1 Introduction
1.1 Inverse theorems
The uniformity norms first appeared in Gowers’ work on arithmetic progressions [Reference Gowers3] and [Reference Gowers2]. For a map
$\unicode[STIX]{x1D711}:G\rightarrow \mathbb{C}$
defined on a finite abelian group, the second uniformity norm is given by
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D711}\Vert _{U^{2}}^{4}=\frac{1}{|G|^{3}}\mathop{\sum }_{x-y+z-w=0}\unicode[STIX]{x1D711}(x)\overline{\unicode[STIX]{x1D711}(y)}\unicode[STIX]{x1D711}(z)\overline{\unicode[STIX]{x1D711}(w)}.\end{eqnarray}$$
It is simple to show that a function with large
$U^{2}$
-norm is correlated to some character
$\unicode[STIX]{x1D712}:G\rightarrow {\mathcal{U}}_{1}\subseteq \mathbb{C}$
in the sense that
$(1/|G|)\sum _{x\in G}\unicode[STIX]{x1D711}(x)\overline{\unicode[STIX]{x1D712}(x)}$
is relatively large. This result was generalized by Gowers and Hatami to non-abelian finite groups and matrix-valued maps. More precisely, they proved the following theorem.
Theorem 1.1 (Gowers and Hatami [Reference Gowers and Hatami4]).
Let
$G$
be a finite group, let
$c\in (0,1]$
and let
$\unicode[STIX]{x1D711}:G\rightarrow \mathbb{M}_{n}$
with
$\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}\leqslant 1$
such that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D711}\Vert _{U^{2}}^{4}:=\frac{1}{|G|^{3}}\mathop{\sum }_{xy^{-1}zw^{-1}=e}\operatorname{Tr}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(w)^{\ast })\geqslant cn.\end{eqnarray}$$
Then there are
$m\in [(c/(2-c))n,((2-c)/c)n]$
,
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}_{m}$
and maps
$U,V:\mathbb{C}^{n}\rightarrow \mathbb{C}^{m}$
such that
$$\begin{eqnarray}\frac{1}{|G|}\mathop{\sum }_{x\in G}\operatorname{Tr}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }U)\geqslant t(c)n,\end{eqnarray}$$
where
$t(c)=\max \{c^{8}/(2-c)^{8},c^{2}/4\}$
. Moreover, if
$n\leqslant m$
we can take
$U$
and
$V$
to be isometries and if
$n\geqslant m$
we can take
$U$
and
$V$
to be co-isometries.
The proof of Theorem 1.1 presented in [Reference Gowers and Hatami4] relies heavily on properties of the Fourier transform of
$\unicode[STIX]{x1D711}$
, which has to be generalized in two ways from the usual Fourier transform since
$\unicode[STIX]{x1D711}$
is defined on a possibly non-abelian group and takes matrix values. This Fourier theory works well for finite (or compact) groups, but it is hard to generalize to infinite, discrete groups. Our interest in the above theorem is in particular due to the fact that it can be used to prove a certain stability result for
$\unicode[STIX]{x1D700}$
-representations with respect to the
$p$
-norm coming from the trace on
$\mathbb{M}_{n}$
, which we will explain in depth later. In a private communication with Gowers and Hatami, the second named author of this paper provided an alternative, operator algebraic proof of this stability result. A strength of this approach is that it is possible to generalize the proof, replacing
$G$
with an (infinite) amenable group and
$\mathbb{M}_{n}$
with a semi-finite von Neumann algebras
${\mathcal{M}}$
. Gowers and Hatami ask in [Reference Gowers and Hatami4] whether this approach can be accommodated to a proof of their above theorem. In this article, we answer this question affirmatively—this is our main result and can be found in Theorems 3.1, 4.1 and 4.2 which constitute generalizations of Theorem 1.1 to various settings.
Our main theorem (Theorem 3.1) is formulated in a quite general way and it might be hard to compare the proofs, so for clarity’s sake, we will now outline the main differences between them. Gowers and Hatami use the singular value decomposition of the Fourier transform of
$\unicode[STIX]{x1D711}$
to cherry-pick the irreducible representations that are correlated to
$\unicode[STIX]{x1D711}$
and put these together to a representation
$\unicode[STIX]{x1D70B}$
together with maps
$U,V:\mathbb{C}^{n}\rightarrow \mathbb{C}^{m}$
such that
$(1/|G|)\sum _{x\in G}\operatorname{Tr}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }U)$
is comparatively big. A subtle difficulty in their proof is that they have no control over the operator norms of
$U$
and
$V$
so they make use of a series of clever arguments to alter
$U$
and
$V$
so that they satisfy
$\Vert U\Vert _{\text{op}},\Vert V\Vert _{\text{op}}\leqslant 1$
. Once these estimates are achieved, an extreme point argument provides the necessary isometries or co-isometries. On the other hand, the crux of our proof is to use the Stinespring dilation theorem, which is a fundamental theorem about completely positive maps. It immediately brings into existence
$\unicode[STIX]{x1D70B}$
and the maps
$U,V:\mathbb{C}^{n}\rightarrow \mathbb{C}^{m}$
so that
$(1/|G|)\sum _{x\in G}\operatorname{Tr}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }U)$
is big, but the main advantage of using Stinespring’s theorem is that it automatically gives us the estimates
$\Vert U\Vert _{\text{op}},\Vert V\Vert _{\text{op}}\leqslant 1$
which makes the proof considerably shorter.
1.2 Stability theorems
As mentioned, the inverse theorem can be used to prove a so-called stability theorem for
$\unicode[STIX]{x1D700}$
-representations. Let us define what we mean by an
$\unicode[STIX]{x1D700}$
-representation, or, more generally, an
$\unicode[STIX]{x1D700}$
-homomorphism.
Definition 1.2. Let
$\unicode[STIX]{x1D700}>0$
, let
$G$
and
$H$
be groups and let
$d$
be a metric on
$H$
. An
$\unicode[STIX]{x1D700}$
-homomorphism is a map
$\unicode[STIX]{x1D711}:G\rightarrow H$
such that
$$\begin{eqnarray}d(\unicode[STIX]{x1D711}(gh),\unicode[STIX]{x1D711}(g)\unicode[STIX]{x1D711}(h))<\unicode[STIX]{x1D700}\quad \text{for all}~g,h\in G.\end{eqnarray}$$
In accordance with nomenclature for ordinary homomorphisms, we call
$\unicode[STIX]{x1D711}$
an
$\unicode[STIX]{x1D700}$
-representation if
$H$
consists of operators on a Hilbert space. A natural and very general question about
$\unicode[STIX]{x1D700}$
-homomorphisms, which for instance was stated by Ulam in [Reference Ulam13], can be formulated in the following way.
Question 1.3. Consider a class
$\mathscr{G}$
of groups together with a class
$\mathscr{H}$
of metric groups. Given
$\unicode[STIX]{x1D6FF}>0$
is there an
$\unicode[STIX]{x1D700}>0$
such that for all
$G\in \mathscr{G}$
and
$(H,d)\in \mathscr{H}$
and
$\unicode[STIX]{x1D700}$
-homomorphisms
$\unicode[STIX]{x1D711}:G\rightarrow H$
there is a genuine homomorphism
$\unicode[STIX]{x1D70B}:G\rightarrow H$
such that
$d(\unicode[STIX]{x1D711}(g),\unicode[STIX]{x1D70B}(g))<\unicode[STIX]{x1D6FF}$
for all
$g\in G$
?
If the question has a positive answer, we say that the class
$\mathscr{G}$
is stable with respect to
$\mathscr{H}$
. This question has been studied in various settings, and the answer highly depends much on which classes of groups and metrics one considers. The case where
$\mathscr{G}$
is the class of finite (or, more generally, compact) groups and
$\mathscr{H}$
consists of unitary groups equipped with the metric induced from the operator norm was treated in [Reference Grove, Karcher and Ruh5]. Later Kazhdan generalized this result to amenable groups.
Theorem 1.4 (Kazhdan [Reference Kazhdan10]).
Let
$0<\unicode[STIX]{x1D700}<\frac{1}{200}$
, let
$G$
be a countable, discrete, amenable group and let
${\mathcal{H}}$
be a Hilbert space. Let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{U}}({\mathcal{H}})$
be an
$\unicode[STIX]{x1D700}$
-representation with respect to the metric coming from the operator norm
$\Vert \cdot \Vert _{\text{op}}$
. Then there exists a representation
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}({\mathcal{H}})$
such that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D711}(g)-\unicode[STIX]{x1D70B}(g)\Vert _{\text{op}}<2\unicode[STIX]{x1D700},\quad g\in G.\end{eqnarray}$$
On the other hand, there are many examples in the literature of non-amenable groups that are not stable. A result by Rademacher [Reference Rademacher11] shows that
$\text{SL}(2,\mathbb{Z})$
is not stable with respect to the class of unitary groups equipped with any unitarily invariant norm. Since
$\text{SL}(2,\mathbb{Z})$
is a quotient of the non-abelian free group
$\mathbb{F}_{2}$
, the latter is also not stable with respect to any unitarily invariant norm. This was later proven in a different way by Johnson [Reference Johnson7]. More recently [Reference Burger, Ozawa and Thom1], it was proven that groups containing a non-abelian free group are not stable with respect to the operator norm. To the best of our knowledge, it is still open whether the stability of
$\unicode[STIX]{x1D700}$
-representations is a characterizing property for amenable groups. Although we will not discuss this question here, let us state it properly.
Question 1.5. Does there exist a non-amenable group
$G$
such that for all
$\unicode[STIX]{x1D6FF}>0$
there exists
$\unicode[STIX]{x1D700}>0$
such that if
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{U}}({\mathcal{H}})$
is an
$\unicode[STIX]{x1D700}$
-representation with respect to the operator norm there is a genuine representation
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}({\mathcal{H}})$
satisfying
$\Vert \unicode[STIX]{x1D711}(g)-\unicode[STIX]{x1D70B}(g)\Vert _{\text{op}}<\unicode[STIX]{x1D6FF}$
?
