dimension, C∗ algebra

2000 Mathematics Subject Classification: Primary 46L10, Secondary 46L54

1. Introduction

The theory of free probability and free entropy was developed by Voiculescu from 1990s. It played a crucial role in the recent study of finite von Neumann algebras (see [1], [3], [4], [5], [6], [7], [8], [11], [14], [15], [23], [24], [25]). The analogue of free entropy dimension in C∗ algebra context, the notion of topological free entropy dimension of of $n-$tuple of elements in a unital C∗ algebra, was also introduced by Voiculescu in [26].

After introducing the concept of topological free entropy dimension of $n$-tuple of elements in a unital C∗ algebra, Voiculescu discussed some of its properties including subadditivity and change of variables in [26]. In this paper, we will add one basic property into the list: topological free entropy dimension of one variable. More specifically, suppose $x$ is a self-adjoint element in a unital C∗ algebra $A$ and $\sigma(x)$ is the spectrum of $x$ in $\mathcal{A}$. Then topological free entropy dimension of $x$ is equal to $1-\frac{1}{n}$ where $n$ is the cardinality of the set $\sigma(x)$ (see Theorem 4.1).

In [26], Voiculescu showed that (i) if $x_{1},\ldots,x_{n}$ is a family of free semicircular elements in a unital C∗ algebra with a tracial state, then $\delta_{top}(x_{1},\ldots,x_{n})=n$, where $\delta_{top}(x_{1},\ldots,x_{n})$ is the topological free entropy dimension of $x_{1},\ldots,x_{n}$; (ii) if $x_{1},\ldots,x_{n}$ is the universal $n$-tuple of self-adjoint contractions, then $\delta_{top}(x_{1},\ldots,x_{n})=n$. Except in these two cases, very few has been known on the values of topological free entropy dimensions in other C∗ algebras. Using the inequality between topological free entropy dimension and Voiculescu’s free dimension capacity, we are able to obtain an estimation of upper-bound of topological free entropy dimension for a unital C∗ algebra with a unique tracial state (see Theorem 5.1). The lower-bound of topological free entropy dimension is also obtained for infinite dimensional simple unital C∗ algebra with a unique tracial state (see Theorem 5.2). As a corollary, we know that the topological free entropy dimension of any family of self-adjoint generators of an irrational rotation C∗ algebra or UHF algebra or $C_{red}^{*}(F_{2})\otimes_{min}C_{red}^{*}(F_{2})$ is equal to 1 (see Theorem 5.3, 5.4, 5.5). For these C∗ algebras, the value of the topological free entropy dimension is independent of the choice of generators.

The rest of the paper is devoted to study another invariant associated to $n$-tuple of elements in C∗ algebras. This invariant, called topological free orbit dimension, is an analogue of free orbit dimension in finite von Neumann algebras (see [11]). We show that the topological free orbit dimension of a self-adjoint element in a unital C∗ algebra is equal to, according to some measurement, the packing dimension of the spectrum of $x$ (see Theorem 7.1).

The organization of the paper is as follows. In the section 2, we recall the definition of topological free entropy dimension. Some technical lemmas are proved in section 3. In section 4, we compute the topological free entropy dimension of one self-adjoint element in a unital C∗ algebra. In section 5, we study the relationship between topological free entropy dimension and free capacity dimension of a unital C∗ algebra. Then we show that topological free entropy dimension of of any family of generators of an infinite dimensional simple unital C∗ algebra with a unique tracial state is always greater than or equal to 1. The concept of topological free orbit dimension of $n$-tuple of elements in a C∗ algebra is introduced in section 6. Its value for one variable is computed in section 7.

2. Definitions and preliminary

In this section, we are going to recall Voiculescu’s definition of topological free entropy dimension of $n$-tuple of elements in a unital C∗ algebra.

2.1. A Covering of a set in a metric space

Suppose $(X,d)$ is a metric space and $K$ is a subset of $X$. A family of balls in $X$ is called a covering of $K$ if the union of these balls covers $K$ and the centers of these balls lie in $K$.

2.2. Covering numbers in complex matrix algebra

$(\mathcal{M}_{k}(\mathbb{C}))^{n}$

Let $\mathcal{M}_{k}(\mathbb{C})$ be the $k\times k$ full matrix algebra with entries in $\mathbb{C}$, and $\tau_{k}$ be the normalized trace on $\mathcal{M}_{k}(\mathbb{C})$, i.e., $\tau_{k}=\frac{1}{k}Tr$, where $Tr$ is the usual trace on $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{U}(k)$ denote the group of all unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{M}_{k}(\mathbb{C})^{n}$ denote the direct sum of $n$ copies of $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{M}_{k}^{s.a}(\mathbb{C})$ be the subalgebra of $\mathcal{M}_{k}(\mathbb{C})$ consisting of all self-adjoint matrices of $\mathcal{M}_{k}(\mathbb{C})$. Let $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$ be the direct sum of $n$ copies of $\mathcal{M}_{k}^{s.a}(\mathbb{C})$. Let $\|\cdot\|$ be an operator norm on $\mathcal{M}_{k}(\mathbb{C})^{n}$ defined by

[TABLE]

for all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$. Let $\|\cdot\|_{2}$ denote the trace norm induced by $\tau_{k}$ on $\mathcal{M}_{k}(\mathbb{C})^{n}$, i.e.,

[TABLE]

for all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$.

For every $\omega>0$, we define the $\omega$-$\|\cdot\|$-ball $Ball(B_{1},\ldots,B_{n};\omega,\|\cdot\|)$ centered at $(B_{1},\ldots,B_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ to be the subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$ consisting of all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ such that

[TABLE]

Definition 2.1**.**

Suppose that $\Sigma$ is a subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$. We define the covering number $\nu_{\infty}(\Sigma,\omega)$ to be the minimal number of $\omega$-$\|\cdot\|$-balls that consist a covering of $\Sigma$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$.

For every $\omega>0$, we define the $\omega$-$\|\cdot\|_{2}$-ball $Ball(B_{1},\ldots,B_{n};\omega,\|\cdot\|_{2})$ centered at $(B_{1},\ldots,B_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ to be the subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$ consisting of all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ such that

[TABLE]

Definition 2.2**.**

Suppose that $\Sigma$ is a subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$. We define the covering number $\nu_{2}(\Sigma,\omega)$ to be the minimal number of $\omega$-$\|\cdot\|_{2}$-balls that consist a covering of $\Sigma$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$.

2.3. Noncommutative polynomials

In this article, we always assume that $\mathcal{A}$ is a unital C∗-algebra. Let $x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}$ be self-adjoint elements in $\mathcal{A}$. Let $\mathbb{C}\langle X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}\rangle$ be the unital noncommutative polynomials in the indeterminates $X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}$. Let $\{P_{r}\}_{r=1}^{\infty}$ be the collection of all noncommutative polynomials in $\mathbb{C}\langle X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}\rangle$ with rational complex coefficients. (Here “rational complex coefficients” means that the real and imaginary parts of all coefficients of $P_{r}$ are rational numbers).

Remark 2.1**.**

We alsways assume that $1\in\mathbb{C}\langle X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}\rangle$.

2.4. Voiculescu’s Norm-microstates Space

For all integers $r,k\geq 1$, real numbers $R,\epsilon>0$ and noncommutative polynomials $P_{1},\ldots,P_{r}$, we define

[TABLE]

to be the subset of $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n+m}$ consisting of all these

[TABLE]

satisfying

[TABLE]

and

[TABLE]

Remark 2.2**.**

In the definition of norm-microstates space, we use the following assumption. If

[TABLE]

where $z_{1},\ldots,z_{n+m}$ denotes $x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}$ and $\alpha_{0},\alpha_{i_{1}\cdots i_{s}}$ are in $\mathbb{C}$, then

[TABLE]

where $Z_{1},\ldots,Z_{n+m}$ denotes $A_{1},\ldots,A_{n},B_{1},\ldots,B_{m}$ and $I_{k}$ is the identity matrix in $\mathcal{M}_{k}(\mathbb{C})$.

Remark 2.3**.**

In the original definition of norm-microstates space in [26], the parameter $R$ was not introduced. Note the following observation: Let $R>\max\{\|x_{1}\|,\ldots,\|x_{n}\|,\|y_{1}\|,\ldots,\|y_{m}\|\}$. When $r$ is large enough so that

[TABLE]

and $0<\epsilon<R-\max\{\|x_{1}\|,\ldots,\|x_{n}\|,\|y_{1}\|,\ldots,\|y_{m}\|\}$, we have

[TABLE]

for all $k\geq 1$, where $\Gamma_{(top)}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{m};k,\epsilon,P_{1},\ldots,P_{r})$ is the norm-microstates space defined in [26]. Thus our definition agrees with the one in [26] for large $R$, $r$ and small $\epsilon$.

