---
abstract: 'We give an example of a non-trivial asymptotic representation of the reduced $C^*$-algebra of a free group. This example allows to evaluate the asymptotic tensor $C ^*$-norm of some elements in tensor product $C^*$-algebras and to show semi-invertibility of the non-invertible extension of $C^*_r(\mathbb F_2)$ considered by Haagerup and Thorbjørnsen.'
address: 'Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119992, Russia [*and *]{} Harbin Institute of Technology, Harbin, P. R. China'
author:
- 'V. Manuilov'
---

Introduction
============

Asymptotic homomorphisms of $C^*$-algebras were first defined and studied in [@C-H] in relation to topological properties of $C^*$-algebras. The most important and the best known case is the case of asymptotic homomorphisms from a suspended $C^*$-algebra $SA$ to the $C^*$-algebra ${{\mathbb K}}$ of compact operators, since the homotopy classes of those are the $K$-homology of $A$, the $E$-theory. Asymptotic homomorphisms to other $C^*$-algebras are less known. For example, it is known that any asymptotic homomorphism to the Calkin algebra is homotopic to a genuine homomorphism [@M-Crelle]. Even less is known about asymptotic homomorphisms to $\mathbb B(H)$, where there is no topological obstruction (recall that the $K$-groups of $\mathbb B(H)$ are trivial). Such asymptotic homomorphisms are called [*asymptotic representations*]{} and were first studied in relation to the asymptotic tensor product $C^*$-algebras [@MT6] and to semi-invertibility of $C^*$-algebra extensions [@MT7].

The aim of this paper is to give an example of an asymptotic representation of the reduced group $C^*$-algebra of the free group $\mathbb F_2$ on two generators. This was made possible due to two interesting families of representations. The first family was constructed by M. Pimsner and D. Voiculescu in [@PV], cf. next section, and the second family (sequence) was discovered by U. Haagerup and S. Thorbjørnsen in [@Haagerup-T] by free probability methods. We combine these two families into an asymptotic representation of $C^*_r(\mathbb F_2)$.

U. Haagerup and S. Thorbjørnsen used their sequence of representations [@Haagerup-T] to construct an extension of $C^*_r(\mathbb F_2)$ which is not invertible in the BDF functor ${\operatorname{Ext}}(C^*_r(\mathbb F_2))$ of L. Brown, R. Douglas and P. Fillmore [@BDF]. We are able to show that their extension is semi-invertible [@MT3], and represents the trivial element in the modified ${\operatorname{Ext}}$ functor. Another application of our asymptotic representation is evaluation of asymptotic tensor $C^*$-norm on tensor products of $C^*_r(\mathbb F_2)$ with some other $C^*$-algebras related to the free group. We also show that such asymptotic tensor products behave differently from the case of groups with the property T of Kazhdan.

A family of representations of $\mathbb F_2$
============================================

For a free group $\mathbb F_2$, M. Pimsner and D. Voiculescu [@PV] have constructed a family $(\pi_s)_{s\in[0,1]}$ of representations on $l^2(\mathbb F_2)$ with the following properties:

1.  $\pi_0=\lambda$ is the regular representation;

2.  $\pi_1=\theta\oplus\lambda^{(2)}$, where $\theta$ is the one-dimensional trivial representation and $\lambda^{(2)}=\lambda\oplus\lambda$;

3.  the map $s\mapsto\pi_s(g)$ is norm-continuous for any $g\in\mathbb
    F_2$;

4.  $\pi_s(g)-\pi_0(g)$ is compact for any $g\in\mathbb F_2$ and any $s\in[0,1]$.

Importance of this homotopy, which connects the regular representation with a representation containing the trivial representation as a direct summand, was emphasized by E. C. Lance [@Lance]. A natural specific example of such a family $\pi_s$ of representations of $\mathbb F_2$ with additional properties is given in [@Pytlik-Szwarc].