It is also worth mentioning that a version of Question 1.3 was considered in [Reference Johnson6] for Banach algebras and
$\unicode[STIX]{x1D700}$
-multiplicative functionals.
In this paper, our focus is
$\unicode[STIX]{x1D700}$
-representations of amenable groups with respect to some norm different from the operator norm. An important and motivating example is the case where
$({\mathcal{M}},\unicode[STIX]{x1D70F})$
is a von Neumann algebra equipped with a faithful, normal trace and
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{U}}({\mathcal{M}})$
is an
$\unicode[STIX]{x1D700}$
-representation with respect to the
$p$
-norm
$\Vert T\Vert _{p}=\unicode[STIX]{x1D70F}(|X|^{p})^{1/p}$
. Gowers and Hatami use their inverse theorem to prove a stability result for
$\unicode[STIX]{x1D700}$
-representations
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{U}}_{n}$
in the case where
$G$
is finite and
${\mathcal{U}}_{n}$
is equipped with the
$p$
-norm coming from the trace. For
$p=2$
, their result can be stated in the following way.
Theorem 1.6 (Gowers and Hatami [Reference Gowers and Hatami4]).
Let
$G$
be a finite group, let
$0<\unicode[STIX]{x1D700}<\frac{1}{16}$
and let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{U}}_{n}$
be an
$\unicode[STIX]{x1D700}$
-representation with respect to the normalized
$2$
-norm
$\Vert \cdot \Vert _{2}$
. Then there exists
$m\in [n,(1-4\unicode[STIX]{x1D700}^{2})^{-1}n]$
together with a representation
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}_{m}$
and an isometry
$U:\mathbb{C}^{n}\rightarrow \mathbb{C}^{m}$
such that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D711}(g)-U^{\ast }\unicode[STIX]{x1D70B}(g)U\Vert _{2}<42\unicode[STIX]{x1D700},\quad g\in G.\end{eqnarray}$$
In the same way as Gowers and Hatami, we deduce a stability result for amenable groups with respect to unitary groups of von Neumann algebras equipped with any unitarily invariant, ultraweakly lower semi-continuous semi-norm (see Definition 2.4). In particular, our result encompasses the results mentioned in this section. Note that the above theorem does not quite answer Question 1.3. If
$n<(1-4\unicode[STIX]{x1D700}^{2})/4\unicode[STIX]{x1D700}^{2}$
, it follows that
$m=n$
, but, as Gowers and Hatami point out, in order to get a result which holds uniformly for all
$n$
and
$\unicode[STIX]{x1D700}$
independently, one needs to allow the dimension of the approximating representation
$\unicode[STIX]{x1D70B}$
to differ from the dimension of
$\unicode[STIX]{x1D711}$
. This is, loosely speaking, because the normalized trace norm is insensitive to low-dimensional perturbations. More precisely, let
$G$
be a countable, discrete, amenable group with left-invariant mean
$\mathbb{E}$
and let
${\mathcal{M}}$
be a finite factor equipped with the 2-norm
$\Vert \cdot \Vert _{2}$
associated with the faithful, normal tracial state. Let
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}({\mathcal{M}})$
be a representation such that
$\unicode[STIX]{x1D70B}(G)$
generates
${\mathcal{M}}$
as a von Neumann algebra and let
$P\in {\mathcal{M}}$
be a projection with
$\unicode[STIX]{x1D70F}(P)=1-\unicode[STIX]{x1D700}$
for some
$0<\unicode[STIX]{x1D700}<\frac{1}{4}$
. Then the cutdown
$\unicode[STIX]{x1D711}:G\rightarrow P{\mathcal{M}}P$
given by
$\unicode[STIX]{x1D711}(g)=P\unicode[STIX]{x1D70B}(g)P,g\in G$
satisfies
$\Vert \unicode[STIX]{x1D711}(gh)-\unicode[STIX]{x1D711}(g)\unicode[STIX]{x1D711}(h)\Vert _{2}<\unicode[STIX]{x1D700}$
, but cannot be close to any representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}P)$
. Indeed, if such a
$\unicode[STIX]{x1D70C}$
existed with
$\Vert \unicode[STIX]{x1D711}(g)-\unicode[STIX]{x1D70C}(g)\Vert _{2}<\frac{1}{2}$
for all
$g\in G$
, then the operator
$\mathbb{E}_{x}\unicode[STIX]{x1D70C}(x)\unicode[STIX]{x1D70B}(x)^{\ast }\in P{\mathcal{M}}$
would be a non-zero intertwiner of the representations
$\unicode[STIX]{x1D70C}$
and
$\unicode[STIX]{x1D70B}$
, but since
$\unicode[STIX]{x1D70B}$
is a factor representation, this is only possible if
$P=1_{{\mathcal{M}}}$
which is not the case since
$\unicode[STIX]{x1D700}>0$
. To paraphrase Gowers and Hatami: the representation that approximates
$\unicode[STIX]{x1D711}$
is in some sense
$\unicode[STIX]{x1D70B}$
, but
$\unicode[STIX]{x1D70B}$
happens to be of the wrong dimension. Note that
$\unicode[STIX]{x1D711}$
does not take values in
${\mathcal{U}}(P{\mathcal{M}}P)$
, but this can be corrected for since
$\Vert \unicode[STIX]{x1D711}(g)^{\ast }\unicode[STIX]{x1D711}(g)-P\Vert _{2}<\unicode[STIX]{x1D700}$
. Also, the fact that
$\Vert \cdot \Vert _{2}$
is not normalized on
$P{\mathcal{M}}P$
can be corrected for by replacing
$\Vert \cdot \Vert _{2}$
with
$(1/\sqrt{1-\unicode[STIX]{x1D700}})\Vert \cdot \Vert _{2}$
. The bottom line is, that if
$G$
is an amenable group with either irreducible representations of arbitrarily high dimension or a representation whose image generates a (necessarily hyperfinite)
$\text{II}_{1}$
factor, then there are non-trivial
$\unicode[STIX]{x1D700}$
-representations of
$G$
, but the above result of [Reference Gowers and Hatami4] and our generalization, Theorem 5.2, show that the above construction is the only way of producing non-trivial
$\unicode[STIX]{x1D700}$
-representations of amenable groups.
2 Preliminaries
Throughout this article, all groups considered are assumed to be discrete and countable. Hilbert spaces are assumed separable and thus von Neumann algebras have separable preduals. For a Hilbert space
${\mathcal{H}}$
, we let
$\mathbb{B}({\mathcal{H}})$
denote the algebra of bounded linear operators on
${\mathcal{H}}$
. If
$\dim {\mathcal{H}}=n$
, we write
$\mathbb{M}_{n}=\mathbb{B}({\mathcal{H}})$
. Given a subset
${\mathcal{S}}\subseteq \mathbb{B}({\mathcal{H}})$
, we let
${\mathcal{S}}^{\prime }$
denote the commutant of
${\mathcal{S}}$
, that is, the set of operators
$T\in \mathbb{B}({\mathcal{H}})$
that commute with all
$S\in {\mathcal{S}}$
, i.e.
$ST=TS$
. For a projection
$P\in \mathbb{B}({\mathcal{H}})$
, we let
$P^{\bot }:=1_{{\mathcal{H}}}-P$
be the projection on the orthogonal complement of
$P{\mathcal{H}}$
. Whenever we consider an amenable group
$G$
, we will implicitly fix a symmetric bi-invariant mean
$\mathbb{E}\in \ell ^{\infty }(G)^{\ast }$
, and we shall write
$\mathbb{E}_{x}f(x):=\mathbb{E}(f)$
, i.e.,
$\mathbb{E}(1)=1$
,
$\mathbb{E}(f)\geqslant 0$
for
$f\geqslant 0$
and
Let
${\mathcal{M}}$
be a von Neumann algebra. We let
${\mathcal{U}}({\mathcal{M}})$
denote the unitary group of
${\mathcal{M}}$
. If
${\mathcal{M}}=\mathbb{M}_{n}$
, we write
${\mathcal{U}}_{n}={\mathcal{U}}({\mathcal{M}})$
. Since we will be dealing with different norms, we will use
$\Vert \cdot \Vert _{\text{op}}$
to denote the operator norm on
${\mathcal{M}}$
. We define
${\mathcal{M}}_{\infty }:={\mathcal{M}}\,\bar{\otimes }\,\mathbb{B}(\ell ^{2}(\mathbb{N}))$
and view
${\mathcal{M}}$
as a corner of
${\mathcal{M}}_{\infty }$
. More precisely, we implicitly fix a rank-1 projection
$E\in \mathbb{B}(\ell ^{2}(\mathbb{N}))$
and identify
$$\begin{eqnarray}{\mathcal{M}}\simeq (1_{{\mathcal{M}}}\otimes E){\mathcal{M}}_{\infty }(1_{{\mathcal{M}}}\otimes E),\end{eqnarray}$$
where
$1_{{\mathcal{M}}}$
is the unit of
${\mathcal{M}}$
. Consistent with this identification, we write
$1_{{\mathcal{M}}}$
instead of
$1_{{\mathcal{M}}}\otimes E$
. We denote the unit of
${\mathcal{M}}_{\infty }$
by
$1_{\infty }$
. Recall that
${\mathcal{M}}$
is the dual of the Banach space of its normal functionals, which we will denote
${\mathcal{M}}_{\ast }$
. The associated weak* topology on
${\mathcal{M}}$
is called the ultraweak or
$\unicode[STIX]{x1D70E}$
-weak topology. Given an amenable group
$G$
and a map
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
such that
$\sup _{x\in G}\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}<\infty$
, we can define the mean
$\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)\in {\mathcal{M}}\simeq ({\mathcal{M}}_{\ast })^{\ast }$
by the formula
$$\begin{eqnarray}f(\mathbb{E}_{x}\unicode[STIX]{x1D711}(x))=\mathbb{E}_{x}f(\unicode[STIX]{x1D711}(x)),\quad f\in {\mathcal{M}}_{\ast }.\end{eqnarray}$$
The characteristic function on an interval
$[a,b]\subseteq \mathbb{R}$
we denote by
$\unicode[STIX]{x1D712}_{[a,b]}$
. We also recall the following definition.