In the later sections, we need to construct the ultraproduct of some matrix algebras, it will be convenient for us to include the parameter “R” in the definition of norm-microstate space.

Define the norm-microstates space of $x_{1},\ldots,x_{n}$ in the presence of $y_{1},\ldots,y_{m}$, denoted by

[TABLE]

as the projection of $\Gamma^{(top)}_{R}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{m};k,\epsilon,P_{1},\ldots,P_{r})$ onto the space $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$ via the mapping

[TABLE]

2.5. Voiculescu’s topological free entropy dimension (see [26])

Define

[TABLE]

to be the covering number of the set $\Gamma^{(top)}_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{m};k,\epsilon,P_{1},\ldots,P_{r})$ by $\omega$-$\|\cdot\|$-balls in the metric space $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$ equipped with operator norm.

Definition 2.3**.**

Define

[TABLE]

*The topological entropy dimension *** of $x_{1},\ldots,x_{n}$ in the presence of $y_{1},\ldots,y_{m}$ is defined by

[TABLE]

Remark 2.4**.**

Let $M>\max\{\|x_{1}\|,\ldots,\|x_{n}\|,\|y_{1}\|,\ldots,\|y_{m}\|\}$ be some positive number. By Remark 2.3, we know

[TABLE]

2.6. C∗ algebra ultraproduct and von Neumann

algebra ultraproduct

Suppose $\{\mathcal{M}_{k_{m}}(\mathbb{C})\}_{m=1}^{\infty}$ is a sequence of complex matrix algebras where $k_{m}$ goes to infinity when $m$ approaches infinity. Let $\gamma$ be a free ultrafilter in $\beta(\mathbb{N})\setminus\mathbb{N}$. We can introduce a unital C∗ algebra $\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})$ as follows:

[TABLE]

We can also introduce the norm closed two sided ideals $\mathcal{I}_{\infty}$ and $\mathcal{I}_{2}$ as follows.

[TABLE]

Definition 2.4**.**

The C∗ algebra ultraproduct of $\{\mathcal{M}_{k_{m}}(\mathbb{C})\}_{m=1}^{\infty}$ along the ultrfilter $\gamma$, denoted by $\prod_{m=1}^{\gamma}\mathcal{M}_{k_{m}}(\mathbb{C})$, is defined to be the quotient algebra of $\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})$ by the ideal $\mathcal{I}_{\infty}$. The image of $(Y_{m})_{m=1}^{\infty}\in\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})$ in the quotient algebra is denoted by $[(Y_{m})_{m}]$.

Definition 2.5**.**

The von Neumann algebra ultraproduct of $\{\mathcal{M}_{k_{m}}(\mathbb{C})\}_{m=1}^{\infty}$ along the ultrfilter $\gamma$, also denoted by $\prod_{m=1}^{\gamma}\mathcal{M}_{k_{m}}(\mathbb{C})$ if no confusion arises, is defined to be the quotient algebra of $\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})$ by the ideal $\mathcal{I}_{2}$. The image of $(Y_{m})_{m=1}^{\infty}\in\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})$ in the quotient algebra is denoted by $[(Y_{m})_{m}]$.

Remark 2.5**.**

The von Neumann algebra ultraproduct $\prod_{m=1}^{\gamma}\mathcal{M}_{k_{m}}(\mathbb{C})$ is a finite factor (see [16]).

2.7. Topological free

entropy dimension of elements in a non-unital C∗ algebra

Topological free entropy dimension can also be defined for $n$-tuple of elements in a non-unital C∗ algebra. Suppose that $\mathcal{A}$ is a non-unital C∗-algebra. Let $x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}$ be self-adjoint elements in $\mathcal{A}$. Let $\mathbb{C}\langle X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}\rangle\ominus\mathbb{C}$ be the noncommutative polynomials in the indeterminates $X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}$ without constant terms. Let $\{P_{r}\}_{r=1}^{\infty}$ be the collection of all noncommutative polynomials in $\mathbb{C}\langle X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}\rangle\ominus\mathbb{C}$ with rational complex coefficients. Then norm-mocrostate space

[TABLE]

can be defined similarly as in section 2.4. So topological free entropy dimension

[TABLE]

can also be defined similarly as in section 2.5.

In the paper, we will focus on the case when $\mathcal{A}$ is a unital C∗ algebra.

3. Some technical lemmas

3.1.

Suppose $x$ is a self-adjoint element in a unital C∗ algebra $\mathcal{A}$. Let $\sigma(x)$ be the spectrum of $x$ in $\mathcal{A}$.

Theorem 3.1**.**

Let $R>\|x\|$. For any $\omega>0$, we have the following.

1. (1)

There are some integer $n\geq 1$ and distinct real numbers $\lambda_{1},\lambda_{2},\cdots,\lambda_{n}$ in $\sigma(x)$ satisfying (i) $|\lambda_{i}-\lambda_{j}|\geq\omega$ for all $1\leq i\neq j\leq n$; and (ii) for any $\lambda$ in $\sigma(x)$, there is some $\lambda_{j}$ with $1\leq j\leq n$ such that $|\lambda-\lambda_{j}|\leq\omega$. 2. (2)

There are some $r_{0}>0$ and $\epsilon_{0}>0$ such that the following holds: when $r>r_{0}$, $\epsilon<\epsilon_{0}$, for any $A$ in $\Gamma^{(top)}_{R}(x;k,\epsilon,P_{1},\ldots,P_{r})$, there are positive integers $1\leq k_{1},\ldots,k_{n}\leq k$ with $k_{1}+k_{2}+\cdots+k_{n}=k$ and some unitary matrix $U$ in $\mathcal{M}_{k}(\mathbb{C})$ satisfying

[TABLE]

where $I_{k_{j}}$ is the $k_{j}\times k_{j}$ identity matrix in $\mathcal{M}_{k_{j}}(\mathbb{C})$ for $1\leq j\leq n$.

Proof.

The proof of part (1) is trivial. We will only prove part (2). Assume that the result in (2) does not hold. Then there is some $\omega>0$ so that the following holds: for all $m\geq 1$, there are $k_{m}\geq 1$ and some $A_{m}$ in $\Gamma^{(top)}_{R}(x;k_{m},\frac{1}{m},P_{1},\ldots,P_{m})$ such that

[TABLE]

for every $1\leq s_{1},\ldots,s_{n}\leq k_{m}$ with $s_{1}+\cdots+s_{n}=k_{m}$ and every unitary matrix $U$ in $\mathcal{M}_{k_{m}}(\mathbb{C})$.

Let $\gamma$ be a free ultrafilter in $\beta(\mathbb{N})\setminus\mathbb{N}$. Let $\mathcal{B}=\prod_{m=1}^{\gamma}\mathcal{M}_{k_{m}}(\mathbb{C})$ be the C∗ algebra ultraproduct of $\{\mathcal{M}_{k_{m}}(\mathbb{C})\}_{m=1}^{\infty}$ along the ultrafilter $\alpha$, i.e. $\prod_{m=1}^{\gamma}\mathcal{M}_{k_{m}}(\mathbb{C})$ is the quotient algebra of the C∗ algebra $\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})$ by $\mathcal{I}_{\infty}$, the [math]-ideal of the norm $\|\cdot\|$, where $\mathcal{I}_{\infty}=\{(A_{m})_{m=1}^{\infty}\in\prod_{m=1}^{\infty}\mathcal{M}_{k_{m}}(\mathbb{C})\ |\ \lim_{m\rightarrow\gamma}\|A_{m}\|=0\}$. Let $a=[(U^{*}A_{m}U)_{m=1}^{\infty}]$ be a self-adjoint element in $\mathcal{B}$. By mapping $x$ to $a$, there is a unital $*$-isomorphism from the C∗ subalgebra generated by $\{I_{\mathcal{A}},x\}$ in $\mathcal{A}$ onto the C∗ subalgebra generated by $\{I_{\mathcal{B}},a\}$ in $\mathcal{B}$. Thus $\sigma(x)=\sigma(a)$. It is not hard to see that Hausdorff-dist$(\sigma(U^{*}A_{m}U),\sigma(a))\rightarrow 0$ as $m$ goes to $\gamma$, which contradicts with the results in part (1) and ($*$).