For a representation $\pi$ of $\mathbb F_2$ let us denote by $C^*_\pi(\mathbb F_2)$ the $C^*$-algebra generated by all $\pi(g)$, $g\in\mathbb F_2$. We write $\pi(a)\in C^*_\pi(\mathbb
F_2)$ for $a\in C^*(\mathbb F_2)$ and make no distinction between $a$ and its image in $C^*_\pi(\mathbb F_2)$ when there is no confusion.

\[exact\] $C^*_{\pi_s}(\mathbb F_2)$ is an exact $C^*$-algebra for any $s\in[0,1]$.

For $s=0$ it is known that $C^*_r(\mathbb F_2)$ is exact, so let us consider the case $s\in(0,1]$. It follows from the property (4) of the family $(\pi_s)$ that $$C^*_{\pi_s}(\mathbb F_2)\subset C^*_r(\mathbb F_2)+{{\mathbb K}}$$ for any $s\in(0,1]$. The extension $$0\to{{\mathbb K}}\to C^*_r(\mathbb F_2)+{{\mathbb K}}\to C^*_r(\mathbb F_2)\to 0$$ is obviously split. So, by results of E. Kirchberg (cf. [@Kirchberg]), $C^*_r(\mathbb F_2)+{{\mathbb K}}$ is exact, hence its $C^*$-subalgebras are exact too.

Example of an asymptotic representation of $C^*_r(\mathbb F_2)$
===============================================================

Recall that, by the Fell’s absorbtion principle [@Fell], for any representation $\sigma$ of $\mathbb F_2$, one has $\lambda\otimes\sigma\cong\lambda^{(\dim\sigma)}$, so, for any sequence $\{\sigma_n\}_{n\in\mathbb N}$ of finitedimensional representations of $\mathbb F_2$, $s\mapsto
\pi_s\otimes(\oplus_{n=1}^\infty\sigma_n)$ is a homotopy that connects $\lambda^{(\infty)}$ with $\lambda^{(\infty)}\oplus(\oplus_{n=1}^\infty\sigma_n)$.

It was shown in [@Haagerup-T] that there exists a sequence $\{\sigma_n\}_{n\in\mathbb N}$ of finitedimensional representations of $\mathbb F_2$ such that $$\label{Haagerup}
\|\lambda(a)\|=\lim\sup\nolimits_{n\to\infty}\|\sigma_n(a)\|$$ for any $a\in\mathbb C[\mathbb F_2]$. From now on let us fix this sequence. Put, for $t\in[n,n+1]$, $$\rho_t=(\pi_0\otimes\sigma_1)\oplus\ldots\oplus(\pi_0\otimes\sigma_n)\oplus
(\pi_{t-n}\otimes\sigma_{n+1})\oplus(\pi_1\otimes\sigma_{n+2})\oplus\ldots.$$ Then we get a family $(\rho_t)_{t\in[0,\infty)}$ of representations of $\mathbb F_2$.

Recall that an asymptotic representation of a $C^*$-algebra $A$ is an asymptotic homomorphism [@C-H] into the $C^*$-algebra $\mathbb B(H)$ of bounded operators on a Hilbert space, i.e. a family of maps $(\varphi_t)_{t\in[0,\infty)}:A\to\mathbb B(H)$ such that the map $t\mapsto\varphi_t(a)$ is continuous for any $a\in A$ and $\varphi_t$ behaves asymptotically like a $*$-homomorphism, as $t\to\infty$. Any continuous family of (genuine) representations gives an asymptotic representation. A more general example of an asymptotic representation can be obtained from a continuous family $\mu_t$, $t\in[0,\infty)$, of representations of a $C^*$-algebra $E$ such that $q:E\to A$ is a quotient $*$-homomorphism if this family satisfies the condition $\lim_{t\to\infty}\|\mu_t(e)\|\leq\|q(e)\|$ for any $e\in E$. If $\chi:A\to E$ is a Bartle–Graves continuous lifting map [@BG] then $\mu_t\circ\chi:A\to\mathbb B(H)$ is the required asymptotic representation.

\[Th\] The family $(\rho_t)_{t\in[0,\infty)}$ defines an asymptotic representation of $C^*_r(\mathbb F_2)$.