Definition 2.1. Let
$G$
be a group and let
${\mathcal{H}}$
be a Hilbert space. A map
$\unicode[STIX]{x1D711}:G\rightarrow \mathbb{B}({\mathcal{H}})$
is called positive definite if for all finite sets
$F\subseteq G$
, the matrix
$[\unicode[STIX]{x1D711}(xy^{-1})]_{x,y\in F}\in \mathbb{B}({\mathcal{H}}^{\oplus |F|})$
is positive as an operator, i.e., if
$$\begin{eqnarray}\mathop{\sum }_{x,y\in F}\langle \unicode[STIX]{x1D711}(xy^{-1})\unicode[STIX]{x1D709}_{y},\unicode[STIX]{x1D709}_{x}\rangle \geqslant 0,\end{eqnarray}$$
for all
$\unicode[STIX]{x1D709}_{x}\in {\mathcal{H}},x\in F$
.
We start out with a rather simple observation, which turns out to be central to this paper. The relevance of the following proposition to stability of
$\unicode[STIX]{x1D700}$
-representations was noted by Shtern in [Reference Shtern12].
Proposition 2.2. Let
$G$
be an amenable group and let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
be given such that
$\sup _{x\in G}\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}<\infty$
. Then the map
$\tilde{\unicode[STIX]{x1D711}}:G\rightarrow {\mathcal{M}}$
defined by
$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}(x)=\mathbb{E}_{y}\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast },\quad x\in G\end{eqnarray}$$
is positive definite.
Proof. Let
$F\subseteq G$
be finite and let
$\unicode[STIX]{x1D709}_{x}\in {\mathcal{H}},x\in F$
. Then
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{x,y\in F}\langle \tilde{\unicode[STIX]{x1D711}}(xy^{-1})\unicode[STIX]{x1D709}_{y},\unicode[STIX]{x1D709}_{x}\rangle & = & \displaystyle \mathop{\sum }_{x,y\in F}\mathbb{E}_{z}\langle \unicode[STIX]{x1D711}(xy^{-1}z)\unicode[STIX]{x1D711}(z)^{\ast }\unicode[STIX]{x1D709}_{y},\unicode[STIX]{x1D709}_{x}\rangle \nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{x,y\in F}\mathbb{E}_{z}\langle \unicode[STIX]{x1D711}(xz)\unicode[STIX]{x1D711}(yz)^{\ast }\unicode[STIX]{x1D709}_{y},\unicode[STIX]{x1D709}_{x}\rangle \nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{x,y\in F}\mathbb{E}_{z}\langle \unicode[STIX]{x1D711}(yz)^{\ast }\unicode[STIX]{x1D709}_{y},\unicode[STIX]{x1D711}(xz)^{\ast }\unicode[STIX]{x1D709}_{x}\rangle \nonumber\\ \displaystyle & = & \displaystyle \mathbb{E}_{z}\langle \mathop{\sum }_{y\in F}\unicode[STIX]{x1D711}(yz)^{\ast }\unicode[STIX]{x1D709}_{y},\mathop{\sum }_{x\in F}\unicode[STIX]{x1D711}(xz)^{\ast }\unicode[STIX]{x1D709}_{x}\rangle \geqslant 0,\nonumber\end{eqnarray}$$
since
$\langle \unicode[STIX]{x1D709},\unicode[STIX]{x1D709}\rangle \geqslant 0$
for all
$\unicode[STIX]{x1D709}\in {\mathcal{H}}$
and
$\mathbb{E}$
is positive.◻
A fundamental fact about positive definite maps is Stinespring’s dilation theorem. We will use a formulation which is essentially [Reference Kasparov9, Theorem 3]. For the reader’s convenience, we recall the proof.
Theorem 2.3 (Kasparov [Reference Kasparov9]).
Let
$G$
be a group and let
${\mathcal{M}}$
be a von Neumann algebra. For every positive definite map
$$\begin{eqnarray}\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}\simeq 1_{{\mathcal{M}}}{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}\end{eqnarray}$$
there exist
$U\in {\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
and a representation
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}({\mathcal{M}}_{\infty })$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(g)=U^{\ast }\unicode[STIX]{x1D70B}(g)U,\quad g\in G.\end{eqnarray}$$
In particular
$\Vert U\Vert _{\text{op}}^{2}=\Vert \unicode[STIX]{x1D711}(1)\Vert _{\text{op}}$
.
Proof. Let
${\mathcal{M}}\subseteq \mathbb{B}({\mathcal{H}})$
be a normal representation of
${\mathcal{M}}$
and consider the vector space
${\mathcal{A}}:=C_{\text{fin}}(G,{\mathcal{H}})$
of finitely supported maps
$G\rightarrow {\mathcal{H}}$
equipped with the sesquilinear form
$$\begin{eqnarray}\langle f,g\rangle _{\unicode[STIX]{x1D711}}=\mathop{\sum }_{x,y\in G}\langle \unicode[STIX]{x1D711}(y^{-1}x)f(x),g(y)\rangle _{{\mathcal{H}}},\end{eqnarray}$$
for
$f,g\in {\mathcal{A}}$
. By positive definiteness of
$\unicode[STIX]{x1D711}$
, this is a positive semidefinite sesquilinear form, so by separation and completion, we get a Hilbert space
$\tilde{{\mathcal{H}}}$
where
$G$
acts as unitaries by the formula
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{0}(g)[f]=[g.f],\quad f\in {\mathcal{A}},\end{eqnarray}$$
where
$g.f(x):=f(g^{-1}x)$
is the left translation action and
$[f]$
denotes the equivalence class of
$f$
. Furthermore, let
$U_{0}:{\mathcal{H}}\rightarrow \tilde{{\mathcal{H}}}$
be given by
$$\begin{eqnarray}U_{0}(\unicode[STIX]{x1D709})=[\unicode[STIX]{x1D6FF}_{e}\unicode[STIX]{x1D709}],\quad \unicode[STIX]{x1D709}\in {\mathcal{H}}.\end{eqnarray}$$
One sees straightforwardly that
$U_{0}^{\ast }([f])=\sum _{x\in G}\unicode[STIX]{x1D711}(x)f(x)$
for
$f\in {\mathcal{A}}$
and from this it is clear that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(g)=U_{0}^{\ast }\unicode[STIX]{x1D70B}_{0}(g)U_{0}.\end{eqnarray}$$
We also define an action of the commutant
${\mathcal{M}}^{\prime }\subseteq \mathbb{B}({\mathcal{H}})$
on
${\mathcal{A}}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D70C}(T)[f]=[Tf],\quad T\in {\mathcal{M}}^{\prime },f\in {\mathcal{A}}.\end{eqnarray}$$
In order to extend
$\unicode[STIX]{x1D70C}$
to a normal representation of
${\mathcal{M}}^{\prime }$
on
$\tilde{{\mathcal{H}}}$
, we have to check that
$\unicode[STIX]{x1D70C}$
is well defined and bounded. Note that since
$T$
commutes with
$\unicode[STIX]{x1D711}$
and
$[\unicode[STIX]{x1D711}(y^{-1}x)]_{x,y\in F}$
is positive for any
$F$
, the operator
$S:=\operatorname{diag}(T^{\ast })[\unicode[STIX]{x1D711}(y^{-1}x)]_{x,y\in F}\operatorname{diag}(T)$
is positive; in fact
$0\leqslant S\leqslant \Vert T\Vert _{\text{op}}^{2}[\unicode[STIX]{x1D711}(y^{-1}x)]_{x,y\in F}$
, so
$$\begin{eqnarray}\displaystyle \Vert [Tf]\Vert _{\tilde{{\mathcal{H}}}}^{2} & = & \displaystyle \mathop{\sum }_{x,y\in F}\langle T^{\ast }\unicode[STIX]{x1D711}(y^{-1}x)Tf(x),f(y)\rangle \nonumber\\ \displaystyle & = & \displaystyle \langle Sf,f\rangle \leqslant \Vert T\Vert _{\text{op}}^{2}\mathop{\sum }_{x,y\in F}\langle \unicode[STIX]{x1D711}(y^{-1}x)f(x),f(y)\rangle ,\nonumber\end{eqnarray}$$
where
$F=\operatorname{supp}f$
, so
$\unicode[STIX]{x1D70C}(T)$
extends to an operator on
$\tilde{{\mathcal{H}}}$
. It is now easy to see that
$\unicode[STIX]{x1D70C}$
is a normal representation of
${\mathcal{M}}^{\prime }$
. Thus, by the representation theory for von Neumann algebras, there is an isometry
$V:\tilde{{\mathcal{H}}}\rightarrow {\mathcal{H}}\otimes \ell ^{2}(\mathbb{N})$
such that
$\unicode[STIX]{x1D70C}(T)=V^{\ast }(T\otimes 1)V$
and
$VV^{\ast }\in ({\mathcal{M}}^{\prime }\otimes 1)^{\prime }$
. Clearly
$\unicode[STIX]{x1D70B}_{0}(G)\subseteq \unicode[STIX]{x1D70C}({\mathcal{M}}^{\prime })^{\prime }$
, so
$$\begin{eqnarray}\unicode[STIX]{x1D70B}(g):=V\unicode[STIX]{x1D70B}_{0}(g)V^{\ast }+1-VV^{\ast }\in ({\mathcal{M}}^{\prime }\otimes 1)^{\prime }={\mathcal{M}}_{\infty },\end{eqnarray}$$
is a unitary representation of
$G$
which, together with the map
$U=VU_{0}$
, has the desired properties.◻
We will consider a special class of semi-norms on the von Neumann algebra
${\mathcal{M}}$
.
