\IfFormatAtLeastTF{2025-06-01}{\DocumentMetadata{lang=en-US,pdfversion=2.0,tagging=on}}{} \documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \IfFileExists{lmodern.sty}{\usepackage{lmodern}}{\PassOptionsToPackage{expansion=false}{microtype}} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{etoolbox} \usepackage[letterpaper,textwidth=5.6in,textheight=\textheight,hmarginratio=1:1]{geometry} \usepackage[nopatch=footnote]{microtype} \usepackage{xcolor} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \newif\ifanonymous \anonymousfalse \ifanonymous \newcommand{\paperauthor}{Anonymous} \else \input{coderepo.tex} \newcommand{\paperauthor}{\paperauthorname} \fi \hypersetup{pdftitle={Non-MF Groups},pdfauthor={\paperauthor},pdfsubject={MF groups and operator-norm matrix approximation},pdfkeywords={MF groups, norm matrix coronas, MF radicals, property (T), asymptotic representations}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{doi:#1}}} % Machine-readable source metadata attaching a printed statement to an % accompanying Lean~4 declaration. 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This manuscript is under revision. \end{center}} \begin{document} \frenchspacing \title{Non-MF Groups} \author{\paperauthor} \subjclass[2020]{Primary 46L05; Secondary 20F65, 22D10, 16S50} \keywords{MF groups, norm matrix coronas, property~(T), asymptotic representations, Leavitt algebra} \begin{abstract} \addtolength{\leftskip}{1.5pc}\addtolength{\rightskip}{1.5pc} \leanverified{Manuscript/OneSidedMFRadical/CountableNonMF}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptNotEveryCountableGroupIsMF} Some groups are not MF. For example, $H=\EL_{12}(L_{\F_2}(1,2))$ is a simple property-\textup{(T)} group where every homomorphism from $H$ to an MF group is trivial, so $C^*_{\mathrm r}(H)$ is a separable stably finite $C^*$-algebra that is not MF\@. Generally, a normal property-\textup{(T)} subgroup contained in the commutators of a one-sided compression of a property-\textup{(T)} subgroup is killed by every homomorphism to an MF group. The same criterion gives a finitely presented torsion-free property-\textup{(T)} group with no nontrivial homomorphism to an MF group. \end{abstract} \maketitle \draftnotice \section{Introduction} The counterexample is \[ H=\EL_{12}(L_{\F_2}(1,2)), \] the elementary group of $12\times12$ matrices over the binary Leavitt algebra. Khanh--Thanh show that $H$ is isomorphic to the unit group of the binary Leavitt algebra~\cite[Proposition~4.2 and Corollary~4.4]{KhanhThanh}, which OpenAI used to construct the first nonsofic group~\cite[Chapter~3]{OAI}. A countable group $G$ is \emph{MF}~\cite{CDE} if there are positive integers $d_n$ and maps $V_n\colon G\to\U(d_n)$, with $V_n(1)=1$, such that \[ \opnorm{V_n(gh)-V_n(g)V_n(h)}\longrightarrow0 \qquad(g,h\in G), \] and \[ \limsup_n\opnorm{V_n(g)-1}>0 \qquad(g\in G\setminus\{1\}). \] In other words, $G$ is MF if there are finite unitary matrices, one for each element of $G$, that multiply correctly up to an error tending to zero in operator norm while every element other than the identity stays a fixed distance from the identity matrix. The separation may equivalently be required along the full sequence with a constant independent of $g$~\cite[Propositions~2 and~7]{Korchagin}. The strong convergence convention of~\cite{GaoEtAl,Schafhauser} also requires the models to reproduce the operator norms of the left regular representation, so a group that is not MF as defined here is not MF in that convention either. Equivalently, $G$ embeds in the unitary group of the norm matrix corona \[ \mathcal Q_{\mathbf d} =\prod_nM_{d_n}(\C)\big/\bigoplus_nM_{d_n}(\C), \] where $\prod_nM_{d_n}(\C)$ is the $C^*$-algebra of bounded sequences, as every product of $C^*$-algebras below, and $\bigoplus_nM_{d_n}(\C)$ is its ideal of sequences with $\opnorm{x_n}\to0$. The quotient $\mathcal Q_{\mathbf d}$ is the corona algebra of $\bigoplus_nM_{d_n}(\C)$, since $\prod_nM_{d_n}(\C)$ is its multiplier algebra. We call a homomorphism $G\to\U(\mathcal Q_{\mathbf d})$ a \emph{corona homomorphism}. A separable $C^*$-algebra is MF if it embeds in a norm matrix corona~\cite{BK}. The groups here fail to be MF because they contain a subgroup that is conjugated into itself. For $L\le G$ put \[ \operatorname{Comp}_G(L)=\{u\in G:uLu^{-1}\le L\}. \] An element $c$ of the centralizer $C_G(L)$ commutes with $L$, so $ucu^{-1}$ commutes with $uLu^{-1}$ but need not commute with the rest of $L$. The normal subgroup of $G$ generated by the commutators of all such elements $ucu^{-1}$ with elements of $L$ is \begin{equation}\label{eq:intrinsic-defect} \mathfrak D_G(L) =\normal{ [ucu^{-1},\ell]: u\in\operatorname{Comp}_G(L),\ c\in C_G(L),\ \ell\in L }_G. \end{equation} \begin{mainthm}[one-sided compression criterion] \label{thm:compression-criterion} Let $G$ be countable and let $L\le G$ have property~\textup{(T)}. If $K\trianglelefteq G$ has property~\textup{(T)} and $K\le\mathfrak D_G(L)$, then every homomorphism from $G$ to an MF group is trivial on $K$. In particular, a nontrivial such $K$ makes $G$ non-MF, and if $G$ has property~\textup{(T)} and $\mathfrak D_G(L)=G$, then every homomorphism from $G$ to an MF group is trivial. \leanverified{Manuscript/OneSidedMFRadical/PrintedCriterion}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptOneSidedCompressionCriterion} \leanverified{Manuscript/OneSidedMFRadical/PrintedCriterion}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptFullRadicalKillsMFTargets} \end{mainthm} Asymptotic multiplicativity in operator norm makes the conjugation maps $\operatorname{Ad}(V_n(g))\colon x\mapsto V_n(g)xV_n(g)^*$ an asymptotic unitary representation of $G$ on $M_{d_n}(\C)$ with its normalized Hilbert--Schmidt inner product, because $\opnorm{\operatorname{Ad}(A)-\operatorname{Ad}(B)}\le2\opnorm{A-B}$. Property~\textup{(T)} applies to unitary representations on Hilbert spaces, so it applies to this one, and its conclusions are estimates in Hilbert--Schmidt norm. The unitary $U$ of the compressor $u$ satisfies $U^*PU\le P$ for the Kazhdan projection $P$ of $L$, because $uLu^{-1}\le L$ has fewer constraints and more fixed vectors, and stable finiteness of the matrix corona forces $U^*PU=P$. So conjugation by $u$ preserves the Hilbert--Schmidt asymptotic commutant of $L$, and every element of $\mathfrak D_G(L)$ tends to $1$ in Hilbert--Schmidt norm in every operator norm asymptotic representation (Corollary~\ref{cor:defect-hs}). Property~\textup{(T)} of $K$ turns this into triviality in operator norm: a corona homomorphism nontrivial on $K$ compresses to the corner where $K$ has no fixed vectors, and the Kazhdan inequality there contradicts the Hilbert--Schmidt triviality (Theorem~\ref{thm:normal-kazhdan}).\footnote{A Hilbert--Schmidt bound on the multiplicative defect does not give an operator norm bound on the conjugation maps, so this argument does not apply to sofic or hyperlinear approximations.