We are going to show that $\lim_{t\to\infty}\|\rho_t(a)\|=\|\lambda(a)\|$ for any $a\in\mathbb C[\mathbb F_2]$. It follows from the definition of $\rho_t$ that $\lim_{t\to\infty}\|\rho_t(a)\|$ is equal to the maximum of the following three numbers: $$\begin{aligned}
M_1(a)&=&\lim\nolimits_{n\to\infty}\max\nolimits_{1\leq i\leq
n}\|(\pi_0\otimes\sigma_i)(\Delta (a))\|;\\
M_2(a)&=&\sup\nolimits_{s\in[0,1]}\lim\nolimits_{n\to\infty}\|(\pi_s\otimes\sigma_n)(\Delta(a))\|;\\
M_3(a)&=&\lim\sup\nolimits_{n\to\infty}\|(\pi_1\otimes\sigma_n)(\Delta(a))\|,
 \end{aligned}$$ where $\Delta$ is the map given by the diagonal homomorphism $\mathbb F_2\to\mathbb F_2\otimes\mathbb F_2$. It follows from the Fell’s absorbtion principle that $\pi_0\otimes\sigma_n\cong\lambda^{(\dim\sigma_n)}$ and $\pi_1\otimes\sigma_n\cong\lambda^{(\dim\sigma_n)}\oplus\sigma_n$, hence $M_1(a)=\|\lambda(a)\|$ and $$M_3(a)=\max\bigl\lbrace\|\lambda(a)\|,
\,\lim\sup\nolimits_{n\to\infty}\|\sigma_n(a)\|\bigr\rbrace=\|\lambda(a)\|$$ by (\[Haagerup\]), so it remains to estimate $M_2(a)$.

Consider the ideal $I=C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb
F_2)\cap{{\mathbb K}}$ in $C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb F_2)$. The equality (\[Haagerup\]) implies that $$\label{isomo}
C^*_r(\mathbb F_2)\cong C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb
F_2)/I.$$

By Proposition \[exact\], for any $s\in[0,1]$ there is an $C^*$-algebra isomorphism $$\label{ravny}
C^*_{\pi_s}(\mathbb
F_2)\otimes(C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb F_2)/I)=
(C^*_{\pi_s}(\mathbb F_2)\otimes
C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb
F_2))/(C^*_{\pi_s}(\mathbb F_2)\otimes I),$$ where $\otimes$ denotes the minimal tensor $C^*$-product, in other words, the norm on the left-hand side of (\[ravny\]) is equal to the norm on the right-hand side. Let $q$ denote the quotient $*$-homomorphism from $C^*_{\pi_s}(\mathbb F_2)\otimes
C^*_{\oplus_{n=1}^\infty \sigma_n}(\mathbb F_2)$ to the $C^*$-algebra in (\[ravny\]) and let $c\in C^*_{\pi_s}(\mathbb
F_2)\odot C^*_{\oplus_{n=1}^\infty \sigma_n}(\mathbb F_2)$, where $\odot$ denotes the algebraic tensor product. Then the norm of $q(c)$ in the right-hand side $C^*$-algebra equals $$\|q(c)\|=\lim\sup\nolimits_{n\to\infty}\|(\pi_s\otimes\sigma_n)(c)\|.$$