Definition 2.4. Let
${\mathcal{M}}$
be a von Neumann algebra. A unitarily invariant semi-norm on
${\mathcal{M}}$
is a semi-norm
$\Vert \cdot \Vert$
on an (algebraic) ideal
${\mathcal{A}}\subseteq {\mathcal{M}}$
such that for all
$U,V\in {\mathcal{U}}({\mathcal{M}})$
and
$T\in {\mathcal{M}}$
it holds that
The semi-norm
$\Vert \cdot \Vert$
is called ultraweakly lower semi-continuous if the unit ball
$\{T\in {\mathcal{M}}\mid \Vert T\Vert \leqslant 1\}$
is closed in the ultraweak topology.
We consider such seminorms as defined on all of
${\mathcal{M}}$
by assigning the value
$\infty$
outside of the ideal
${\mathcal{A}}$
. An important example of unitarily invariant ultraweakly lower semi-continuous semi-norms occurs in the case where
${\mathcal{M}}$
is a semi-finite von Neumann algebra equipped with a normal trace
$\unicode[STIX]{x1D70F}$
. In this case, we define the
$p$
-semi-norms by
$$\begin{eqnarray}\Vert T\Vert _{p}:=\unicode[STIX]{x1D70F}(|T|^{p})^{1/p},\quad T\in {\mathcal{M}}.\end{eqnarray}$$
The tracial property implies unitary invariance and the fact that
$\unicode[STIX]{x1D70F}$
is normal implies that
$\Vert \cdot \Vert _{p}$
is ultraweakly lower semi-continuous. If
$\unicode[STIX]{x1D70F}$
is faithful, this is a norm. In the following, we will list the basic properties of ultraweakly semi-continuous, unitarily invariant semi-norms that we will use throughout this paper. We start by a basic proposition.
Proposition 2.5. Let
$R,S\in {\mathcal{M}}$
and assume
$0\leqslant R\leqslant S$
. Then there exists
$T\in {\mathcal{M}}$
with
$\Vert T\Vert _{\text{op}}\leqslant 1$
and
$R=S^{1/2}TS^{1/2}$
.
Proof. Let
$A_{n}=\unicode[STIX]{x1D712}_{[1/n,\infty )}(S^{1/2})S^{-1/2}$
. We note that
$$\begin{eqnarray}\displaystyle \Vert R^{1/2}A_{n}\Vert _{\text{op}}^{2} & = & \displaystyle \Vert A_{n}RA_{n}\Vert _{\text{op}}\leqslant \Vert A_{n}SA_{n}\Vert _{\text{op}}\nonumber\\ \displaystyle & = & \displaystyle \Vert \unicode[STIX]{x1D712}_{[1/n,\infty )}(S^{1/2})\Vert _{\text{op}}\leqslant 1,\nonumber\end{eqnarray}$$
so
$R^{1/2}A_{n}$
has an ultraweak limit point, say
$A\in {\mathcal{M}}$
with
$\Vert A\Vert _{\text{op}}\leqslant 1$
. We also note that the increasing sequence
$\unicode[STIX]{x1D712}_{[1/n,\infty )}(S^{1/2})$
converges even strongly to
$\unicode[STIX]{x1D712}_{(0,\infty )}(S^{1/2})$
. By a similar calculation as above, we have that
$$\begin{eqnarray}\Vert R^{1/2}\unicode[STIX]{x1D712}_{\{0\}}(S^{1/2})\Vert _{\text{op}}\leqslant \Vert S^{1/2}\unicode[STIX]{x1D712}_{\{0\}}(S^{1/2})\Vert _{\text{op}}=0,\end{eqnarray}$$
so for some ultraweakly convergent subnet, we have that
$$\begin{eqnarray}AS^{1/2}=\lim _{\unicode[STIX]{x1D6FC}}R^{1/2}\unicode[STIX]{x1D712}_{[1/n_{\unicode[STIX]{x1D6FC}},\infty )}(S^{1/2})=R^{1/2}\unicode[STIX]{x1D712}_{(0,\infty )}(S^{1/2})=R^{1/2}.\end{eqnarray}$$
Letting
$T=A^{\ast }A$
, we reach the desired conclusion.◻
Proposition 2.6. Let
${\mathcal{M}}$
be a von Neumann algebra and let
$\Vert \cdot \Vert$
be a unitarily invariant semi-norm on
${\mathcal{M}}$
. Then, for all
$R,S,T\in {\mathcal{M}}$
, we have that
if
$0\leqslant R\leqslant S,$
then
Proof. We begin with the proof of (♠). First assume
$\Vert R\Vert _{\text{op}},\Vert S\Vert _{\text{op}}<1$
. By (a strengthening of) the Russo–Dye theorem (see [Reference Kadison and Pedersen8]),
$R$
and
$S$
are convex combinations of unitaries in
${\mathcal{M}}$
, that is,
$R=\sum _{i=1}^{n}\unicode[STIX]{x1D706}_{i}U_{i}$
and
$S=\sum _{i=1}^{m}\unicode[STIX]{x1D707}_{i}V_{i}$
with
$\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D707}_{i}\in [0,1],U_{i},V_{i}\in {\mathcal{U}}({\mathcal{M}})$
and
$\sum _{i=1}^{n}\unicode[STIX]{x1D706}_{i}=\sum _{i=1}^{m}\unicode[STIX]{x1D707}_{i}=1$
. Thus
$$\begin{eqnarray}\Vert RTS\Vert \leqslant \mathop{\sum }_{i=1}^{n}\mathop{\sum }_{j=1}^{m}\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D707}_{j}\Vert U_{i}TV_{j}\Vert =\mathop{\sum }_{i=1}^{n}\unicode[STIX]{x1D706}_{i}\mathop{\sum }_{j=1}^{m}\unicode[STIX]{x1D707}_{j}\Vert T\Vert =\Vert T\Vert .\end{eqnarray}$$
Now let
$R$
and
$S$
be arbitrary and let
$\unicode[STIX]{x1D700}>0$
. Then
$R^{\prime }:=(\Vert R\Vert _{\text{op}}+\unicode[STIX]{x1D700})^{-1}R$
and
$S^{\prime }:=(\Vert S\Vert _{\text{op}}+\unicode[STIX]{x1D700})^{-1}S$
have an operator norm strictly less than 1, so we get that
$$\begin{eqnarray}\Vert RTS\Vert =(\Vert R\Vert _{\text{op}}+\unicode[STIX]{x1D700})(\Vert S\Vert _{\text{op}}+\unicode[STIX]{x1D700})\Vert R^{\prime }TS^{\prime }\Vert \leqslant (\Vert R\Vert _{\text{op}}+\unicode[STIX]{x1D700})(\Vert S\Vert _{\text{op}}+\unicode[STIX]{x1D700})\Vert T\Vert .\end{eqnarray}$$
Since this holds for all
$\unicode[STIX]{x1D700}>0$
, the result follows.
Now, for (♢), by the polar decomposition, we have that
$T=U|T|$
and
$|T|=U^{\ast }T$
for a partial isometry
$U\in {\mathcal{M}}$
. Thus, according to (♠), we have that
$$\begin{eqnarray}\Vert T\Vert =\Vert U|T|\Vert \leqslant \Vert |T|\Vert =\Vert U^{\ast }T\Vert \leqslant \Vert T\Vert ,\end{eqnarray}$$
so
$\Vert T\Vert =\Vert |T|\Vert$
. By taking adjoints on both sides of the equations, we also get that
$\Vert T^{\ast }\Vert =\Vert |T|\Vert$
.
Proceeding with (♣), using the polar decomposition as above, we get
$$\begin{eqnarray}\displaystyle \Vert T^{\ast }T\Vert & = & \displaystyle \Vert |T|^{2}\Vert =\Vert |T|\,|T|^{\ast }\Vert =\Vert U^{\ast }TT^{\ast }U\Vert \leqslant \Vert TT^{\ast }\Vert \nonumber\\ \displaystyle & = & \displaystyle \Vert U|T|\,|T|U^{\ast }\Vert \leqslant \Vert T^{\ast }T\Vert .\nonumber\end{eqnarray}$$
Finally, we prove (♡). Let
$R\leqslant S$
. By Proposition 2.5 we determine
$T\in {\mathcal{M}},\Vert T\Vert _{\text{op}}\leqslant 1$
such that
$R=S^{1/2}TS^{1/2}$
. Thus it follows from (♣) and (♠) that
$$\begin{eqnarray}\Vert R\Vert =\Vert S^{1/2}TS^{1/2}\Vert =\Vert T^{1/2}ST^{1/2}\Vert \leqslant \Vert S\Vert .\Box\end{eqnarray}$$
Some consequences, which we will use throughout the proofs, are the following.