∎

3.2.

In this subsection, we will use the following notation.

1. (i)

Let $n,m$ be some positive integers with $n\geq m$. 2. (ii)

Let $\delta$, $\theta$ be some positive numbers. 3. (iii)

Let $\{\lambda_{1},\lambda_{2},\ldots,\lambda_{m}\}\cup\{\lambda_{m+1},\ldots,\lambda_{n}\}$ be a family of real numbers such that

[TABLE] 4. (iv)

Let $k$ be a positive integer such that $k-(n-m)$ is divided by $m$. We let

[TABLE] 5. (v)

We let

[TABLE]

be a diagonal matrix in $\mathcal{M}_{n-m}(\mathbb{C})$ and

[TABLE]

be a block-diagonal matrix in $\mathcal{M}_{k}(\mathbb{C}),$ where $I_{t}$ is the identity matrix in $\mathcal{M}_{t}(\mathbb{C})$. 6. (vi)

We let $A$ be defined as above and

[TABLE] 7. (vii)

Assume that $\{e_{ij}\}_{i,j=1}^{k}$ is a canonical basis of $\mathcal{M}_{k}(\mathbb{C})$. We let

[TABLE]

where $[\frac{i}{m}]$, or $[\frac{j}{m}]$, denotes the largest integer $\leq$ $[\frac{i}{m}]$, or $[\frac{j}{m}]$ respectively.

Lemma 3.1**.**

We follow the notation as above. Suppose   $\|U_{1}AU_{1}^{*}-U_{2}AU_{2}^{*}\|_{2}\leq\delta$ for some unitary matrices $U_{1}$ and $U_{2}$ in $\mathcal{U}(k)$. Then the following hold.

1. (1)

There exists some $S\in V_{2}$ such that $\|S\|_{2}\leq 1$ and

[TABLE] 2. (2)

If $n=m$, then there is a unitary matrix $W$ in $V_{2}$ such that

[TABLE]

Proof.

Assume that

[TABLE]

where $U_{i,j}$ is a $t\times t$ matrix, $U_{i,m+1}$ a $t\times(n-m)$ matrix, $U_{m+1,j}$ a $(n-m)\times t$ matrix for $1\leq i,j\leq m$ and $U_{m+1,m+1}$ is a $(n-m)\times(n-m)$ matrix.

(1)  Let

[TABLE]

It is easy to see that $S$ is in $V_{2}$, $\|S\|_{2}\leq 1$ and

[TABLE]

Hence

[TABLE]

It follows that

[TABLE]

(2)  If $n=m$, then

[TABLE]

By the construction of $S$, we can assume $S=WH$ is a polar decomposition of $S$ in $V_{2}$ for some unitary matrix $W$ and positive matrix $H$ in $V_{2}$. Again by the construction of $S$, we know that $\|S\|\leq 1$, whence $\|H\|\leq 1.$ From the proven fact that $\|U_{2}^{*}U_{1}-S\|_{2}\leq\frac{\delta}{\theta},$ we know that

[TABLE]

Thus

[TABLE]

It follows that

[TABLE]

∎

Lemma 3.2**.**

We have the following results.

1. (1)

For every $U\in\mathcal{U}(k)$, let

[TABLE]

Then the volume of $\Sigma(U)$ is bounded above by

[TABLE]

where $\mu$ is the normalized Haar measure on the unitary group $\mathcal{U}(k)$ and $C,C_{1}$ are some constants independent of $k,\delta,\theta$. 2. (2)

When $n=m$, for every $U\in\mathcal{U}(k)$, let

[TABLE]

Then

[TABLE]

Proof.

(1)  By computing the covering number of the set $\{S\ |\ S\in V_{2},\text{ such that }\|S\|_{2}\leq 1\}$ by $\delta/\theta$-$\|\cdot\|_{2}$-balls in $\mathcal{M}_{k}(\mathbb{C})$, we know

[TABLE]

where $C$ is a universal constant. Thus the covering number of the set $\Sigma(U)$ by the $4\delta/\theta$-$\|\cdot\|_{2}$-balls in $\mathcal{M}_{k}(\mathbb{C})$ is bounded by

[TABLE]

But the ball of radius $4\delta/\theta$ in $\mathcal{U}(k)$ has the volume bounded by

[TABLE]

where $C_{1}$ is a universal constant. Thus

[TABLE]

(2) A slight adaption of the proof of part (1) gives us the proof of part (2). ∎

Lemma 3.3**.**

Let $\Omega(A)$ be defined as in (vi) at the beginning of this subsection.

1. (1)

The covering number of $\Omega(A)$ by the $\frac{1}{2}\delta$-$\|\cdot\|_{2}$-balls in $\mathcal{M}_{k}(\mathbb{C})$ is bounded below by

[TABLE] 2. (2)

If $n=m$, then

[TABLE]

Proof.

(1)  For every $U\in\mathcal{U}(k)$, define

[TABLE]

By preceding lemma, we have

[TABLE]

A “parking” (or exhausting) argument will show the existence of a family of unitary elements $\{U_{i}\}_{i=1}^{N}\subset\mathcal{U}(k)$ such that

[TABLE]

and

[TABLE]

From the definition of each $\Sigma(U_{j})$, it follows that

[TABLE]

By Lemma 3.1, we know that

[TABLE]

which implies that

[TABLE]

(2) is similar as (1). ∎

3.3.

We have following theorem.

Theorem 3.2**.**

Let $n\geq m$, $\delta,\theta>0$ and $\{\lambda_{1},\lambda_{2},\ldots,\lambda_{m}\}\cup\{\lambda_{m+1},\ldots,\lambda_{n}\}$ be a family of real numbers such that

[TABLE]

for all $1\leq i<j\leq m$. Let $k$ be a positive integer such that $k-(n-m)$ is divided by $m$ and

[TABLE]

Let

[TABLE]

be a diagonal matrix in $\mathcal{M}_{n-m}(\mathbb{C})$ and

[TABLE]

be a block-diagonal matrix in $\mathcal{M}_{k}(\mathbb{C}),$ where $I_{t}$ is the identity matrix in $\mathcal{M}_{t}(\mathbb{C})$. We let

[TABLE]

Then the covering number of $\Omega(A)$ by the $\frac{1}{2}\delta$-$\|\cdot\|$-balls in $\mathcal{M}_{k}(\mathbb{C})$ is bounded below by

[TABLE]

where $C,C_{1}$ are some universal constants.

When $n=m$, we have

[TABLE]

Proof.

Note that

[TABLE]

The result follows directly from preceding lemma. ∎

3.4.

The following proposition, whose proof is skipped, is an easy extension of Lemma 3.3.

Proposition 3.1**.**

Let $m,k$ be some positive integers and $\theta,\delta$ be some positive numbers. Let $T_{1},T_{2},\ldots,T_{m+1}$ is a partition of the set $\{1,2,\ldots,k\}$, i.e. $\cup_{i=1}^{m+1}T_{i}=\{1,2,\ldots,k\}$ and $T_{i}\cap T_{j}=\emptyset$ for $1\leq i\neq j\leq m+1$. Let $\lambda_{1},\ldots,\lambda_{k}$ be some real numbers such that, if $1\leq j_{1}\neq j_{2}\leq m$ then

[TABLE]

Let $A=diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ be a self-adjoint matrix in $\mathcal{M}_{k}(\mathbb{C})$ and

[TABLE]

be a subset of $\mathcal{M}_{k}(\mathbb{C})$.

Let $s_{j}$ be the cardinality of the set $T_{j}$ for $1\leq j\leq m+1$. Then the covering number of $\Omega(A)$ by the $\frac{1}{2}\delta$-$\|\cdot\|_{2}$-balls in $\mathcal{M}_{k}(\mathbb{C})$ is bounded below by

[TABLE]

where $C,C_{1}$ are some universal constants.

4. Topological free entropy dimension of one variable

Suppose $x$ is a self-adjoint element of a unital C∗ algebra $\mathcal{A}$. In this section, we are going to compute the topological entropy dimension of $x$.

4.1. Upperbound

Proposition 4.1**.**

Suppose $x$ in $\mathcal{A}$ is a self-adjoint element with the spectrum $\sigma(x)$. Then

[TABLE]

where $n$ is the cardinality of $\sigma(x)$. Here we assume that $\frac{1}{\infty}=0.$

Proof.

By [26], we know that the inequality always holds when $n$ is infinity. We need only to show that

[TABLE]

when $n<\infty$.