Now let $c=\Delta(a)$, where $a\in \mathbb C[\mathbb F_2]$ and $\Delta$ is considered as the diagonal homomorphism, $\Delta:C^*(\mathbb F_2)\to C^*_{\pi_s}(\mathbb F_2)\otimes
C^*_{\oplus_{n=1}^\infty \sigma_n}(\mathbb F_2)$. Then the norm of $q(c)=q(\Delta(a))$ in the right-hand side of (\[ravny\]) equals $$\|q(\Delta(a))\|=\lim\sup\nolimits_{n\to\infty}\|(\pi_s\otimes\sigma_n)(a)\|.$$ Because of (\[ravny\]) we can evaluate the same norm using the left-hand side of (\[ravny\]): $$q(\|\Delta(a))\|=\|(\pi_s\otimes\lambda)(a)\|$$ (we use here the isomorphism (\[isomo\])), but the latter equals $\|\lambda(a)\|$ by the Fell’s absorbtion principle, so $$\label{e1}
\lim\sup\nolimits_{n\to\infty}\|(\pi_s\otimes\sigma_n)(a)\|=
\|\lambda(a)\|.$$ Let us show that convergence in (\[e1\]) is uniform. This would follow from equicontinuity of the sequence $\{f_n\}_{n\in\mathbb
N}$ of functions, where $f_n(s)=\|(\pi_s\otimes\sigma_n)(b)\|$, $s\in[0,1]$, for any finite sum $b=\sum_{i,j}\alpha_{ij}g_i\otimes
g_j\in C^*_{\pi_s}(\mathbb F_2)\odot C^*_{\sigma_n}(\mathbb F_2)$, where $\alpha_{i,j}\in\mathbb C$, $g_i\in\mathbb F_2$. Equicontinuity follows from the obvious estimate $$|f_n(s')-f_n(s)|\leq\sum\nolimits_{i,j}|\alpha_{ij}|\cdot\|\pi_{s'}(g_i)-\pi_s(g_i)\|\cdot\|\sigma_n(g_j)\|,$$ which is independent of $n$ since $\|\sigma_n(g_j)\|=1$ for all $n$ and $j$. Finally we can conclude that $$M_2(a)\leq
\sup\nolimits_{s\in[0,1]}\lim\sup\nolimits_{n\to\infty}\|(\pi_s\otimes\sigma_n)(a)\|=\|\lambda(a)\|.$$

An asymptotic representation $(\varphi_t)_{t\in[0,\infty)}:A\to\mathbb B(H)$ is [*representation-like*]{} if, for any $a_1,\ldots,a_n\in A$ and for any $\varepsilon>0$ there exists $T$ such that for any $t>T$ there is an isometry $U_t$ that satisfies $\|\varphi_t(a_i)-U_t\pi(a_i)U_t^*\|<\varepsilon$, $i=1,\ldots,n$, where $\pi:A\to\mathbb B(H)$ is some faithful representation.

We show below that the asymptotic representation $(\rho_t)_{t\in[0,\infty)}$ from Theorem \[Th\] is not representation-like.

Applications
============

Semi-invertibility of extensions
--------------------------------

Let $E=C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb F_2)+{{\mathbb K}}$. Then $$\label{extension}
0\to{{\mathbb K}}\to E\to E/{{\mathbb K}}=C^*_{\oplus_{n=1}^\infty\sigma_n}(\mathbb
F_2)/I\cong C^*_r(\mathbb F_2)\to 0$$ is an extension of $C^*_r(\mathbb F_2)$ by ${{\mathbb K}}$ constructed in [@Haagerup-T], where it was proved that this extension is non-invertible. Here we show that this extension is semi-invertible [@MT3], moreover, it represents the trivial element in the group ${\operatorname{Ext}}^{-1/2}(C^*_r(\mathbb F_2),{{\mathbb K}})$ [@MT3]. Let us briefly recall these definitions. An extension $0\to {{\mathbb K}}\to E\stackrel{p}\to A\to 0$, of $A$ by ${{\mathbb K}}$, is [*asymptotically split*]{} if there exists an asymptotic homomorphism $(\varphi_t)_{t\in[0,\infty)}:A\to E$ such that $p\circ\varphi_t={\operatorname{id}}_A$ for any $t\in[0,\infty)$. An extension of $A$ by ${{\mathbb K}}$ is [*semi-invertible*]{} if its direct sum with some other extension of $A$ by ${{\mathbb K}}$ is asymptotically split. Two extensions of $A$ by ${{\mathbb K}}$ are [*stably equivalent*]{} if their direct sums with some semi-invertible extension of $A$ by ${{\mathbb K}}$ are unitarily equivalent. Finally, the group ${\operatorname{Ext}}^{-1/2}(A,{{\mathbb K}})$ is the group of classes of stable equivalence of semi-invertible extensions of $A$ by ${{\mathbb K}}$. The only difference of ${\operatorname{Ext}}^{-1/2}(A,{{\mathbb K}})$ from the group ${\operatorname{Ext}}(A,{{\mathbb K}})^{-1}$ of invertibles in the classical BDF functor is that we use asymptotically split extensions instead of split ones.