Corollary 2.7. Let
${\mathcal{M}}$
be a von Neumann algebra and let
$\Vert \cdot \Vert :{\mathcal{M}}\rightarrow \mathbb{R}\,\cup \{\infty \}$
be a unitarily invariant semi-norm on
${\mathcal{M}}$
. Let
$S,T,P\in {\mathcal{M}}$
with
$\Vert S\Vert _{\text{op}},\Vert T\Vert _{\text{op}}\leqslant 1$
and
$P\geqslant S^{\ast }S,T^{\ast }T$
. Then
Proof. Since
$P-S^{\ast }S,P-T^{\ast }T$
and
$(S-T)^{\ast }(S-T)$
are positive, we get that
$$\begin{eqnarray}\displaystyle 0\leqslant P-S^{\ast }S & {\leqslant} & \displaystyle P-S^{\ast }S+P-T^{\ast }T+(S-T)^{\ast }(S-T)\nonumber\\ \displaystyle & = & \displaystyle P-S^{\ast }S+P-T^{\ast }T+S^{\ast }S+T^{\ast }T-S^{\ast }T-T^{\ast }S\nonumber\\ \displaystyle & = & \displaystyle P-S^{\ast }T+P-T^{\ast }S.\nonumber\end{eqnarray}$$
Similarly, we have
$$\begin{eqnarray}0\leqslant P-T^{\ast }T\leqslant P-S^{\ast }T+P-T^{\ast }S,\end{eqnarray}$$
so by (♡) and (♢), using that
$(P-T^{\ast }S)^{\ast }=P-S^{\ast }T$
(since
$P$
is positive), the result follows.◻
Corollary 2.8. Let
${\mathcal{M}}$
be a von Neumann algebra, let
$\Vert \cdot \Vert$
be a unitarily invariant semi-norm on
${\mathcal{M}}$
and let
$S,T\in {\mathcal{M}}$
. Then
Proof. Using polar decomposition again, we can find an operator
$U$
with
$\Vert U\Vert _{\text{op}}\leqslant 1$
so that
$U^{\ast }S^{\ast }T\geqslant 0$
and
$\Vert S^{\ast }T\Vert =\Vert U^{\ast }S^{\ast }T\Vert$
. Note that this implies that
$U^{\ast }S^{\ast }T=T^{\ast }SU$
. Thus
$$\begin{eqnarray}\displaystyle 0 & {\leqslant} & \displaystyle (SU-T)^{\ast }(SU-T)=U^{\ast }S^{\ast }SU+T^{\ast }T-U^{\ast }S^{\ast }T-T^{\ast }SU\nonumber\\ \displaystyle & = & \displaystyle U^{\ast }S^{\ast }SU+T^{\ast }T-2U^{\ast }S^{\ast }T.\nonumber\end{eqnarray}$$
Thus
$U^{\ast }S^{\ast }T\leqslant \frac{1}{2}(U^{\ast }S^{\ast }SU+T^{\ast }T)$
, so the inequality follows from (♡) and (♠). The last equality is (♣). ◻
Lemma 2.9. Let
${\mathcal{M}}$
be a von Neumann algebra, let
$P,Q\in {\mathcal{M}}$
be projections and let
$S\in P{\mathcal{M}}Q$
with
$\Vert S\Vert _{\text{op}}\leqslant 1$
. Then there are partial isometries
$V_{1},V_{2}\in P{\mathcal{M}}Q$
such that
$$\begin{eqnarray}S={\textstyle \frac{1}{2}}(V_{1}+V_{2}).\end{eqnarray}$$
Proof. We write
$S=U|S|$
where
$U\in P{\mathcal{M}}Q$
is a partial isometry. Now let
$V_{\pm }:=|S|\pm i\sqrt{Q-|S|^{2}}$
. We note that
$V_{\pm }$
are unitaries in
$Q{\mathcal{M}}Q$
, and
$$\begin{eqnarray}S={\textstyle \frac{1}{2}}(UV_{+}+UV_{-}).\end{eqnarray}$$
Thus
$V_{1}=UV_{+}$
and
$V_{2}=UV_{-}$
are partial isometries with the desired properties.◻
The only place where we use ultraweak lower semi-continuity is in the following lemma. In fact, this extra assumption on the semi-norm
$\Vert \cdot \Vert$
is only necessary if the group
$G$
is infinite.
Lemma 2.10. Let
$G$
be an amenable group, let
${\mathcal{M}}$
be a von Neumann algebra, let
$\Vert \cdot \Vert$
be an ultraweakly lower semi-continuous semi-norm on
${\mathcal{M}}$
and let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
such that
$\sup _{x\in G}\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}<\infty$
. Then
Proof. For
$\unicode[STIX]{x1D707}\in \ell ^{1}(G)\subseteq \ell ^{\infty }(G)^{\ast }$
with
$\unicode[STIX]{x1D707}(x)\geqslant 0,x\in G$
, we can define
$\unicode[STIX]{x1D707}(\unicode[STIX]{x1D711}):=\sum _{x\in G}\unicode[STIX]{x1D707}(x)\unicode[STIX]{x1D711}(x)$
. We note that this sum converges in operator norm and hence also in the ultraweak topology. Furthermore, for finite
$F\subseteq G$
, by the triangle inequality, we have that
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{x\in F}\unicode[STIX]{x1D707}(x)\unicode[STIX]{x1D711}(x)\biggr\|\leqslant \mathop{\sum }_{x\in F}\unicode[STIX]{x1D707}(x)\Vert \unicode[STIX]{x1D711}(x)\Vert \leqslant \unicode[STIX]{x1D707}_{x}(\Vert \unicode[STIX]{x1D711}(x)\Vert ).\end{eqnarray}$$
By lower semi-continuity, we get that
$\Vert \unicode[STIX]{x1D707}(\unicode[STIX]{x1D711})\Vert \leqslant \unicode[STIX]{x1D707}_{x}(\Vert \unicode[STIX]{x1D711}(x)\Vert )$
.
Now let
$\unicode[STIX]{x1D707}_{i}\in \ell ^{1}(G)$
be a net of positive functions with
$\Vert \unicode[STIX]{x1D707}_{i}\Vert _{1}=1$
converging to
$\mathbb{E}$
in the weak* topology on
$\ell ^{\infty }(G)^{\ast }$
. For all
$f\in {\mathcal{M}}_{\ast }$
we have that
$$\begin{eqnarray}f(\unicode[STIX]{x1D707}_{i}(\unicode[STIX]{x1D711}))=f\biggl(\mathop{\sum }_{x\in G}\unicode[STIX]{x1D707}_{i}(x)\unicode[STIX]{x1D711}(x)\biggr)=\mathop{\sum }_{x\in G}\unicode[STIX]{x1D707}_{i}(x)f(\unicode[STIX]{x1D711}(x))\rightarrow \mathbb{E}_{x}f(\unicode[STIX]{x1D711}(x)),\end{eqnarray}$$
so
$\unicode[STIX]{x1D707}_{i}(\unicode[STIX]{x1D711})$
converges to
$\mathbb{E}(\unicode[STIX]{x1D711})$
in the ultraweak topology, whence we conclude that
$$\begin{eqnarray}\Vert \mathbb{E}_{x}\unicode[STIX]{x1D711}(x)\Vert \leqslant \liminf _{i}\Vert \unicode[STIX]{x1D707}_{i}(\unicode[STIX]{x1D711})\Vert \leqslant \liminf _{i}(\unicode[STIX]{x1D707}_{i})_{x}(\Vert \unicode[STIX]{x1D711}(x)\Vert )=\mathbb{E}_{x}\Vert \unicode[STIX]{x1D711}(x)\Vert .\Box\end{eqnarray}$$
This concludes the preliminary section and we turn our attention to the main theorem of this article.
3 The main theorem
We now state and prove our main theorem.
Theorem 3.1. Let
$\unicode[STIX]{x1D700}>0$
, let
$G$
be an amenable group,
${\mathcal{M}}$
a von Neumann algebra and let
$\Vert \cdot \Vert$
be a unitarily invariant, ultraweakly lower semi-continuous semi-norm
$\Vert \cdot \Vert$
on
${\mathcal{M}}_{\infty }$
. Let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
be any map and assume that
$\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}\leqslant 1$
for all
$x\in G$
and that
$$\begin{eqnarray}\displaystyle & \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }\Vert <\unicode[STIX]{x1D700}, & \displaystyle \nonumber\\ \displaystyle & \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(xz)^{\ast }\Vert <\unicode[STIX]{x1D700}. & \displaystyle \nonumber\end{eqnarray}$$
Then there exists a projection
$P\in {\mathcal{M}}_{\infty }$
, partial isometries
$U,V\in P({\mathcal{M}}_{\infty })1_{{\mathcal{M}}}$
and a representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
such that
$$\begin{eqnarray}\mathbb{E}_{x}\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U\Vert <44\unicode[STIX]{x1D700},\end{eqnarray}$$
and
$$\begin{eqnarray}\Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert <20\unicode[STIX]{x1D700},\qquad \Vert P-UU^{\ast }\Vert <15\unicode[STIX]{x1D700},\qquad \Vert P-VV^{\ast }\Vert <85\unicode[STIX]{x1D700}.\end{eqnarray}$$
Before commencing the proof, a few comments are in order. The two inequalities in the assumptions replace the assumption
$\Vert \unicode[STIX]{x1D711}\Vert _{U^{2}}\geqslant cn$
in Theorem 1.1 and are more or less a direct adaptation of this latter inequality to our situation. Note that for amenable groups, unlike for finite groups, the equality
$\mathbb{E}_{x}\mathbb{E}_{y}f(x,y)=\mathbb{E}_{y}\mathbb{E}_{x}f(x,y)$
does not hold in general. For instance, in the case
$G=\mathbb{Z}$
the function
$$\begin{eqnarray}f(x,y)=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if}~|x|\leqslant |y|,\\ 0\quad & \text{otherwise},\end{array}\right.\end{eqnarray}$$
satisfies
$\mathbb{E}_{x}\mathbb{E}_{y}f(x,y)=1$
and
$\mathbb{E}_{y}\mathbb{E}_{x}f(x,y)=0$
for all invariant means
$\mathbb{E}$
on
$\ell ^{\infty }(\mathbb{Z})$
! Therefore, the inequalities in the assumptions in the theorem are in general different. One reason for working with an abstract semi-norm instead of, say, the trace
$p$
-norm, is that the proof becomes conceptually simple; the long computations on the next couple of pages are nothing but repeated applications of the basic facts about ultraweakly lower semi-continuous, unitarily invariant semi-norms that we collected and proved in §2. In order to underline this point, and hopefully to the convenience of the reader, we indicate the usage of the (in)equalities (†), (°), (♠), (♢), (♣), (♡), (♯), (♭) and (‡) to the right of the (in)equality, where it is used. Usage of the triangle inequality will be indicated by (
$\triangle$
). The lines without any indications should be self-explanatory from the definitions or remarks during the proof.