Assume that $\lambda_{1},\ldots,\lambda_{n}$ are in the spectrum of $x$ in $\mathcal{A}$.

Let $R>\|x\|$. By Theorem 3.1, for every $\omega>0$, there are $r_{0}>0$ and $\epsilon_{0}>0$ such that, for all $r>r_{0},\epsilon<\epsilon_{0}$,

[TABLE]

there are some $1\leq k_{1},\ldots,k_{n}\leq k$, with $k_{1}+\cdots+k_{n}=k$ and a unitary matrix $U$ in $\mathcal{M}_{k}(\mathbb{C})$ satisfying

[TABLE]

Let

[TABLE]

By Corollary 12 in [21] or Theorem 3 in [2], the covering number of $\Omega(k_{1},\ldots,k_{n-1},k_{n})$ by $\omega$-$\|\cdot\|$-balls in $\mathcal{M}_{k}(\mathbb{C})$ is upperbounded by

[TABLE]

where $C_{2}$ is a constant which does not depend on $k,k_{1},\ldots,k_{n}$ (may depend on $n$ and $\|x\|$).

Let $\mathcal{I}$ be the set consisting of all these $(k_{1},\ldots,k_{n})$ in $\mathbb{Z}^{n}$ such that $1\leq k_{1},\ldots,k_{n}\leq k$ and $k_{1}+\cdots+k_{n}=k$. Then the cardinality of the set $\mathcal{I}$ is equal to

[TABLE]

Note that

[TABLE]

for all $1\leq k_{1},\ldots,k_{n}\leq k$ with $k_{1}+\cdots+k_{n}=k$; and by ($**$)

[TABLE]

is contained in $2\omega$-neighborhood of the set

[TABLE]

It follows that the covering number of the set

[TABLE]

by $3\omega$-$\|\cdot\|$-balls in $\mathcal{M}_{k}(\mathbb{C})$ is upperbounded by

[TABLE]

Thus

[TABLE]

∎

4.2. Lower-bound

We follow the notation from last subsection.

Proposition 4.2**.**

Suppose that $x$ is a self-adjoint element with the finite spectrum $\sigma(x)$ in $\mathcal{A}$. Then

[TABLE]

where $n$ is the cardinality of the set $\sigma(x)$.

Proof.

Suppose that $\lambda_{1},\ldots,\lambda_{n}$ are distinct spectrum of $x$. There is some positive number $\theta$ such that

[TABLE]

Assume $k=nt$ for some positive integer $t$. Let

[TABLE]

be a diagonal matrix in $\mathcal{M}_{k}(\mathbb{C})$ where $I_{t}$ is the $t\times t$ identity matrix. It is easy to see that, for all $R>\|x\|$, $r\geq 1$ and $\epsilon>0$, we have

[TABLE]

For any $\omega>0$, applying Theorem 3.2 for $n=m$ and $\delta=\frac{1}{2}\omega$, we have

[TABLE]

Note that $k=nt=mt$ and $\theta$ is some fixed number. A quick computation shows that

[TABLE]

∎

Proposition 4.3**.**

Suppose that $x$ is a self-adjoint element in $\mathcal{A}$ with infinite spectrum. Then

[TABLE]

Proof.

For any $0<\theta<1$, there are $\lambda_{1},\ldots,\lambda_{m}$ in the spectrum of $x$, $\sigma(x)$, satisfying (i)

[TABLE]

and (ii) for any $\lambda$ in $\sigma(x)$, there is some $\lambda_{j}$ with $|\lambda-\lambda_{j}|\leq\theta$. By functional calculus, for any $R>\|x\|$, $r\geq 1$ and $\epsilon>0$, there are some positive integer $n\geq m$ and real numbers $\lambda_{m+1},\ldots,\lambda_{n}$ in $\sigma(x)$ satisfying: for every $t\geq 1$ the matrix

[TABLE]

is in

[TABLE]

where we assume that $k=mt+n-m$. For any $\omega>0,$ let $\delta=\frac{1}{2}\omega$. By Theorem 3.2, we know that

[TABLE]

Thus

[TABLE]

Then,

[TABLE]

When $\theta$ goes to [math], $m$ goes to infinity as $\sigma(x)$ has infinitely many elements. Therefore,

[TABLE]

∎

4.3. Topological free entropy dimension in one variable case

By Proposition 4.1, Proposition 4.2 and Proposition 4.3, we have the following result.

Theorem 4.1**.**

Suppose $x$ is a self-adjoint element in a unital C∗ algebra $A$. Then

[TABLE]

where $n$ is the cardinality of the set $\sigma(x)$ and $\sigma(x)$ is the set of spectrum of $x$ in $\mathcal{A}$. Here we assume that $\frac{1}{\infty}=0$.

5. Topological free entropy dimension of $n$-tuple in unital C∗ algebras

5.1. An equivalent definition of topological free entropy dimension

Suppose that $\mathcal{A}$ is a unital C∗ algebra and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}$ are self-adjoint elements in $\mathcal{A}$. For every $R,\epsilon>0$ and positive integers $r,k$, let

[TABLE]

be Voiculescu’s norm-microstate space defined in section 2.4.

Define

[TABLE]

to be the covering number of the set $\Gamma^{(top)}_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{m};k,\epsilon,P_{1},\ldots,P_{r})$ by $\omega$-$\|\cdot\|_{2}$-balls in the metric space $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$ equipped with trace norm (see Definition 2.2).

Definition 5.1**.**

Define

[TABLE]

And

[TABLE]

The following proposition was pointed out by Voiculescu in [26]. For the sake of completeness, we also include a proof here.

Proposition 5.1**.**

Suppose that $\mathcal{A}$ is a unital C∗ algebra and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}$ are self-adjoint elements in $\mathcal{A}$. Then

[TABLE]

where $\delta_{top}(x_{1},\ldots,\ x_{n}:y_{1},\ldots,y_{m})$ is the topological free entropy dimension of $x_{1},\ldots,x_{n}$ in presence of $y_{1},\ldots,y_{m}$.

Proof.

This is an easy consequence of Lemma 1 in [21]. Let $\lambda$ be the Lebesgue measure on $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$. Let, for every $\omega>0$,

[TABLE]

It follows from the results in [21] or Theorem 8 in [2] that, for some $M_{1},M_{2}$ independent of $k,\omega$ such that

[TABLE]

For every $\omega>0$ and any subset set $K$ of $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$ , let

[TABLE]

Note the following fact:

[TABLE]

It follows from Lemma 1 in [21] that

[TABLE]

and

[TABLE]

Combining with the equalities (5.1.1), we get

[TABLE]

and

[TABLE]

Therefore, we have

[TABLE]

It is a well-known fact (for example see Theorem 8 in [2]) that

[TABLE]

for some universal constant $C_{3}>0$. Hence

[TABLE]

Let $K$ be $\Gamma^{(top)}_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{m};k,\epsilon,P_{1},\ldots,P_{r})$. By the definitions of $\tilde{\delta}_{top}$ and $\delta_{top}$, we have

[TABLE]

∎

5.2. Upper-bound of topological free entropy dimension in a unital C∗ algebra

Let us recall Voiculescu’s definition of free dimension capacity in [26].

Definition 5.2**.**

Suppose that $\mathcal{A}$ is a unital C∗ algebra with a family of self-adjoint generators $x_{1},\ldots,x_{n}$. Suppose that $TS(\mathcal{A})$ is the set consisting of all tracial states of $\mathcal{A}$. If $TS(\mathcal{A})\neq\varnothing$, define Voiculescu’s free dimension capacity $\kappa\delta(x_{1},\ldots,x_{n})$ of $x_{1},\ldots,x_{n}$ as follows,

[TABLE]

where $\delta_{0}(x_{1},\ldots,x_{n}:\tau)$ is Voiculescu’s (von Neumann algebra) free entropy dimension of $x_{1},\ldots,x_{n}$ in $\langle\mathcal{A},\tau\rangle$.

The relationship between topological free entropy dimension of a unital C∗ algebra with a unique tracial state and its free dimension capacity is indicated by the following result.

Theorem 5.1**.**

Suppose that $\mathcal{A}$ is a unital C∗ algebra with a family of self-adjoint generators $x_{1},\ldots,x_{n}$. Suppose that $TS(\mathcal{A})$ is the set consisting of all tracial states of $\mathcal{A}$. If $TS(\mathcal{A})$ is a set with a single element, then

[TABLE]

To prove the preceding theorem, we need the following lemma.