} \leanverified{Manuscript/OneSidedMFRadical/PrintedRemarks}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptFullRadicalsDoNotEntailNonsoficity} The stable letter of an ascending HNN extension of a property-\textup{(T)} group along a proper self-embedding is a compressor. The group $H$ realizes the compression inside a simple group, where one nontrivial commutator $[ucu^{-1},\ell]$ normally generates everything. Let $L_{\F_2}(1,2)$ denote the binary Leavitt algebra, the unital $\F_2$-algebra generated by $s_0,s_1,t_0,t_1$ subject to \begin{equation}\label{eq:leavitt} t_i s_j=\delta_{ij}, \qquad s_0t_0+s_1t_1=1. \end{equation} For a unital ring $R$, write $e_{ij}(a)=1+aE_{ij}$ and let $\EL_n(R)$ be the subgroup of $\GL_n(R)$ generated by the elements $e_{ij}(a)$, $i\ne j$. \begin{mainthm}[the binary Leavitt group]\label{thm:headline} Put $R=L_{\F_2}(1,2)$ and $H=\EL_{12}(R)$. Then $H$ is finitely generated, nontrivial, simple, has property~\textup{(T)}, and every homomorphism from $H$ to an MF group is trivial. So $H$ is not MF, and $C^*_{\mathrm r}(H)$ is separable and stably finite but not MF, while $C^*_{\max}(H)$ is not even finite: it contains a proper isometry. \leanverified{Manuscript/OneSidedMFRadical/ReducedCStarConsequence}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptPrintedHeadline} \leanverified{Manuscript/OneSidedMFRadical/HeadlineTheorem}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptBinaryLeavittHeadline} \leanverified{Manuscript/OneSidedMFRadical/ReducedCStarConsequence}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptReducedCStarConsequence} \leanverified{Manuscript/OneSidedMFRadical/ReducedCStarConsequence}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptNotEverySeparableStablyFiniteCStarAlgebraIsMF} \end{mainthm} \begin{mainthm}[a torsion-free finitely presented example] \label{thm:torsion-free} There is a two-generated, finitely presented, torsion-free, acylindrically hyperbolic group $Q$ with property~\textup{(T)} such that every homomorphism from $Q$ to an MF group is trivial. In particular, no nontrivial quotient of $Q$ is MF\@. \end{mainthm} Section~\ref{sec:torsion-free} builds $Q$ from Fournier-Facio's torsion-free property-\textup{(T)} group~\cite[\S2]{FFF} and Hull's small cancellation theorem~\cite{Hull}. \subsection*{Relation to prior work} Blackadar and Kirchberg introduced MF algebras and proved that a separable $C^*$-algebra is NF if and only if it is nuclear and MF~\cite{BK}. Whether every separable stably finite $C^*$-algebra is MF, the MF problem~\cite[\S6]{GoldbringHart}, was answered negatively through the failure of the Connes embedding problem~\cite{MIPRE}\cite[Proposition~6.1 and Remark~6.2]{GoldbringHart}. Theorem~\ref{thm:headline} gives a counterexample among reduced group $C^*$-algebras. Whether every countable group is MF was a longstanding open problem~\cite{BDL}. \leanverified{Manuscript/OneSidedMFRadical/ReducedCStarConsequence}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptSeparableStablyFiniteNonMFCStarAlgebraExists} Theorem~\ref{thm:compression-criterion} uses the configuration of the nonsofic construction~\cite[Proposition~2.3]{OAI}: a property-\textup{(T)} subgroup $L$ conjugated into itself by an element $u$, and an element $c$ commuting with $L$ whose conjugate $ucu^{-1}$ does not. There, soficity and the rigidity theorems of Kun~\cite{Kun16} and Kun--Thom~\cite{KT19} force the centralizing subgroup to be locally embeddable into finite groups, and a copy of Thompson's group $V$ gives the contradiction. Here the rigidity is the Kazhdan projection of $L$ in a norm matrix corona, which stable finiteness of the corona forces to commute with the compressor, and the contradiction is a commutator $[ucu^{-1},\ell]\ne1$ that every operator norm asymptotic representation sends to $1$ in Hilbert--Schmidt norm. The group $H$ is the unit group of the binary Leavitt algebra, the group of~\cite{OAI}, and Theorem~\ref{thm:torsion-free} reuses Fournier-Facio's configuration~\cite{FFF}. Bachner--Dogon--Lubotzky showed that for groups of Deligne type, operator--Hilbert--Schmidt stability would imply that the group is not MF~\cite[Proposition~1.5]{BDL}. The compression relation replaces stability. For a short elementary construction of non-MF generalized wreath products from one-sided compression, see~\cite{Eckhardt}. Leavitt introduced the algebras $L_K(1,n)$~\cite{Leavitt}. The binary Leavitt algebra is purely infinite simple~\cite{AbramsAranda}, is an exchange ring~\cite{AraExchange}, and has center $\F_2$~\cite{ArandaCrow}. Preusser's sandwich theorem then gives simplicity of $H$~\cite{Preusser}, and property~\textup{(T)} is Ershov--Jaikin-Zapirain's theorem~\cite[Theorem~1.1]{EJZ}. \section{One-sided compression} \label{sec:compression-radical} \subsection{Corona homomorphisms} The image of a corona homomorphism from $G$ is a countable subgroup of a corona unitary group, so MF, and every countable MF group embeds in the unitary group of a norm matrix corona. \leanverified{Manuscript/OneSidedMFRadical/PrintedDefinitions}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptCoronaImagesAndTargets} \begin{lemma}\label{prop:mf-residual-calculus} Let $G$ be countable and $K\le G$. Every homomorphism from $G$ to an MF group is trivial on $K$ if and only if every corona homomorphism from $G$ is trivial on $K$. \leanverified{Manuscript/OneSidedMFRadical/LiteralMFClosure}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptMFResidualCalculusLiteral} \end{lemma} \begin{proof} A corona homomorphism is a homomorphism to an MF group, and a homomorphism to an MF group composed with an embedding of that group into a corona unitary group is a corona homomorphism with the same kernel. \end{proof} \subsection{Kazhdan transport in normalized Hilbert--Schmidt norm} For $a\in M_d(\C)$ write \[ \hsnorm{a}=\tr_d(a^*a)^{1/2}, \qquad \tr_d=\frac1d\operatorname{Tr}. \] An \emph{operator norm asymptotic representation} of a countable group $G$ is a sequence of maps $V_n\colon G\to\U(d_n)$ such that $V_n(1)=1$ and \[ \opnorm{V_n(gh)-V_n(g)V_n(h)}\longrightarrow0 \qquad(g,h\in G). \] The elements $g$ with $\hsnorm{V_n(g)-1}\to0$ form a normal subgroup of $G$, because $\hsnorm{a}\le\opnorm{a}$, the Hilbert--Schmidt norm is invariant under adjoints and unitary conjugation, and $\opnorm{V_n(g^{-1})-V_n(g)^*}\to0$: \begin{gather*} \hsnorm{V_n(gh)-1}\le\hsnorm{V_n(g)-1}+\hsnorm{V_n(h)-1}+o(1),\\ \hsnorm{V_n(g^{-1})-1}=\hsnorm{V_n(g)-1}+o(1),\qquad \hsnorm{V_n(ghg^{-1})-1}=\hsnorm{V_n(h)-1}+o(1). \end{gather*} For $L\le G$, let \[ \mathcal C_2(V,L) =\left\{(x_n)\ \middle|\ \begin{aligned} &\sup_n\opnorm{x_n}<\infty,\\ &\hsnorm{[V_n(\ell),x_n]}\longrightarrow0 &&(\ell\in L) \end{aligned}\right\} \] be the bounded Hilbert--Schmidt asymptotic commutant of $V(L)$. Coordinatewise conjugation by $V_n(g)$ defines a bijection, denoted $\operatorname{Ad}(V(g))$, of the bounded matrix sequences. \begin{lemma}\label{lem:stable-finite} For every sequence of positive integers $(m_n)$, the norm matrix corona \[ \prod_nM_{m_n}(\C)\big/\bigoplus_nM_{m_n}(\C) \] is stably finite. Consequently, unitarily equivalent projections in a norm matrix corona are equal whenever one is dominated by the other. \leanverified{Manuscript/OneSidedMFRadical/StableFiniteness}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptNormMatrixCoronaStableFinite} \end{lemma} \begin{proof} Write $A$ for the displayed corona. If $v^*v=1$ in $A$ and $(x_n)$ is a bounded lift of $v$, then $\opnorm{x_n^*x_n-1}\to0$, so for all large $n$ the matrix $u_n=x_n(x_n^*x_n)^{-1/2}$ is unitary and $\opnorm{u_n-x_n}\to0$; after assigning arbitrary unitary values to the finitely many remaining coordinates, $(u_n)$ is a unitary lift of $v$, and $vv^*=1$. So $A$ is finite, and so is $M_k(A)$, which is the corona with matrix sizes $km_n$. Finally, if $p\le q$ are equivalent projections in a stably finite algebra $B$ and $w$ is a partial isometry with $w^*w=q$ and $ww^*=p$, then $w\in qBq$ is an isometry in the finite corner $qBq$, so $p=ww^*=q$. \end{proof} \begin{lemma}[one-sided order for the Kazhdan projection] \label{lem:kazhdan-projection-order} Let $L$ have property~\textup{(T)}, let $B$ be a unital $C^*$-algebra, and let $\pi\colon L\to\U(B)$ be a homomorphism. Denote by $P\in B$ the image of the Kazhdan projection under the extension $C^*_{\max}(L)\to B$. If $U\in\U(B)$ satisfies \[ U\pi(L)U^*\subseteq\pi(L), \] then \[ U^*PU\le P. \] \leanverified{Manuscript/OneSidedMFRadical/KazhdanProjectionOrderLiteral}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptLiteralMaximalCStarKazhdanProjectionOrder} \end{lemma} \begin{proof} Represent $B$ faithfully and nondegenerately on a Hilbert space $\mathcal H$. Then $P$ is the orthogonal projection onto $\operatorname{Fix}\pi(L)$. If $\xi$ is $L$-fixed and $\ell\in L$, then $\pi(\ell)U^*\xi=U^*(U\pi(\ell)U^*)\xi=U^*\xi$ because $U\pi(\ell)U^*$ belongs to $\pi(L)$. So $U^*\operatorname{Fix}\pi(L)\subseteq\operatorname{Fix}\pi(L)$, and the range projection $U^*PU$ is dominated by $P$ in $B(\mathcal H)$, and then in $B$ by faithfulness. \end{proof} \begin{theorem}[one-sided Kazhdan transport]\label{thm:transport} Let $L\le G$ have property~\textup{(T)}, and let $u\in G$ satisfy $uLu^{-1}\le L$. Let $(V_n)$ be an operator norm asymptotic representation of $G$. Then \[ \operatorname{Ad}(V(u))\bigl(\mathcal C_2(V,L)\bigr) =\mathcal C_2(V,L). \] \leanverified{Manuscript/OneSidedMFRadical/TransportCommutantEquality}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptOneSidedKazhdanTransportCommutantEquality} \end{theorem} \begin{proof} Fix $x=(x_n)\in\mathcal C_2(V,L)$ and a free ultrafilter $\omega$, and let $\mathcal K_\omega$ be the ultraproduct of the Hilbert spaces $(M_{d_n}(\C),\hsnorm{\,\cdot\,})$. For unitaries $A,B$, \[ \opnorm{\operatorname{Ad}(A)-\operatorname{Ad}(B)}\le2\opnorm{A-B}, \] so the maps $\operatorname{Ad}(V_n(g))$ are asymptotically multiplicative in operator norm, and \[ \sigma(g)=\lim_\omega\operatorname{Ad}(V_n(g)) \] is a unitary representation of $G$ on $\mathcal K_\omega$ with $\xi=[x_n]_\omega\in\operatorname{Fix}\sigma(L)$. The same estimate makes \[ \widetilde\sigma(g)=[\operatorname{Ad}(V_n(g))]_n\in\U(\mathcal B), \qquad \mathcal B=\prod_nB(M_{d_n}(\C))\big/\bigoplus_nB(M_{d_n}(\C)), \] a homomorphism, and $\mathcal B$ is a norm matrix corona with coordinate sizes $d_n^2$ after a choice of matrix units. Let $P\in\mathcal B$ be the image of the Kazhdan projection of $L$~\cite{AkemannWalter} under $C^*_{\max}(L)\to\mathcal B$. On $\mathcal K_\omega$ its range is $\operatorname{Fix}\sigma(L)$. Put $U=\widetilde\sigma(u)$. Since $uLu^{-1}\le L$, \[ U\widetilde\sigma(L)U^*\subseteq\widetilde\sigma(L), \qquad\text{so}\qquad U^*PU\le P \] by Lemma~\ref{lem:kazhdan-projection-order}, and \[ U^*PU=P \] by Lemma~\ref{lem:stable-finite}, since the two projections are unitarily equivalent. So $U$ commutes with $P$, and \[ \begin{aligned} &U\xi,\ U^*\xi\in\operatorname{Fix}\sigma(L),\qquad\text{that is,}\\ &\lim_\omega\hsnorm{[V_n(\ell),V_n(u)^{\pm1}x_nV_n(u)^{\mp1}]}=0 \qquad(\ell\in L). \end{aligned} \] Since $\omega$ was arbitrary, the limits hold along the full sequence: a subsequence staying above some $\varepsilon>0$ would lie in a free ultrafilter and give a nonzero ultralimit. So $\operatorname{Ad}(V(u))$ and its inverse both preserve $\mathcal C_2(V,L)$. \end{proof} \begin{corollary}\label{cor:defect-hs} Let $L\le G$ have property~\textup{(T)} and let $(V_n)$ be an operator norm asymptotic representation of $G$. Then \[ \hsnorm{V_n(d)-1}\longrightarrow0 \qquad(d\in\mathfrak D_G(L)). \] \leanverified{Manuscript/OneSidedMFRadical/DefectHS}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptCompressionDefectHSInvisible} \leanverified{Manuscript/OneSidedMFRadical/PrintedDefectShadow}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptPrintedDefectShadowInclusion} \end{corollary} \begin{proof} Let $u\in\operatorname{Comp}_G(L)$, $c\in C_G(L)$, and $\ell\in L$, and write $\mathcal C_2=\mathcal C_2(V,L)$. Then \[ (V_n(c))\in\mathcal C_2 \ \Longrightarrow\ (V_n(u)V_n(c)V_n(u)^*)\in\mathcal C_2 \ \Longrightarrow\ (V_n(ucu^{-1}))\in\mathcal C_2, \] the first membership because $c$ commutes with $L$, the first implication by Theorem~\ref{thm:transport}, and the second because $\opnorm{V_n(u)V_n(c)V_n(u)^*-V_n(ucu^{-1})}\to0$. So $\hsnorm{[V_n(\ell),V_n(ucu^{-1})]}\to0$, that is, $\hsnorm{V_n([ucu^{-1},\ell])-1}\to0$. The elements with this property form a normal subgroup of $G$, so it contains $\mathfrak D_G(L)$. \end{proof} \subsection{From Hilbert--Schmidt to operator norm} A corona homomorphism $\rho$ cannot simply be restricted to a corner: a projection $q$ commuting with $\rho(G)$ has lifts $q_n$ that only asymptotically commute with the lifts of $\rho(g)$, so the compressions $q_n\rho(g)q_n$ are only approximately unitary. \begin{lemma}[central corona corners]\label{lem:central-corona-corner} Let \(\rho\colon G\to\U(\mathcal Q_{\mathbf d})\) be a homomorphism from a countable group, and let \(q\in\mathcal Q_{\mathbf d}\) be a nonzero projection commuting with \(\rho(G)\). Then, after passing to an infinite coordinate subsequence, there are nonzero projections \(q_n\in M_{d_n}(\C)\), integers \(r_n=\operatorname{rank}(q_n)\), unitary identifications \(J_n\colon\C^{r_n}\to q_n\C^{d_n}\), and an operator norm asymptotic representation \[ W_n\colon G\longrightarrow\U(r_n) \] with \(W_n(1)=I_{r_n}\) such that, in the corona over the retained coordinates, the class of \((J_nW_n(g)J_n^*)\) is the coordinate restriction of \(q\rho(g)\), a unitary of the corner \(q\mathcal Q_{\mathbf d}q\) with unit \(q\). \leanverified{Manuscript/OneSidedMFRadical/CentralCoronaCorner}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptCentralCoronaCorner} \end{lemma} \begin{proof} Lift \(q\) to projections \(q_n\) by spectral rounding, and lift each \(\rho(g)\) to unitaries \(U_n(g)\) by polar correction, with \(U_n(1)=I_{d_n}\). Since \(q\ne0\), infinitely many \(q_n\) are nonzero. Retain those coordinates and put \(r_n=\operatorname{rank}(q_n)\). The commutation relation in the corona gives \(\opnorm{q_nU_n(g)-U_n(g)q_n}\to0\) for each \(g\), so \cite[Proposition~2.4]{BDL} with \(p=\infty\) gives an operator norm asymptotic representation \(\widetilde W_n\) of \(G\) by unitaries of the corner \(q_nM_{d_n}(\C)q_n\) with \(\opnorm{\widetilde W_n(g)-q_nU_n(g)q_n}\to0\). Replacing \(\widetilde W_n(1)\) by \(q_n\) changes it by \(o(1)\). Choose unitaries \(J_n\colon\C^{r_n}\to q_n\C^{d_n}\) and put \(W_n(g)=J_n^*\widetilde W_n(g)J_n\). Conjugation by \(J_n\) preserves operator norms and multiplicative defects, so \((W_n)\) is