Proof of Theorem 3.1.
Define
$\tilde{\unicode[STIX]{x1D711}}:G\rightarrow {\mathcal{M}}$
by
$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}(x):=\mathbb{E}_{y}\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast },x\in G,\end{eqnarray}$$
which is positive definite by Proposition 2.2. Thus, by the Stinespring dilation theorem (Proposition 2.3), there exists a unitary representation
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{U}}({\mathcal{M}}_{\infty })$
together with
$U\in {\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
with
$\Vert U\Vert _{\text{op}}\leqslant 1$
such that
$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}(g)=U^{\ast }\unicode[STIX]{x1D70B}(g)U,\quad g\in G.\end{eqnarray}$$
Define
$V:=\mathbb{E}_{x}\unicode[STIX]{x1D70B}(x)^{\ast }U\unicode[STIX]{x1D711}(x)\in {\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
and
$A:=\mathbb{E}_{x}\unicode[STIX]{x1D70B}(x)UU^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }=\mathbb{E}_{x}\unicode[STIX]{x1D70B}(x)^{\ast }UU^{\ast }\unicode[STIX]{x1D70B}(x)\in \unicode[STIX]{x1D70B}(G)^{\prime }$
. We see that
So
$U$
and
$V$
satisfy the desired inequality, but they need not be partial isometries and we have a priori no control over their range projections. In order to correct for that, we observe that
$0\leqslant A\leqslant 1_{\infty }$
and
so, since
$\unicode[STIX]{x1D70B}(x)$
and
$(1-A)^{1/2}$
commute,
Now let
$P:=\unicode[STIX]{x1D712}_{[1/2,1]}(A)$
which is a projection that commutes with
$\unicode[STIX]{x1D70B}$
, so we can consider the representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
given by
$\unicode[STIX]{x1D70C}(g):=P\unicode[STIX]{x1D70B}(g)P$
. Since
$|\unicode[STIX]{x1D712}_{[1/2,1]}(t)-t|\leqslant 2(t-t^{2})$
for all
$t\in [0,1]$
, we have that
and hence for all
$x\in G$
, we have
Since
$\unicode[STIX]{x1D70B}(x)=\unicode[STIX]{x1D70C}(x)+P^{\bot }\unicode[STIX]{x1D70B}(x)P^{\bot },x\in G$
, we get that
Replace
$U$
and
$V$
by
$U_{0}=PU,V_{0}=PV\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
. We still need to turn
$U_{0}$
and
$V_{0}$
into partial isometries. We write the polar decomposition of
$U_{0}=S|U_{0}|$
and define
$U_{1}:=S\unicode[STIX]{x1D712}_{[1/2,1]}(|U_{0}|)$
. This is a partial isometry, and we calculate
so
and we conclude
We proceed by estimating:
so
Similarly, we replace
$V_{0}=|V_{0}^{\ast }|T$
with
$V_{1}:=\unicode[STIX]{x1D712}_{[1/2,1]}(|V_{0}^{\ast }|)T$
and get a partial isometry. In order to get the remaining estimates, we first note that
which entails that
so
Finally, we conclude that
Now the proof is complete by renaming
$U_{1}$
and
$V_{1}$
to
$U$
and
$V$
.◻
Remark 3.2. We remark that it follows from the estimates that
that is, after inflating
$\unicode[STIX]{x1D711}$
a bit, it is approximated on average by
$\unicode[STIX]{x1D70C}$
.
Remark 3.3. We note for later use that if one does not require
$V$
to be a partial isometry, then we can end the above proof earlier and get the better estimate
$$\begin{eqnarray}\mathbb{E}_{x}\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U\Vert <10\unicode[STIX]{x1D700}.\end{eqnarray}$$
4 The inverse theorem
In this section we explain how Theorem 1.1 follows from our Theorem 3.1. More precisely, in the case where
$c=1-\unicode[STIX]{x1D700}$
is sufficiently close to 1, the result is a direct consequence of Theorem 3.1 in the case where
$\Vert x\Vert =\operatorname{Tr}(x^{\ast }x)^{1/2}$
(we will, though, achieve a coarser lower bound on the trace), but for smaller
$c$
, although the techniques will be similar, we will need to accommodate our proof a bit. Note that given a map
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
, expressions of the form
$$\begin{eqnarray}\mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast })\end{eqnarray}$$
or
$$\begin{eqnarray}\mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(xz)^{\ast })\end{eqnarray}$$
define non-negative real numbers. For the first, note that we have
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast })=\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }\unicode[STIX]{x1D711}(x))\end{eqnarray}$$
by the trace property. Now, it is easy to see from the proof of Proposition 2.2 that
$K:G\times G\rightarrow {\mathcal{M}}$
given by
$$\begin{eqnarray}K(x,y):=\mathbb{E}_{z}(\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }\unicode[STIX]{x1D711}(x))\end{eqnarray}$$
is a positive definite operator-valued kernel. Thus, by complete positivity of
$\unicode[STIX]{x1D70F}:{\mathcal{M}}\rightarrow \mathbb{C}$
, the composition
$\unicode[STIX]{x1D70F}\circ K:G\times G\rightarrow \mathbb{C}$
is a positive definite kernel as well. Using the representation theorem for positive definite kernels, there exists a Hilbert space valued function
$\unicode[STIX]{x1D6FC}:G\rightarrow {\mathcal{H}}$
such that for all
$x,y\in G$
, we have
$(\unicode[STIX]{x1D70F}\circ K)(x,y)=\langle \unicode[STIX]{x1D6FC}(x),\unicode[STIX]{x1D6FC}(y)\rangle$
and we can conclude that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast })\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }\unicode[STIX]{x1D711}(x))\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbb{E}_{x}\mathbb{E}_{y}(\unicode[STIX]{x1D70F}\circ K)(x,y)=\mathbb{E}_{x}\mathbb{E}_{y}\langle \unicode[STIX]{x1D6FC}(x),\unicode[STIX]{x1D6FC}(y)\rangle =\langle \mathbb{E}_{x}\unicode[STIX]{x1D6FC}(x),\mathbb{E}_{y}\unicode[STIX]{x1D6FC}(y)\rangle \geqslant 0.\nonumber\end{eqnarray}$$
For the second expression, we compute more easily
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(xz)^{\ast })\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\mathbb{E}_{y}(\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast })\mathbb{E}_{z}(\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(xz)^{\ast }))\geqslant 0.\nonumber\end{eqnarray}$$
We will now study what can be said about
$\unicode[STIX]{x1D711}$
in the presence of lower bounds on those quantities. Theorem 4.1 will cover what is called the 99% regime and Theorem 4.2 covers what is called the 1% regime.
Theorem 4.1. Let
$\unicode[STIX]{x1D700}>0$
, let
$G$
be an amenable group, let
${\mathcal{M}}$
be a von Neumann algebra and fix a normal trace
$\unicode[STIX]{x1D70F}$
on
${\mathcal{M}}_{\infty }$
with
$\unicode[STIX]{x1D70F}(1_{{\mathcal{M}}})=1$
. Let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
be a map with
$\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}\leqslant 1$
for all
$x\in G$
. Assume that
$$\begin{eqnarray}\displaystyle & \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast })\geqslant 1-\unicode[STIX]{x1D700}, & \displaystyle \nonumber\\ \displaystyle & \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(xz)^{\ast })\geqslant 1-\unicode[STIX]{x1D700}. & \displaystyle \nonumber\end{eqnarray}$$
Then there exist a projection
$P\in {\mathcal{M}}_{\infty }$
, partial isometries
$U,V\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
and a representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
such that
$$\begin{eqnarray}\displaystyle & \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)\geqslant 1-63\unicode[STIX]{x1D700}^{1/2}, & \displaystyle \nonumber\\ \displaystyle & \Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert _{2}<29\unicode[STIX]{x1D700}^{1/2},\qquad \Vert P-UU^{\ast }\Vert _{2}<22\unicode[STIX]{x1D700}^{1/2}, & \displaystyle \nonumber\\ \displaystyle & \Vert P-VV^{\ast }\Vert _{2}<121\unicode[STIX]{x1D700}^{1/2}. & \displaystyle \nonumber\end{eqnarray}$$
Proof. We consider the semi-norm
$\Vert T\Vert _{2}:=\unicode[STIX]{x1D70F}(T^{\ast }T)^{1/2},T\in {\mathcal{M}}_{\infty }$
. This is an ultraweakly lower semi-continuous unitarily invariant semi-norm on
${\mathcal{M}}_{\infty }$
. It follows that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }\Vert _{2}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant (\mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}(\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }\Vert _{2}^{2}))^{1/2}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}((2-2\operatorname{Re}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast }))^{1/2})\leqslant (2\unicode[STIX]{x1D700})^{1/2}.\nonumber\end{eqnarray}$$
Similarly
$$\begin{eqnarray}\mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\Vert 1_{{\mathcal{M}}}-\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(z)\unicode[STIX]{x1D711}(xz)^{\ast }\Vert _{2}\leqslant (2\unicode[STIX]{x1D700})^{1/2}.\end{eqnarray}$$
Hence, from Theorem 3.1 there exist
$P\in {\mathcal{M}}_{\infty }$
and partial isometries
$U,V\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
together with
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
such that
and furthermore
$\Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert _{2}<20(2\unicode[STIX]{x1D700})^{1/2}$
,
$\Vert P-UU^{\ast }\Vert _{2}<15(2\unicode[STIX]{x1D700})^{1/2}$
and
$\Vert P-VV^{\ast }\Vert _{2}<85(2\unicode[STIX]{x1D700})^{1/2}$
. It now follows from the Cauchy–Schwarz inequality that
$$\begin{eqnarray}\displaystyle |1-|\unicode[STIX]{x1D70F}(\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)|| & {\leqslant} & \displaystyle |1-\unicode[STIX]{x1D70F}(\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)|\nonumber\\ \displaystyle & = & \displaystyle |\unicode[STIX]{x1D70F}(1_{{\mathcal{M}}}\cdot (1_{{\mathcal{M}}}-\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U))|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \Vert 1_{{\mathcal{M}}}\Vert _{2}\Vert 1_{{\mathcal{M}}}-\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U\Vert _{2}\nonumber\\ \displaystyle & {<} & \displaystyle 44(2\unicode[STIX]{x1D700})^{1/2}.\nonumber\end{eqnarray}$$
By multiplying
$U$
with a complex number of modulus 1, we can assume that
$\unicode[STIX]{x1D70F}(\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)\geqslant 0$
so we get the desired
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)>1-44(2\unicode[STIX]{x1D700})^{1/2}>1-63\unicode[STIX]{x1D700}^{1/2}.\Box\end{eqnarray}$$
Now we turn our attention to the inverse theorem for general
$c\in (0,1]$
. This theorem is specific for the trace and we cannot use Theorem 3.1, but the proof is similar and even a bit shorter. Note that in this case we need fewer assumptions; we only assume one inequality.