Sublemma 5.2.1**.**

Suppose that $\mathcal{A}$ is a unital C∗ algebra with a family of self-adjoint generators $x_{1},\ldots,x_{n}$. Suppose that $TS(\mathcal{A})\neq\varnothing$ is the set consisting of all tracial states of $\mathcal{A}$. Let $R>\max\{\|x_{1}\|,\ldots,\|x_{n}\|\}$ be some positive number. Then for any $m\geq 1$, there is some $r_{m}\in\mathbb{N}$ such that

[TABLE]

where $\Gamma_{R}(x_{1},\ldots,x_{n};k,m,\frac{1}{m};\tau)$ is microstate space of $x_{1},\ldots,x_{n}$ with respect to $\tau$ (see [23]).

Proof of Sublemma 5.2.1: .

We will prove the result by contradiction. Suppose, to the contrary, there is some $m_{0}\geq 1$ so that following holds: for any $r\in\mathbb{N}$, there are some $k_{r}\geq 1$ and some

[TABLE]

satisfying

[TABLE]

Let $\alpha$ be a free ultrafilter in $\beta(\mathbb{N})\setminus\mathbb{N}$. Let $\mathcal{N}=\prod_{r=1}^{\alpha}\mathcal{M}_{k_{r}}(\mathbb{C})$ be the von Neumann algebra ultraproduct of $\{\mathcal{M}_{k_{r}}(\mathbb{C})\}_{r=1}^{\infty}$ along the ultrafilter $\alpha$, i.e. $\prod_{r=1}^{\alpha}\mathcal{M}_{k_{r}}(\mathbb{C})$ is the quotient algebra of the C∗ algebra $\prod_{r=1}^{\infty}\mathcal{M}_{k_{r}}(\mathbb{C})$ by $\mathcal{I}_{2}$, the [math]-ideal of the trace $\tau_{\alpha}$, where $\tau_{\alpha}((A_{r})_{r=1}^{\infty})=\lim_{r\rightarrow\alpha}\frac{Tr(A_{r})}{k_{r}}$. Let, for each $1\leq j\leq n$, $a_{j}=[(A_{r}^{(j)})_{r=1}^{\infty}]$ be a self-adjoint element in $\mathcal{N}$. By mapping $x_{j}$ to $a_{j}$, there is a unital $*$-homomorphism $\psi$ from the C∗ algebra $\mathcal{A}$ onto the C∗ subalgebra generated by $\{a_{1},\ldots,a_{n}\}$ in $\mathcal{N}$.

Let $\tau_{0}$ be the tracial state on $\mathcal{A}$ which is induced by $\tau_{\alpha}$ on $\psi(\mathcal{A})$, i.e.

[TABLE]

It follows that when $r$ is large enough,

[TABLE]

which contradicts with the inequality (5.2.1). This complete the proof. ∎

Proof of Theorem 5.1: .

Let $R>\max\{\|x_{1}\|,\ldots,\|x_{n}\|\}$. Let $\tau$ be the unique trace of $\mathcal{A}$. By Sublemma 5.2.1, for any $m\geq 1$, there is $r\in\mathbb{N}$ such that

[TABLE]

Therefore, for any $1>\omega>0$, we have

[TABLE]

Now it is easy to check that

[TABLE]

By Proposition 5.1, we know that

[TABLE]

∎

Remark 5.1**.**

Combining Theorem 5.1 with the results in [11] or [14], we will be able to compute the upper-bound of topological free entropy dimension for a large class of unital C∗ algebras. For example, $\delta_{top}(x_{1},\ldots,x_{n})\leq 1$ if $x_{1},\ldots,x_{n}$ is a family of self-adjoint operators that generates an irrational rotation algebra $\mathcal{A}$.

5.3. Lower-bound of topological free entropy dimension in a unital C∗ algebra

In this subsection, we assume that $\mathcal{A}$ is a finitely generated, infinite dimensional, unital simple C∗ algebra with a unique tracial state $\tau$. Assume that $x_{1},\ldots,x_{n}$ is a family of self-adjoint generators of $\mathcal{A}$. Let $H$ be the Hilbert space $L^{2}(\mathcal{A},\tau)$. Without loss of generality, we might assume that $\mathcal{A}$ is faithfully represented on the Hilbert space $H$. Let $\mathcal{M}$ be the von Neumann algebra generated by $\mathcal{A}$ on $H$. It is not hard to see that $\mathcal{M}$ is a diffuse von Neumann algebra with a tracial state $\tau$.

For each positive integer $m$, there is a family of mutually orthogonal projections $p_{1},\ldots,p_{m}$ in $\mathcal{M}$ such that $\tau(p_{j})=1/m$ for $1\leq j\leq m$. Let

[TABLE]

Let $\{P_{r}(x_{1},\ldots,x_{n})\}_{r=1}^{\infty}$ be defined as in section 2.3. Thus $\{P_{r}(x_{1},\ldots,x_{n})\}_{r=1}^{\infty}$ is dense in $\mathcal{M}$ with respect to the strong operator topology. Hence, for each $m\geq 1$, there is some self-adjoint element $P_{r_{m}}(x_{1},\ldots,x_{n})$ in $\mathcal{A}$ such that

[TABLE]

where $\|a\|_{2}=\sqrt{\tau(a^{*}a)}$ for all $a\in\mathcal{M}$.

Lemma 5.1**.**

Let $\mathcal{A}$ be finitely generated, infinite dimensional, unital simple C∗ algebra with a unique tracial state $\tau$. Assume that $x_{1},\ldots,x_{n}$ is a family of self-adjoint generators of $\mathcal{A}$. Let $H$, $\mathcal{M}$ be defined as above. For each $m\geq 1$, let $y_{m}$ and $P_{r_{m}}(x_{1},\ldots,x_{n})$ be chosen as above. Then

[TABLE]

Proof.

Let $R>\max\{\|P_{r_{m}}(x_{1},\ldots,x_{n})\|,\|x_{1}\|,\ldots,\|x_{n}\|\}$. There exists a positive constant $D>1$ such that

[TABLE]

for all $A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}$ in $\mathcal{M}_{k}(\mathbb{C})$ satisfying $0\leq\|A_{1}\|,\ldots,\|A_{n}\|,\|B_{1}\|,\ldots,\|B_{n}\|\leq R$.

Then it is not hard to verify that, for $\omega>0$,

[TABLE]

for each $r\geq r_{m}$ and $\epsilon<\frac{\omega}{4}$. By definition of $\delta_{top}$ and Remark 2.3, we have

[TABLE]

∎

Definition 5.3**.**

Suppose $\mathcal{A}$ is a unital C∗ algebra and $x_{1},\ldots,x_{n}$ is a family of self-adjoint elements of $\mathcal{A}$ that generates $\mathcal{A}$ as a C∗ algebra. If for any $R>\max\{\|x_{1}\|,\ldots,\|x_{n}\|,\|y_{1}\|,\ldots,\|y_{m}\|\}$, $r>0$, $\epsilon>0$, there is a sequence of positive integers $k_{1}<k_{2}<\cdots$ such that

[TABLE]

then $\mathcal{A}$ is called having approximation property.

Lemma 5.2**.**

Let $\mathcal{A}$ be a finitely generated, infinite dimensional, unital simple C∗ algebra with a unique tracial state $\tau$. Assume that $A$ has approximation property. Assume that $x_{1},\ldots,x_{n}$ is a family of self-adjoint generators of $\mathcal{A}$. Let $H$, $\mathcal{M}$ be defined as above. Let $m$ be a positive integer. Let $y_{m}$ and $P_{r_{m}}(x_{1},\ldots,x_{n})$ be chosen as above. Let $R>\max\{\|P_{r_{m}}(x_{1},\ldots,x_{n})\|,\|x_{1}\|,\ldots,\|x_{n}\|\}$. Then there is some positive integer $r>r_{m}$ so that the following hold: $\forall\ k\geq 1$, if

[TABLE]

then there are some $1\leq k_{1},\ldots,k_{m}\leq k$ with $\frac{1}{m}-\frac{1}{r}\leq\frac{k_{j}}{k}\leq\frac{1}{m}+\frac{1}{r}\text{ for each }\ 1\leq j\leq m$ and $k_{1}+\cdots+k_{m}=k$, and a unitary matrix $U$ in $\mathcal{U}(k)$ satisfying

[TABLE]

Proof.