The extension $($\[extension\]$)$ is semi-invertible. Moreover, it represents the trivial element in the group ${\operatorname{Ext}}^{-1/2}(C^*_r(\mathbb F_2),{{\mathbb K}})$.

Let $q:\mathbb B(H)\to \mathbb Q=\mathbb B(H)/{{\mathbb K}}$ be the quotient map. Then an easy calculation shows that $$q(\rho_t(a))=q\bigl(\oplus_{n=1}^\infty\sigma_n(a)\oplus\lambda^{(\infty)}(a)\bigr).$$ Therefore, the direct sum of the extension (\[extension\]) and of the trivial extension admits an asymptotic lift.

This gives an evidence to the following conjecture:

All extensions of $C^*_r(\mathbb F_2)$ are semi-invertible.

Estimates for the asymptotic tensor $C^*$-norm
----------------------------------------------

In [@MT6] we have introduced the asymptotic tensor $C^*$-norm (one-sided and symmetric) for tensor products of separable $C^*$-algebras. Recall that the left asymptotic tensor $C^*$-norm on the algebraic tensor product $A\odot B$ of two $C^*$-algebras is defined by $$\|c\|_\lambda=\sup\nolimits_{\varphi,\psi}\lim\sup\nolimits_{t\to\infty}\|(\varphi_t\otimes\psi)(c)\|,$$ where $c\in A\odot B$ and the supremum is taken over all asymptotic representations $\varphi=(\varphi_t)_{t\in[0,\infty)}:A\to\mathbb B(H)$ and all representations $\psi:B\to\mathbb B(H)$. The symmetric asymptotic tensor norm $\|\cdot\|_\sigma$ is defined as a similar supremum, where $\psi$ stands for asymptotic representations of $B$.

Now we are able to calculate the left asymptotic norm of some elements in $C^*_r(\mathbb F_2)\odot
C^*_{\oplus_{n=1}^\infty\bar{\sigma}_n}(\mathbb F_2)$, where $\bar{\sigma}$ is the representation contragredient to $\sigma$, and the symmetric asymptotic tensor $C^*$-norm of some elements in $C^*_r(\mathbb F_2)\odot C^*_r(\mathbb F_2)$.

Let $a=\frac{1}{4}(g_1+g_1^{-1}+g_2+g_2^{-1})$, where $g_1,g_2$ are free generators for $\mathbb F_2$.

\[estim\] $\|\Delta(a)\|_\lambda=1$ in $C^*_r(\mathbb F_2)\odot
C^*_{\oplus_{n=1}^\infty\bar{\sigma}_n}(\mathbb F_2)$; $\|\Delta(a)\|_\sigma=1$ in $C^*_r(\mathbb F_2)\odot C^*_r(\mathbb
F_2)$.

Obviously $\|\Delta(a)\|_\lambda\leq \|a\|\leq 1$. Similarly, $\|\Delta(a)\|_\sigma\leq 1$. Let us use the asymptotic representation $\rho_t$ to estimate both norms from below. $$\|\Delta(a)\|_\lambda\geq\lim\sup\nolimits_{t\to\infty}
\|(\rho_t\otimes(\oplus_{n=1}^\infty\bar{\sigma}_n))(\Delta(a))\|\geq
\lim\sup\nolimits_{n\to\infty}\|(\sigma_n\otimes\bar{\sigma}_n)(\Delta(a))\|=1$$ because $\sigma_n\otimes\bar{\sigma}_n$ contains a copy of the trivial representation.

Similarly, $$\|\Delta(a)\|_\sigma\geq\lim\sup\nolimits_{t\to\infty}
\|(\rho_t\otimes\bar{\rho}_t)(\Delta(a))\|\geq
\lim\sup\nolimits_{n\to\infty}\|(\sigma_n\otimes\bar{\sigma}_n)(\Delta(a))\|=1.$$

The asymptotic representation of Theorem \[Th\] is not representation-like.