Theorem 4.2. Let
$c\in (0,1]$
, let
${\mathcal{M}}$
be a von Neumann algebra and let
$\unicode[STIX]{x1D70F}$
be a normal trace on
${\mathcal{M}}_{\infty }$
such that
$\unicode[STIX]{x1D70F}(1_{{\mathcal{M}}})=1$
. Let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{M}}$
be a map with
$\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}\leqslant 1$
for all
$x\in G$
. Assume that
$$\begin{eqnarray}\mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast })\geqslant c.\end{eqnarray}$$
Then there exists a projection
$P\in {\mathcal{M}}_{\infty }$
, partial isometries
$U,V\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
and a representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
such that
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{c}{2}\leqslant \unicode[STIX]{x1D70F}(UU^{\ast })\leqslant \unicode[STIX]{x1D70F}(P)\leqslant \frac{2}{c},\qquad \frac{c}{2}\leqslant \unicode[STIX]{x1D70F}(VV^{\ast })\leqslant \unicode[STIX]{x1D70F}(P)\leqslant \frac{2}{c}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)\geqslant \frac{c}{2}. & \displaystyle \nonumber\end{eqnarray}$$
Proof. The proof begins as the proof of the main theorem. Define the positive definite
$\tilde{\unicode[STIX]{x1D711}}(x):=\mathbb{E}_{y}\unicode[STIX]{x1D711}(xy)\unicode[STIX]{x1D711}(y)^{\ast }$
and use Proposition 2.3 to determine
$U\in {\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
and a representation
$\unicode[STIX]{x1D70B}:G\rightarrow {\mathcal{M}}_{\infty }$
such that
$\tilde{\unicode[STIX]{x1D711}}(x)=U^{\ast }\unicode[STIX]{x1D70B}(x)U$
. Let
$V:=\mathbb{E}_{x}\unicode[STIX]{x1D70B}(x)^{\ast }U\unicode[STIX]{x1D711}(x)\in {\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
. Also, let
$A:=\mathbb{E}_{x}\unicode[STIX]{x1D70B}(x)VV^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }$
, which is a positive element of
${\mathcal{M}}_{\infty }$
, and let
$P:=\unicode[STIX]{x1D712}_{[c/2,1]}(A^{1/2})$
. Since
$A$
clearly commutes with
$\unicode[STIX]{x1D70B}$
, so does
$P$
, and therefore
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
given by
$\unicode[STIX]{x1D70C}(g)=P\unicode[STIX]{x1D70B}(g)P$
for
$g\in G$
is a representation. We have that
$\Vert A^{1/2}\Vert _{\text{op}}^{2}=\Vert A\Vert _{\text{op}}\leqslant \Vert V\Vert _{\text{op}}^{2}\leqslant \Vert U\Vert _{\text{op}}^{2}=\Vert \tilde{\unicode[STIX]{x1D711}}(1)\Vert \leqslant 1$
and by Kadison’s inequality
$\unicode[STIX]{x1D70F}(A^{1/2})^{2}\leqslant \unicode[STIX]{x1D70F}(A)=\unicode[STIX]{x1D70F}(VV^{\ast })=\unicode[STIX]{x1D70F}(V^{\ast }V)\leqslant \unicode[STIX]{x1D70F}(1_{{\mathcal{M}}})=1$
. Thus, by the inequality
$\unicode[STIX]{x1D712}_{[c/2,1]}(t)\leqslant (2/c)t$
for
$t\in [0,1]$
, we get
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(P)\leqslant \frac{2}{c}\unicode[STIX]{x1D70F}(A^{1/2})\leqslant \frac{2}{c},\end{eqnarray}$$
and by the inequality
$t^{2}(1-\unicode[STIX]{x1D712}_{[c/2,1]}(t))\leqslant (c/2)t,$
for
$t\in [0,1]$
, we get
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(AP^{\bot })\leqslant \frac{c}{2}\unicode[STIX]{x1D70F}(A^{1/2})\leqslant \frac{c}{2},\end{eqnarray}$$
so, remembering that
$\unicode[STIX]{x1D70F}$
is normal and thus commutes with the mean, we conclude
$$\begin{eqnarray}\displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }P^{\bot }\unicode[STIX]{x1D70B}(x)^{\ast }U) & = & \displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D70B}(x)^{\ast }U\unicode[STIX]{x1D711}(x)V^{\ast }P^{\bot })\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70F}(VV^{\ast }P^{\bot })=\mathbb{E}_{y}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D70B}(y)VV^{\ast }P^{\bot }\unicode[STIX]{x1D70B}(y)^{\ast })\nonumber\\ \displaystyle & = & \displaystyle \mathbb{E}_{y}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D70B}(y)VV^{\ast }\unicode[STIX]{x1D70B}(y)^{\ast }P^{\bot })=\unicode[STIX]{x1D70F}(AP^{\bot })\leqslant \frac{c}{2}.\nonumber\end{eqnarray}$$
Furthermore, we have that
$$\begin{eqnarray}\displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }U) & = & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }U^{\ast }\unicode[STIX]{x1D70B}(yx^{-1})U)\nonumber\\ \displaystyle & = & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\tilde{\unicode[STIX]{x1D711}}(yx^{-1}))\nonumber\\ \displaystyle & = & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yx^{-1}z)\unicode[STIX]{x1D711}(z)^{\ast })\nonumber\\ \displaystyle & = & \displaystyle \mathbb{E}_{x}\mathbb{E}_{y}\mathbb{E}_{z}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)\unicode[STIX]{x1D711}(y)^{\ast }\unicode[STIX]{x1D711}(yz)\unicode[STIX]{x1D711}(xz)^{\ast })\geqslant c,\nonumber\end{eqnarray}$$
so we conclude
$$\begin{eqnarray}\displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U) & = & \displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }PU)\nonumber\\ \displaystyle & = & \displaystyle \mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }U)-\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70B}(x)^{\ast }P^{\bot }U)\nonumber\\ \displaystyle & {\geqslant} & \displaystyle c-\frac{c}{2}=\frac{c}{2}.\nonumber\end{eqnarray}$$
As in the other proof, note that
$U$
and
$V$
are not partial isometries and they also fail to map into the right Hilbert space, so we have to correct for that. The latter problem is again solved by replacing
$U$
and
$V$
with
$U_{0}:=PU$
and
$V_{0}:=PV$
which both lie in
$P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
. Of course, we still have
$$\begin{eqnarray}\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V_{0}^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U_{0})=\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U)\geqslant \frac{c}{2}.\end{eqnarray}$$
Now since
$\Vert U_{0}\Vert _{\text{op}},\Vert V_{0}\Vert _{\text{op}}\leqslant 1$
, by Lemma 2.9 we have that
$U_{0}=\frac{1}{2}(U_{1}+U_{2})$
and
$V_{0}=\frac{1}{2}(V_{1}+V_{2})$
where
$U_{1},U_{2},V_{1},V_{2}\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
are all partial isometries. Thus, there must be at least one combination of partial isometries, say,
$U_{1}$
and
$V_{1}$
such that
$$\begin{eqnarray}|\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V_{1}^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U_{1})|\geqslant \frac{c}{2}.\end{eqnarray}$$
By multiplying
$U_{1}$
with a complex number of modulus 1, we can assume
$$\begin{eqnarray}\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V_{1}^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U_{1})\geqslant \frac{c}{2}.\end{eqnarray}$$
Let
$B:=\mathbb{E}_{x}\unicode[STIX]{x1D711}(x)V_{1}^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }$
. Then
$\Vert B\Vert _{\text{op}}\leqslant 1$
, so
$BB^{\ast }\leqslant 1_{{\mathcal{M}}}$
, and hence
$\unicode[STIX]{x1D70F}(BB^{\ast })\leqslant 1,$
which gives us
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(U_{1}U_{1}^{\ast })\geqslant \unicode[STIX]{x1D70F}(BB^{\ast })\unicode[STIX]{x1D70F}(U_{1}^{\ast }U_{1})\geqslant |\unicode[STIX]{x1D70F}(BU_{1})|=\mathbb{E}_{x}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D711}(x)V_{1}^{\ast }\unicode[STIX]{x1D70C}(x)^{\ast }U_{1})\geqslant \frac{c}{2}.\end{eqnarray}$$
A similar calculation gives us that
$\unicode[STIX]{x1D70F}(V_{1}V_{1}^{\ast })\geqslant c/2,$
and the proof is complete with
$U_{1}$
and
$V_{1}$
as
$U$
and
$V$
.◻
5 Stability of
$\unicode[STIX]{x1D700}$
-representations
We use our main theorem to prove a stability result for
$\unicode[STIX]{x1D700}$
-representations. The proof actually works for a slightly larger class of maps.