We will prove the result by contradiction. Assume, to the contrary, for all $r\geq r_{m}$ there are some $k_{r}\geq 1$ and some

[TABLE]

satisfying

[TABLE]

for all $1\leq s_{1},\ldots,s_{m}\leq k_{r}$ with $\frac{1}{m}-\frac{1}{r}\leq\frac{s_{j}}{k_{r}}\leq\frac{1}{m}+\frac{1}{r}\text{ for each }\ 1\leq j\leq n$ and $s_{1}+\cdots+s_{m}=k_{r}$, and all unitary matrix $U$ in $\mathcal{U}(k)$.

Let $\alpha$ be a free ultrafilter in $\beta(\mathbb{N})\setminus\mathbb{N}$. Let $\mathcal{N}=\prod_{r=1}^{\alpha}\mathcal{M}_{k_{r}}(\mathbb{C})$ be the von Neumann algebra ultraproduct of $\{\mathcal{M}_{k_{r}}(\mathbb{C})\}_{r=1}^{\infty}$ along the ultrafilter $\alpha$, i.e. $\prod_{r=1}^{\alpha}\mathcal{M}_{k_{r}}(\mathbb{C})$ is the quotient of the C∗ algebra $\prod_{r=1}^{\infty}\mathcal{M}_{k_{r}}(\mathbb{C})$ by $\mathcal{I}_{2}$, the [math]-ideal of the trace $\tau_{\alpha}$, where $\tau_{\alpha}((A_{r})_{r=1}^{\infty})=\lim_{r\rightarrow\alpha}\frac{Tr(A_{r})}{k_{r}}$. Let, for each $1\leq j\leq n$, $a_{j}=[(A_{r}^{(j)})_{r=1}^{\infty}]$ be a self-adjoint element in $\mathcal{N}$. By mapping $x_{j}$ to $a_{j}$, there is a unital $*$-homomorphism $\psi$ from the C∗ algebra $\mathcal{A}$ onto the C∗ subalgebra generated by $\{a_{1},\ldots,a_{n}\}$ in $\mathcal{N}$. Since $\mathcal{A}$ is a simple C∗ algebra and $\psi(I_{\mathcal{A}})=I_{\mathcal{N}}$, $\psi$ actually is a $*$-isomorphism. Since $\mathcal{A}$ has a unique trace $\tau$, $\psi$ induces a $*$-isomorphism (still denoted by $\psi$) from $\mathcal{M}$ onto the von Neumann subalgebra generated by $a_{1},\ldots,a_{n}$ in $\mathcal{N}$. Therefore,

[TABLE]

This contradicts with the definition of $y_{m}$ and inequality (5.3.1). ∎

The following lemma is well-known (for example, see Lemma 4.1 in [23]).

Lemma 5.3**.**

Suppose $A$, or $B$, is a self-adjoint matrix in $\mathcal{M}_{k}^{s.a.}(\mathbb{C})$ with a list of eigenvalues $\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}$, or $\mu_{1}\leq\mu_{2}\leq\cdots\leq\mu_{k}$ respectively. Then

[TABLE]

where $U$ is any unitary matrix in $\mathcal{U}(k)$.

Lemma 5.4**.**

Let $r,m$ be some positive integer with $4<m<r$. Suppose $k_{1},\ldots,k_{m}$ is a family of positive integers such that $\frac{1}{m}-\frac{1}{r}\leq\frac{k_{j}}{k}\leq\frac{1}{m}+\frac{1}{r}$ for all $1\leq j\leq k$ and $k_{1}+\cdots+k_{m}=k$. If $A$ is a self-adjoint matrix in $\mathcal{M}_{k}(\mathbb{C})$ such that, for some unitary matrix $U$ in $\mathcal{U}(k)$,

[TABLE]

then, for any $\omega>0$ we have

[TABLE]

for some constants $C_{1},C>1$ independent of $k,\omega$, where

[TABLE]

Proof.

Suppose that $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{k}$ are the eigenvalues of $A$. For each $1\leq j\leq m$, let

[TABLE]

and

[TABLE]

here we assume that $k_{0}=0$. Let $B=diag(1\cdot I_{k_{1}},\cdots,m\cdot I_{k_{m}})$ be a diagonal matrix in $\mathcal{M}_{k}(\mathbb{C})$. By Lemma 5.3, we have

[TABLE]

where $card(\hat{T}_{j})$ is the cardinality of the set $\hat{T}_{j}$. Thus

[TABLE]

Let $s_{j}=card(T_{j})$ for $1\leq j\leq m$, whence

[TABLE]

Let

[TABLE]

and $s_{m+1}$ be the cardinality of the set $T_{m+1}$. Thus

[TABLE]

It is not hard to see that $T_{1},\ldots,T_{m+1}$ is a partition of the set $\{1,2,\ldots,k\}$. Moreover, if $1\leq j_{1}\neq j_{2}\leq m$ then for any

[TABLE]

we have

[TABLE]

Applying Proposition 3.1 for such $T_{1},\ldots,T_{m},T_{m+1}$, $\theta=1/2$ and $\omega=\delta/2$, we have

[TABLE]

for some constants $C,C_{1}>1$ independent of $k,\omega$.

∎

Lemma 5.5**.**

Let $\mathcal{A}$ be a finitely generated, infinite dimensional, simple unital C∗ algebra with a unique tracial state $\tau$. Assume that $A$ has approximation property. Assume that $x_{1},\ldots,x_{n}$ is a family of self-adjoint generators of $\mathcal{A}$. Let $H$, $\mathcal{M}$ be defined as above. Let $m$ be a positive integer. Let $y_{m}$ and $P_{r_{m}}(x_{1},\ldots,x_{n})$ be chosen as above. Let $R>\max\{\|P_{r_{m}}(x_{1},\ldots,x_{n})\|,\|x_{1}\|,\ldots,\|x_{n}\|\}$. When $r$ is large enough and $\epsilon$ is small enough, for any $\omega>0$, we have

[TABLE]

Proof.

By Lemma 5.2, when $r$ is large enough and $\epsilon$ is small enough, the following hold: $\forall\ k\geq 1$, if

[TABLE]

then there are some $1\leq k_{1},\ldots,k_{m}\leq k$ with $\frac{1}{m}-\frac{1}{r}\leq\frac{k_{j}}{k}\leq\frac{1}{m}+\frac{1}{r}\text{ for each }\ 1\leq j\leq m$ and $k_{1}+\cdots+k_{m}=k$, and a unitary matrix $U$ in $\mathcal{U}(k)$ satisfying

[TABLE]

Combining with Lemma 5.4, we know that if

[TABLE]

then, for any $\omega>0$,

[TABLE]

where

[TABLE]

Note that $\Omega(B)\subset\Gamma^{(top)}_{R}(P_{r_{m}}(x_{1},\ldots,x_{n}):x_{1},\ldots,x_{n};k,r,\epsilon)$. It follows that, for any $\omega>0$,

[TABLE]

∎

Now we have the following result.

Theorem 5.2**.**

Let $\mathcal{A}$ be a finitely generated, infinite dimensional, simple unital C∗ algebra with a unique tracial state $\tau$. Assume that $x_{1},\ldots,x_{n}$ is a family of self-adjoint generators of $\mathcal{A}$. If $\mathcal{A}$ has approximation property, then $\delta_{top}(x_{1},\ldots,x_{n})\geq 1$.

Proof.

Let $H$ be the Hilbert space $L^{2}(\mathcal{A},\tau)$. Without loss of generality, we might assume that $\mathcal{A}$ is faithfully represented on the Hilbert space $H$. Let $\mathcal{M}$ be the von Neumann algebra generated by $\mathcal{A}$ on $H$. It is not hard to see that $\mathcal{M}$ is a diffuse von Neumann algebra with a tracial state $\tau$. For each positive integer $m$, there is a family of mutually orthogonal projections $p_{1},\ldots,p_{m}$ in $\mathcal{M}$ such that $\tau(p_{j})=1/m$ for $1\leq j\leq m$. Let

[TABLE]

Let $\{P_{r}(x_{1},\ldots,x_{n})\}_{r=1}^{\infty}$ be defined as in section 2.3. Thus $\{P_{r}(x_{1},\ldots,x_{n})\}_{r=1}^{\infty}$ is dense in $\mathcal{M}$ with respect to the strong operator topology. Hence, for each $m\geq 1$, there is some self-adjoint element $P_{r_{m}}(x_{1},\ldots,x_{n})$ in $\mathcal{A}$ such that

[TABLE]

By Lemma 5.5, for any $\omega>0$, when $r$ is large enough and $\epsilon$ is small enough, we have for some constants $C_{1},C>1$ independent of $k,\omega$

[TABLE]

Therefore,

[TABLE]

By Proposition 5.1, we get

[TABLE]

By Lemma 5.1,

[TABLE]

Since $m$ is an arbitrary positive integer, we obtain

[TABLE]

∎

5.4. Values of topological free entropy dimensions in some

unital C∗ algebras

In this subsection, we are going to compute the values of topological free entropy dimensions in some unital C∗ algebras by using the results from preceding subsection.