an operator norm asymptotic representation with \(W_n(1)=I_{r_n}\), and \(\opnorm{J_nW_n(g)J_n^*-q_nU_n(g)q_n}\to0\) identifies the class of \((J_nW_n(g)J_n^*)\) with the coordinate restriction of \(q\rho(g)\). \end{proof} \begin{theorem}[normal Kazhdan subgroups]\label{thm:normal-kazhdan} Let \(G\) be countable and let \(K\trianglelefteq G\) have property~\textup{(T)}. If every operator norm asymptotic representation \((V_n)\) of \(G\) satisfies \(\hsnorm{V_n(k)-1}\to0\) for all \(k\in K\), then every corona homomorphism from \(G\) is trivial on \(K\). \leanverified{Manuscript/OneSidedMFRadical/NormalKazhdan}{GroupApproximation.Manuscript.OneSidedMFRadical.manuscriptNormalKazhdanRadical} \end{theorem} \begin{proof} Suppose that a corona homomorphism \(\Theta\colon G\to\U(\mathcal Q_{\mathbf d})\) is nontrivial on \(K\). Replace \(G\) by \(\Theta(G)\) and \(K\) by \(\Theta(K)\): the new \(K\) is a nontrivial normal property-\textup{(T)} subgroup, since property~\textup{(T)} passes to quotients, and it still satisfies the hypothesis, because every operator norm asymptotic representation of \(\Theta(G)\) pulls back along \(G\to\Theta(G)\) to one of \(G\). Let \(p\in\mathcal Q_{\mathbf d}\) be the image of the Kazhdan projection of \(K\), the projection onto the \(K\)-fixed vectors in any faithful representation of the corona. Since \(gKg^{-1}=K\), \[ \Theta(g)p\Theta(g)^*=p\qquad(g\in G), \] and \(q=1-p\) is nonzero, because \(p=1\) would make every element of \(K\) act trivially. Lemma~\ref{lem:central-corona-corner} gives an operator norm asymptotic representation \(W_n\colon G\to\U(r_n)\) whose corona homomorphism \(\widehat\Theta\) is the coordinate restriction of \(g\mapsto q\Theta(g)\). The Kazhdan projection of \(K\) under \(\widehat\Theta\) is the coordinate restriction of \(qp=0\), so \(\widehat\Theta|_K\) has no nonzero fixed vectors, and for a finite Kazhdan set \(S\subseteq K\) with constant \(\kappa>0\), \[ b=\frac1{|S|}\sum_{s\in S} (\widehat\Theta(s)-1)^*(\widehat\Theta(s)-1) \ \ge\ \frac{\kappa^2}{|S|}\,1 \] in \(\mathcal Q_{\mathbf r}\). The coordinates \(b_n=\frac1{|S|}\sum_{s\in S}(W_n(s)-I_{r_n})^*(W_n(s)-I_{r_n})\) represent \(b\), so the negative part of \(b_n-(\kappa^2/|S|)I_{r_n}\) tends to zero in operator norm, and normalized traces on \(M_{r_n}(\C)\) give tends to zero in operator norm, and taking normalized traces on \(M_{r_n}(\C)\) gives \[ \frac1{|S|}\sum_{s\in S}\hsnorm{W_n(s)-I_{r_n}}^{\,2} \ge \frac{\kappa^2}{|S|}-o(1). \] After passing to a subsequence, one fixed \(s_0\in S\) stays a positive Hilbert--Schmidt distance from \(I_{r_n}\), against the hypothesis applied to \((W_n)\) and \(s_0\in K\). \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:compression-criterion}] By Corollary~\ref{cor:defect-hs}, every operator norm asymptotic representation of \(G\) satisfies \(\hsnorm{V_n(k)-1}\to0\) for \(k\in K\le\mathfrak D_G(L)\). Theorem~\ref{thm:normal-kazhdan} then makes every corona homomorphism trivial on \(K\), and Lemma~\ref{prop:mf-residual-calculus} makes every homomorphism to an MF group trivial on \(K\). If \(K\ne1\), then \(G\) does not embed in an MF group, so \(G\) is not MF\@. The case \(K=G=\mathfrak D_G(L)\) gives the last assertion. \end{proof} \subsection{The maximal group \texorpdfstring{$C^*$}{C*}-algebra} Lemma~\ref{lem:kazhdan-projection-order} also applies inside $C^*_{\max}(G)$ itself, where a strict compression produces a proper isometry. \begin{lemma}[strictly dominated projections]\label{lem:proper-isometry} Let $A$ be a unital $C^*$-algebra containing a projection $p$ and a unitary $u$ with $p