Definition 5.1. If
$G$
is amenable, a map
$\unicode[STIX]{x1D711}:G\rightarrow H$
is called a mean
$\unicode[STIX]{x1D700}$
-homomorphism if for all
$g\in G$
$$\begin{eqnarray}\mathbb{E}_{h}d(\unicode[STIX]{x1D711}(gh),\unicode[STIX]{x1D711}(g)\unicode[STIX]{x1D711}(h))<\unicode[STIX]{x1D700}.\end{eqnarray}$$
Note that the notion of a mean
$\unicode[STIX]{x1D700}$
-homomorphism also covers the case of maps from a finite group to a discrete group which satisfy for all
$g\in G$
that the equality
$\unicode[STIX]{x1D711}(gh)=\unicode[STIX]{x1D711}(g)\unicode[STIX]{x1D711}(h)$
holds for most
$h$
. Again we use the terminology a mean
$\unicode[STIX]{x1D700}$
-representation if the group
$H$
consists of operators.
Theorem 5.2. Let
$\unicode[STIX]{x1D700}>0$
, let
$G$
be an amenable group, let
${\mathcal{M}}$
be a von Neumann algebra and let
$\Vert \cdot \Vert$
be a unitarily invariant, ultraweakly lower semi-continuous semi-norm
$\Vert \cdot \Vert$
on
${\mathcal{M}}_{\infty }$
. Let
$\unicode[STIX]{x1D711}:G\rightarrow {\mathcal{U}}({\mathcal{M}})$
be a mean
$\unicode[STIX]{x1D700}$
-representation with respect to the metric coming from
$\Vert \cdot \Vert$
. Then there is a projection
$P\in {\mathcal{M}}_{\infty }$
, a partial isometry
$U\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
and a representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{U}}(P{\mathcal{M}}_{\infty }P)$
such that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D711}(g)-U^{\ast }\unicode[STIX]{x1D70C}(g)U\Vert <71\unicode[STIX]{x1D700},\quad g\in G,\end{eqnarray}$$
and
$$\begin{eqnarray}\Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert <40\unicode[STIX]{x1D700},\qquad \Vert P-UU^{\ast }\Vert <30\unicode[STIX]{x1D700}.\end{eqnarray}$$
Proof. We show that
$\unicode[STIX]{x1D711}$
satisfies the conditions of our Theorem 3.1 with
$2\unicode[STIX]{x1D700}$
instead of
$\unicode[STIX]{x1D700}$
. Note that since
$\unicode[STIX]{x1D711}$
takes unitary values, the condition that
$\Vert \unicode[STIX]{x1D711}(x)\Vert _{\text{op}}\leqslant 1$
is automatically satisfied, for all
$x\in G$
.
and
It follows from Remark 3.3 that there are a projection
$P\in {\mathcal{M}}_{\infty }$
and operators
$U,V\in P{\mathcal{M}}_{\infty }1_{{\mathcal{M}}}$
, such that
$U$
is a partial isometry and
$\Vert V\Vert _{\text{op}}\leqslant 1$
, and a representation
$\unicode[STIX]{x1D70C}:G\rightarrow {\mathcal{M}}$
so that
and
$$\begin{eqnarray}\Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert <40\unicode[STIX]{x1D700},\qquad \Vert P-UU^{\ast }\Vert <30\unicode[STIX]{x1D700}.\end{eqnarray}$$
Thus
Remark 5.3. In fact, the best way to prove this theorem might be to start from scratch and use the methods from the proof of Theorem 3.1 accommodated suitably. This will, however, save us only
$2\unicode[STIX]{x1D700}$
and we end up with the estimate
$\Vert \unicode[STIX]{x1D711}(g)-U^{\ast }\unicode[STIX]{x1D70C}(g)U\Vert <69\unicode[STIX]{x1D700}$
.
This general stability result for
$\unicode[STIX]{x1D700}$
-representations subsumes the theorems advertized in the introduction. These results follow almost immediately (but our estimates are different than the original ones). For convenience, we include some comments to the proofs here.
Proof of Theorem 1.4.
We use Theorem 5.2 in the case where
${\mathcal{M}}\subseteq \mathbb{B}({\mathcal{H}})$
is the von Neumann algebra generated by
$\unicode[STIX]{x1D711}(G)$
and
$\Vert \cdot \Vert$
is the operator norm
$\Vert \cdot \Vert _{\text{op}}$
. If
$\unicode[STIX]{x1D700}<\frac{1}{40}$
, then
$\Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert _{\text{op}},\Vert P-UU^{\ast }\Vert _{\text{op}}<1$
and since
$1_{{\mathcal{M}}}-U^{\ast }U$
and
$P-UU^{\ast }$
are projections, this implies
$1_{{\mathcal{M}}}=U^{\ast }U$
and
$P=UU^{\ast }$
. It follows that
$\unicode[STIX]{x1D70C}^{\prime }:G\rightarrow {\mathcal{U}}({\mathcal{M}})$
given by
$\unicode[STIX]{x1D70C}^{\prime }(x)=U^{\ast }\unicode[STIX]{x1D70C}(x)U$
is a unitary representation, and thus the result follows.◻
Proof of Theorem 1.6.
In the case
${\mathcal{M}}=\mathbb{M}_{n}$
, we can identify
${\mathcal{M}}_{\infty }$
with
$\mathbb{B}(\ell ^{2}(\mathbb{N}))$
in such a way that
$1_{{\mathcal{M}}}$
is a rank-
$n$
projection and we use Theorem 5.2 with
$\Vert \cdot \Vert$
being the 2-norm
$\Vert \cdot \Vert _{2}$
coming from the semi-finite trace
$\unicode[STIX]{x1D70F}$
on
$\mathbb{B}(\ell ^{2}(\mathbb{N}))$
normalized in such a way that rank-1 projections have trace
$1/n$
. The inequalities
$\Vert 1_{{\mathcal{M}}}-U^{\ast }U\Vert _{2}<40\unicode[STIX]{x1D700},\Vert P-UU^{\ast }\Vert _{2}<30\unicode[STIX]{x1D700}$
translate into
$$\begin{eqnarray}|\operatorname{rank}(P)-\operatorname{rank}(1_{{\mathcal{M}}})|<(40^{2}+30^{2})\unicode[STIX]{x1D700}^{2}n=2500\unicode[STIX]{x1D700}^{2}n.\end{eqnarray}$$
First assume
$\operatorname{rank}(P)\geqslant \operatorname{rank}(1_{{\mathcal{M}}})$
. Let
$Q=P-UU^{\ast }$
and
$R=1_{{\mathcal{M}}}-U^{\ast }U$
. Since
$U\in P({\mathcal{M}}_{\infty })1_{{\mathcal{M}}}$
is a partial isometry, there is a partial isometry
$U_{0}=Q({\mathcal{M}}_{\infty })R$
such that
$U^{\prime }:=U+U_{0}\in P({\mathcal{M}}_{\infty })1_{{\mathcal{M}}}$
is an isometry.
$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert \unicode[STIX]{x1D711}(g)-(U^{\prime })^{\ast }\unicode[STIX]{x1D70C}(g)U^{\prime }\Vert _{2}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \Vert \unicode[STIX]{x1D711}(g)-U^{\ast }\unicode[STIX]{x1D70C}(g)U\Vert _{2}+\Vert U_{0}^{\ast }\unicode[STIX]{x1D70C}(g)U\Vert _{2}+\Vert U^{\ast }\unicode[STIX]{x1D70C}(g)U_{0}\Vert _{2}+\Vert U_{0}^{\ast }\unicode[STIX]{x1D70C}(g)U_{0}\Vert _{2}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 71\unicode[STIX]{x1D700}+3\Vert Q\Vert _{2}<161\unicode[STIX]{x1D700}.\nonumber\end{eqnarray}$$
If
$\operatorname{rank}(P)\leqslant \operatorname{rank}(1_{{\mathcal{M}}})$
, then we pick any projection
$Q\geqslant P$
with
$\operatorname{rank}(Q)=\operatorname{rank}(1_{{\mathcal{M}}})$
and consider the representation
$$\begin{eqnarray}\unicode[STIX]{x1D70C}^{\prime }:G\rightarrow Q({\mathcal{M}}_{\infty })Q,\qquad \unicode[STIX]{x1D70C}^{\prime }(g)=\unicode[STIX]{x1D70C}(g)+Q-P.\end{eqnarray}$$
As above, we can extend
$U\in Q({\mathcal{M}}_{\infty })1_{{\mathcal{M}}}$
to an isometry, which further extends to a unitary
$U^{\prime }:1_{{\mathcal{M}}}(\ell ^{2}(\mathbb{N}))\rightarrow Q(\ell ^{2}(\mathbb{N}))$
and we get that
$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D711}(g)-(U^{\prime })^{\ast }\unicode[STIX]{x1D70C}^{\prime }(g)U^{\prime }\Vert _{2} & {\leqslant} & \displaystyle \Vert \unicode[STIX]{x1D711}(g)-(U^{\prime })^{\ast }\unicode[STIX]{x1D70C}(g)U^{\prime }\Vert _{2}+\Vert (U^{\prime })^{\ast }(Q-P)U^{\prime }\Vert _{2}\nonumber\\ \displaystyle & {<} & \displaystyle (161+\sqrt{2500})\unicode[STIX]{x1D700}=211\unicode[STIX]{x1D700}.\nonumber\end{eqnarray}$$
In both cases we get the desired result. ◻
Acknowledgements
Part of this research was carried out while the first author visited RIMS in Kyoto. He thanks this institution for its hospitality. The results of this article are part of the PhD thesis of the first author. This research was supported by ERC Starting Grant No. 277728 and ERC Consolidator Grant No. 681207. The second author was partially supported by JSPS KAKENHI 264001 and 15H05739.