Theorem 5.3**.**

Let $\mathcal{A}_{\theta}$ be an irrational rotation C∗ algebra. Then

[TABLE]

where $x_{1},\ldots,x_{n}$ is a family of self-adjoint operators that generates $\mathcal{A}_{\theta}$.

Proof.

Note that $\mathcal{A}_{\theta}$ is an infinite dimensional, unital simple C∗ algebra with a unique tracial state $\tau$. By [24] or [11] and Theorem 5.1, we know that

[TABLE]

It follows from [18] that $\mathcal{A}_{\theta}$ has approximation property. Therefore

[TABLE]

Hence

[TABLE]

∎

Theorem 5.4**.**

Let $\mathcal{A}$ be a UHF algebra (uniformly hyperfinite C∗ algebra). Then

[TABLE]

where $x_{1},\ldots,x_{n}$ is a family of self-adjoint operators that generates $\mathcal{A}$.

Proof.

By [17], we know that $\mathcal{A}$ is generated by two self-adjoint elements. It is not hard to see that $\mathcal{A}$ is an infinite dimensional, unital simple C∗ algebra with a unique tracial state $\tau$. By [24] or [11] and Theorem 5.1, we know that

[TABLE]

It is easy to check that $\mathcal{A}$ has approximation property. Therefore

[TABLE]

Hence

[TABLE]

∎

Recall that for any sequence $(\mathcal{A}_{m})_{m=1}^{\infty}$ of C∗ algebras,we can introduce two C∗ algebras

[TABLE]

The norm in the quotient C∗ algebra $\prod_{m}\mathcal{A}_{m}/\sum_{m}\mathcal{A}_{m}$ is given by

[TABLE]

where $\rho$ is the quotient map from $\prod_{m}\mathcal{A}_{m}$ onto $\sum_{m}\mathcal{A}_{m}$.

If $\mathcal{A}$ is an exact C∗ algebra, then the sequence

[TABLE]

is exact. Therefore, we have the following natural identification

[TABLE]

On the other hand, we have the following natural embedding

[TABLE]

and the identification

[TABLE]

Thus we have for any exact C∗ algebra $\mathcal{A}$ a natural embedding

[TABLE]

Lemma 5.6**.**

Suppose that $\mathcal{A}$ and $\mathcal{B}$ are unital C∗ algebras and $\rho$ is an unital embedding

[TABLE]

Suppose that $x_{1},\ldots,x_{n}$ is a family of elements in $\mathcal{A}$. Suppose $r$ is a positive integer and $\{P_{j}(x_{1},\ldots,x_{n})\}_{j=1}^{r}$ is a family of noncommutative polynomials of $x_{1},\ldots,x_{n}$. Then there are some $k\in\mathbb{N}$ and $a_{1}^{(k)},\ldots,a_{n}^{(k)}$ in $\mathcal{M}_{k}(\mathcal{B})$ so that

[TABLE]

Proof.

We might assume that

[TABLE]

By the definition of $\prod_{m}\mathcal{\mathcal{missing}}M_{m}(\mathcal{B})/\sum_{m}\mathcal{\mathcal{missing}}M_{m}(\mathcal{B})$, there are some positive integers $m_{1}\leq m_{2}$ such that

[TABLE]

Let $k=\sum_{j=m_{1}}^{m_{2}}j$ and

[TABLE]

Then, it is not hard to check that

[TABLE]

∎

Theorem 5.5**.**

Let $p\geq 2$ be a positive integer and $F_{p}$ be the free group on $p$ generators. Let $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$ be a minimal tensor product of two reduced C∗ algebras of free groups $F_{p}$. Then

[TABLE]

where $x_{1},\ldots,x_{n}$ is any family of self-adjoint generators of $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$.

Proof.

Note that $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$ is an infinite dimensional, unital simple C∗ algebra with a unique tracial state. By the result from [5] or [11] and Theorem 5.1, Theorem 5.2, to show $\delta_{top}(x_{1},\ldots,x_{n})=1,$ we need only to show that $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$ has approximation property. Therefore, it suffices to show the following: Let $R>\max\{\|x_{1}\|,\ldots,\|x_{n}\|\}$. For any $r\geq 1$, there is some $k\in\mathbb{N}$ so that

[TABLE]

By the result from [9], we know there is a unital embedding

[TABLE]

which induce a unital embedding

[TABLE]

Note that $C_{red}^{*}(F_{p})$ is an exact C∗ algebra. From the explanation preceding the theorem it follows that there is a unital embedding

[TABLE]

By Lemma 5.6, for a family of elements $x_{1},\ldots,x_{n}$ in $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$ and $r\geq 1$, there are some $m\in\mathbb{N}$ and some $a_{1}^{(m)},\ldots,a_{n}^{(m)}$ in $\mathcal{M}_{m}(C_{red}^{*}(F_{p}))$ so that $\max\{\|a_{1}\|,\ldots,\|a_{n}\|\}<R$ and

[TABLE]

On the other hand, by the existence of embedding

[TABLE]

it follows that there is a unital embedding

[TABLE]

But

[TABLE]

Hence for such $a_{1}^{(m)},\ldots,a_{n}^{(m)}$ in $\mathcal{M}_{m}(C_{red}^{*}(F_{p}))$ and $r\geq 1$, by Lemma 5.6, there are some $k\in\mathbb{N}$ and $A_{1},\ldots,A_{n}$ in $\mathcal{M}_{k}(\mathbb{C})$ so that $\max\{\|A_{1}\|,\ldots,\|A_{n}\|\}<R$ and

[TABLE]

Altogether, we have

[TABLE]

which implies that $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$ has approximation property.

Hence

[TABLE]

for any family of self-adjoint elements $x_{1},\ldots,x_{n}$ that generates $C_{red}^{*}(F_{p})\otimes_{min}C_{red}^{*}(F_{p})$. ∎

Theorem 5.6**.**

Suppose that $\mathcal{K}$ be the C∗ algebra consisting of all compact operators on a separable Hilbert space $H$. Suppose $\mathcal{A}=\mathbb{C}\bigoplus\mathcal{K}$ is the unitization of $\mathcal{K}$. If $x_{1},\ldots,x_{n}$ is a family of self-adjoint elements that generate $\mathcal{A}$ as a C∗ algebra, then

[TABLE]

Proof.

By [17], we know that unital C∗ algebra $\mathcal{A}$ is generated by two self-adjoint elements in $\mathcal{A}$. Note that $\mathcal{A}$ has a unique trace $\tau$, which is defined by

[TABLE]

By Theorem 5.1, it is not hard to see that

[TABLE]

where $x_{1},\ldots,x_{n}$ is a family of self-adjoint generators of $\mathcal{A}$.

∎

6. Topological free orbit dimension of C∗ algebras

Assume that $\mathcal{A}$ is a unital C∗-algebra. Let $x_{1},\ldots,x_{n},$ $y_{1},\ldots,y_{m}$ be self-adjoint elements in $\mathcal{A}$. Let $\mathbb{C}\langle X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{m}\rangle$ be the noncommutative polynomials in the indeterminates $X_{1},\ldots,X_{n},$ $Y_{1},\ldots,Y_{m}$. Let $\{P_{r}\}_{r=1}^{\infty}$ be the collection of all noncommutative polynomials in $\mathbb{C}\langle X_{1},\ldots,X_{n},$ $Y_{1},\ldots,Y_{m}\rangle$ with rational coefficients.

6.1. Unitary orbits of balls in $\mathcal{M}_{k}(\mathbb{C})^{n}$

We let $\mathcal{M}_{k}(\mathbb{C})$ be the $k\times k$ full matrix algebra with entries in $\mathbb{C}$, and $\mathcal{U}(k)$ be the group of all unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{M}_{k}(\mathbb{C})^{n}$ denote the direct sum of $n$ copies of $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{M}_{k}^{s.a}(\mathbb{C})$ be the subalgebra of $\mathcal{M}_{k}(\mathbb{C})$ consisting of all self-adjoint matrices of $\mathcal{M}_{k}(\mathbb{C})$. Let $(\mathcal{M}_{k}^{s.a}(\mathbb{C}))^{n}$ be the direct sum of $n$ copies of $\mathcal{M}_{k}^{s.a}(\mathbb{C})$.