It is easy to see that for a representation-like asymptotic representation $(\varphi_t)_{t\in[0,\infty)}:A\to\mathbb B(H)$ one has $$\lim\sup\nolimits_{t\to\infty}\|(\varphi_t\otimes\mu)(c)\|\leq
\|(\pi\otimes\mu)(c)\|$$ for any $C^*$-algebra $B$, any representation $\mu$ of $B$ and any $c\in A\odot B$, where $\pi$ is (any) faithful representation of $A$.

The following corollary shows that there is no asymptotic analogue for the Fell’s absorbtion principle.

$C^*_r(\mathbb F_2)\otimes_\sigma C^*_r(\mathbb F_2)\neq
C^*_r(\mathbb F_2)\otimes C^*_r(\mathbb F_2)$.

It is well-known [@Kesten] that $\|(\lambda\otimes\lambda)(\Delta(a))\|=\|\lambda(a)\|=\frac{\sqrt{3}}{2}<1=\|\Delta(a)\|_\sigma$.

Note that $C^*_r(\mathbb F_2)\otimes_\lambda C^*_r(\mathbb
F_2)\neq C^*_r(\mathbb F_2)\otimes C^*_r(\mathbb F_2)$ due to the Fell’s absorbtion principle.

Note also that the result of Proposition \[estim\] contrasts the analogous result for property T groups, cf. [@MT7]. Let $G$ be a property T group, $g_1,\ldots,g_m$ its generators and $a=\frac{1}{2m}(g_1+g_1^{-1}+\ldots+g_m+g_m^{-1})\in\mathbb C[G]$, $\alpha>0$ the Kazhdan constant (i.e. ${\operatorname{Sp}}(a)\subset[-1,1-\alpha]\cup\{1\}$, cf. [@HRV]). Let $\tau_n$, $n\in\mathbb N$, be a sequence of finitedimensional representations of $G$. The proof of the following proposition is essentially contained in [@MT7].

\[estim2\] $\|\Delta(a)\|_\lambda=1-\alpha$ in $C^*_r(G)\odot
C^*_{\oplus_{n=1}^\infty\tau_n}(G)$.

It is known [@Valette] that $\Delta(a)=x+\Delta(p)$, where $x\leq 1-\alpha$ and $p\in C^*(G)$ is the projection corresponding to the trivial representation of $G$. Let $(\varphi_t)_{t\in[0,\infty)}:C^*_r(G)\to\mathbb B(H)$ be an arbitrary asymptotic representation. The proposition would follow if we show that $$\label{nol}
\lim_{t\to\infty}\bigl(\varphi_t\otimes(\oplus_{n=1}^\infty\tau_n)\bigr)(\Delta(p))=0.$$

Put $$q_t(n)=(\varphi_t\otimes\tau_n)(\Delta(p));\quad
q_t=\oplus_{n=1}^\infty q_t(n).$$ Then $q_t$ is asymptotically a projection, i.e. $\lim_{t\to\infty}\|q_t-q_t^2\|=0$, hence the same holds for each $n$ uniformly: $\lim_{t\to\infty}\|q_t(n)-q_t^2(n)\|=0$. This means that one can determine if $\lim_{t\to\infty}q_t(n)=0$ by looking at $q_t(n)$ for some fixed $t$ (same for all $n$).

Suppose that (\[nol\]) is not true. Then there exists $n$ such that $\lim_{t\to\infty}\|q_t(n)\|=1$. Hence $\lim_{t\to\infty}\|(\varphi_t\otimes\tau_n)(\Delta(a))\|=1$, i.e. the norm of $\Delta(a)$ in $C^*_r(G)\otimes_\lambda
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C^*_{\tau_n}(G)=C^*_r(G)\otimes C^*_{\tau_n}(G)$. But the norm of $\Delta(a)$ in $C^*_r(G)\otimes C^*_{\tau_n}(G)$ cannot equal 1 [@WassT]. This contradiction ends the proof.

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