For every $\omega>0$, we define the $\omega$-orbit-$\|\cdot\|$-ball $\mathcal{U}(B_{1},\ldots,B_{n};\omega)$ centered at $(B_{1},\ldots,B_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ to be the subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$ consisting of all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ such that there exists some unitary matrix $W$ in $\mathcal{U}(k)$ satisfying

[TABLE]

6.2. Norm-microstate space

For all integers $r,k\geq 1$, real numbers $R,\epsilon>0$ and noncommutative polynomials $P_{1},\ldots,P_{r}$, we let

[TABLE]

be as defined as in section 2.4.

6.3. Topological free orbit dimension

Definition 6.1**.**

For $\omega>0$, we define the covering number

[TABLE]

to be the minimal number of $\omega$-orbit–$\|\cdot\|$-balls that cover $\Gamma^{(top)}_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{p};k,\epsilon,P_{1},\ldots,P_{r})$ with the centers of these $\omega$-orbit-$\|\cdot\|$-balls in $\Gamma^{(top)}_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{p};k,\epsilon,P_{1},\ldots,P_{r})$

For each function $f:\ \mathbb{N}\times\mathbb{N}\times\mathbb{R}^{+}\rightarrow\mathbb{R}$, we define,

[TABLE]

and

[TABLE]

where $\mathfrak{k}_{f}(x_{1},,\ldots,x_{n}:y_{1},\ldots,y_{p})$ is called the topological $f(\cdot)$-free-orbit-dimension of  $x_{1},\ldots,x_{n}$ in the presence of $y_{1},\ldots,y_{p}$.

6.4. Topological free entropy dimension and topological free orbit

dimension

The following result follows directly from the definitions of topological free entropy dimension and topological free orbit dimension of $n$-tuple of self-adjoint elements in a C∗ algebra.

Theorem 6.1**.**

Suppose that $\mathcal{A}$ is a unital C∗ algebra and $x_{1},\ldots,x_{n}$ is a family of self-adjoint elements of $\mathcal{A}$. Let $f:\mathbb{N}\times\mathbb{N}\times\mathbb{R}^{+}\rightarrow\mathbb{R}$ be defined by

[TABLE]

for $s,k\in\mathbb{N}$, $\omega>0$. Then

[TABLE]

7. Topological free orbit dimension of one variable

We recall the packing number of a set in a metric space as follows.

Definition 7.1**.**

Suppose that $X$ is a metric space with a metric distance $d$. (i) The packing number of a set $K$ by $\omega$-balls in $X$, denoted by $P(K,\omega)$, is the maximal cardinality of the subsets $F$ in $K$ satisfying for all $a,b$ in $F$ either $a=b$ or $d(a,b)\geq\omega$. (ii) The packing dimension of the set $K$ in $X$, denoted by $d(K)$, is defined by

[TABLE]

7.1. Upper-bound of the topological free orbit dimension of one

variable

Suppose that $x=x^{*}$ is a self-adjoint element in a unital C∗ algebra $\mathcal{A}$ and $\sigma(x)$ is the spectrum of $x$ in $\mathcal{A}$.

For any $\omega>0$, let $m=P(K,\omega)$ be the packing number of $\sigma(x)$ in $\mathbb{R}$. Thus there exists a family of elements $\lambda_{1},\ldots,\lambda_{m}$ in $\sigma(x)$ such that (i) $|\lambda_{i}-\lambda_{j}|\geq\omega$ for all $1\leq i\neq j\leq m$; and (ii) for any $\lambda$ in $\sigma(x)$, there is some $\lambda_{j}$ with $1\leq j\leq m$ satisfying $|\lambda-\lambda_{j}|\leq\omega$.

Lemma 7.1**.**

For any given $R>\|x\|$, when $r$ is large enough and $\epsilon$ is small enough, we have

[TABLE]

Proof.

By Theorem 3.1, there exist some $r_{0}\geq 1$ and $\epsilon_{0}>0$ such that the following holds: when $r>r_{0}$, $\epsilon<\epsilon_{0}$, for any $A$ in $\Gamma^{(top)}_{R}(x;k,\epsilon,P_{1},\ldots,P_{r})$, there are positive integers $1\leq k_{1},\ldots,k_{m}\leq k$ with $k_{1}+\cdots+k_{m}=k$ and some unitary matrix $U$ in $\mathcal{M}_{k}(\mathbb{C})$ satisfying

[TABLE]

where $I_{k_{j}}$ is the $k_{j}\times k_{j}$ identity matrix for $1\leq j\leq m$.

Let

[TABLE]

Let $\mathcal{J}$ be the set consisting of all these $(k_{1},\ldots,k_{m})\in\mathbb{N}^{m}$ with $k_{1}+\cdots+k_{m}=k$. Then the cardinality of the set $\mathcal{J}$ is equal to

[TABLE]

Then

[TABLE]

is contained in $2\omega$-neighborhood of the set

[TABLE]

It follows that

[TABLE]

Therefore,

[TABLE]

∎

7.2. Lower-bound

Suppose that $x=x^{*}$ is a self-adjoint element in a unital C∗ algebra $\mathcal{A}$ and $\sigma(x)$ is the spectrum of $x$ in $\mathcal{A}$.

Lemma 7.2**.**

We have

[TABLE]

Proof.

For any $\omega>0$, let $m=P(K,\omega)$ be the packing number of $\sigma(x)$ in $\mathbb{R}$. Thus there exists a family of elements $\lambda_{1},\ldots,\lambda_{m}$ in $\sigma(x)$ such that (i) $|\lambda_{i}-\lambda_{j}|\geq\omega$ for all $1\leq i\neq j\leq m$; and (ii) for any $\lambda$ in $\sigma(x)$, there is some $\lambda_{j}$ with $1\leq j\leq m$ satisfying $|\lambda-\lambda_{j}|\leq\omega$.

For any $R>\|x\|$, $r\geq 1$ and $\epsilon>0$, by functional calculus, there are $\lambda_{m+1},\ldots,\lambda_{n}$ in $\sigma(x)$ such that for every $1\leq t_{1},\ldots,t_{m}\leq k-n$ with $2nt_{1}+\ldots+2nt_{m}=k-n$, the matrix

[TABLE]

where we assume that $2n|(k-n)$.

Let $\mathcal{J}$ be the set consisting of all these $(t_{1},\ldots,t_{m})\in\mathbb{N}^{m}$ with $2nt_{1}+\ldots+2nt_{m}=k-n$. Then the cardinality of the set $\mathcal{J}$ is equal to

[TABLE]

By Weyl’s theorem in [27] on the distance of unitary orbits of two self-adjoint matrices, for any two distinct elements

[TABLE]

in $\mathcal{J}$ and any $W$ in $\mathcal{U}(k)$, we have

[TABLE]

where

[TABLE]

are two diagonal self-adjoint matrices in $\mathcal{M}_{k}(\mathbb{C})$. Combining with (7.2.1), we have

[TABLE]

Hence

[TABLE]

∎

7.3. Topological free orbit dimension of one self-adjoint element

Theorem 7.1**.**

Suppose that $x=x^{*}$ is a self-adjoint element in a unital C∗ algebra $\mathcal{A}$ and $\sigma(x)$ is the spectrum of $x$ in $\mathcal{A}$. Let $d(\sigma(x))$ be the packing dimension of the set $\sigma(x)$ in $\mathbb{R}$. Let $f:\mathbb{N}\times\mathbb{N}\times\mathbb{R}^{+}\rightarrow\mathbb{R}$ be defined by

[TABLE]

for $s,k\in\mathbb{N}$, $\omega>0$. Then

[TABLE]

Proof.

The result follows directly from Lemma 7.1, Lemma 7.2 and Definition 7.1. ∎

Theorem 7.2**.**

Suppose that $x=x^{*}$ is a self-adjoint element in a unital C∗ algebra $\mathcal{A}$. Let $f:\mathbb{N}\times\mathbb{N}\times\mathbb{R}^{+}\rightarrow\mathbb{R}$ be defined by

[TABLE]

for $s,k\in\mathbb{N}$, $\omega>0$. Then

[TABLE]

Proof.

The result follows directly from Lemma 7.1 and Definition 7.1. ∎