# On the amenable subalgebras of group von Neumann algebras

Tattwamasi Amrutam  Ben Gurion University of the Negev. Department of
Mathematics. Be’er Sheva, 8410501, Israel.  ,  Yair Hartman  Ben Gurion
University of the Negev. Department of Mathematics. Be’er Sheva, 8410501,
Israel.  and  Hanna Oppelmayer  Universität Innsbruck. Department of
Mathematics. Technikerstrasse 13, 6020 Innsbruck, Austria

###### Abstract.

We approach the study of sub-von Neumann algebras of the group von Neumann
algebra $L(\Gamma)$ for countable groups $\Gamma$ from a dynamical
perspective. It is shown that $L(\Gamma)$ admits a maximal invariant amenable
subalgebra. The notion of invariant probability measures (IRAs) on the space
of sub-algebras is introduced, analogous to the concept of Invariant Random
Subgroups. And it is shown that amenable IRAs are supported on the maximal
amenable invariant sub-algebra.

###### Contents

  1. 1 Introduction and the statement of main results
    1. 1.1 Invariant Random Subalgebras
  2. 2 Preliminaries
    1. 2.1 Topology on the subalgebras of a von Neumann algebra
    2. 2.2 Amenable von Neumann algebras
    3. 2.3 Furstenberg boundary
  3. 3 Maximal amenable $\Gamma$-invariant subalgebra
  4. 4 Amenable Invariant Random Algebras
    1. 4.1 Normal closure of IRAs
    2. 4.2 Closedness of amenable subalgebras
    3. 4.3 Uppersemicontinuity of the Hype-map
    4. 4.4 Amenable IRAs and amenable Radical

##  1\. Introduction and the statement of main results

The study of the space $\text{Sub}(\Gamma)$ of all closed subgroups of a given
group $\Gamma$ \- the Chabouty space \- is nowadays a central study theme in
geometric group theory and ergodic theory.

Interestingly, although this space was already defined in 1950, it experienced
a dramatic revival over the last two decades. During the same era, Effros
[Eff65] defined a topology on the collection of von Neumann subalgebras
$\text{SA}(\mathcal{M})$ of a given von Neumann algebra $\mathcal{M}$, that we
refer to as the Effros-Maréchal topology.

In this paper, we focus on the study of $\text{SA}(L(\Gamma))$ for a countable
discrete group $\Gamma$ from a dynamical perspective: we consider
$\text{SA}(L(\Gamma))$ as a $\Gamma$-space. Indeed, the natural conjugation
action $\Gamma\curvearrowright\text{SA}(L(\Gamma))$ is continuous with respect
to the Effros-Maréchal topology. The space $\text{SA}(L(\Gamma))$ extends
$\text{Sub}(\Gamma)$ (see ex:IRAfromIRS) and hence, is a natural object to
study from the group theoretic point of view. We find that as a
$\Gamma$-space, it is somewhat more structured than the collection of
subalgebras of the general $\mathbb{B}(\mathcal{H})$ (see e.g., [HW98]).

The collection of amenable subgroups has received special attention in the
sense that they reveal information about the group $\Gamma$, e.g., whether
$\Gamma$ has unique trace property or is $C^{*}$-simple etc. Most of our focus
in this paper is devoted to the study of the collection of all amenable
subalgebras $\text{SA}_{\text{am}}(L(\Gamma))$ of $L(\Gamma)$.

We show that $\text{SA}_{\text{am}}(L(\Gamma))$ is closed (see
closedamenablecollection). This result is in contrast with
$\text{SA}_{\text{am}}(\mathbb{B}(\mathcal{H}))$ as shown in [HW98, Theorem
5.4]. closedamenablecollection is similar in spirit to the closedness of
$\text{Sub}_{\text{am}}(\Gamma)$ for discrete group $\Gamma$ (while for
general locally compact groups it is still an open problem (cf. [CM14])).

Our first main result asserts the existence of a maximal amenable invariant
subalgebra of $L(\Gamma)$:

######  Theorem A.

Let $\Gamma$ be a discrete countable group and let $\text{Rad}(\Gamma)$ denote
the maximal normal amenable subgroup of $\Gamma$. Then,
$L(\text{Rad}(\Gamma))$ is the maximal amenable $\Gamma$-invariant subalgebra
of $L(\Gamma)$.

We remark that the classical proof of the existence of $\text{Rad}(\Gamma)$
does not clearly generalize to subalgerbas. In our proof, we use two main
tools. The first is Furman’s [Fur03] beautiful characterization of
$\text{Rad}(\Gamma)$ as the kernel of the action on the Furstenberg Boundary.
The other tool used is singularity (cf. singularity), which has been exploited
in the past for various rigidity results (see e.g., [KK17, HKB23, BBHP22,
AH22] etc.). Moreover, our result answers [CDS22, Conjecture 5.9] in positive.

###  1.1. Invariant Random Subalgebras

A significant part of the study of the Chabouty space is devoted to invariant
probability measures, which are invariant under the conjugation action. These
are called Invariant Random Subgroups (IRSs) introduced by [AGV14]. We refer
the readers to the references in [SZ94, ABB+17, Bow14, BGK15, AGV14] for more
details. We specifically mention the result of [BDLW16] proving that IRSs that
are supported on amenable subgroups are almost surely contained in the
amenable radical.

This paper introduces the notion of Invariant Random Subalgebras (IRAs) in
parallel to that of IRSs.

######  Definition 1.1.

iras A Borel probability measure on $\text{SA}(L(\Gamma))$ that is invariant
under the $\Gamma$-conjugation action is called IRA, short for Invariant
Random Sub-Algebra.

Our second main theorem generalizes the result of [BDLW16] in the context of
IRAs.

######  Theorem B.

Let $\mu$ be an $IRA$ on $\text{SA}(L(\Gamma))$ for a countable group
$\Gamma$. If $\mu$-almost every $\mathcal{M}\in\text{SA}(L(\Gamma))$ is
amenable, then $\mu$-a.e $\mathcal{M}\leq L(\text{Rad}(\Gamma))$.

In particular, for groups $\Gamma$ with trivial amenable radical, there are no
amenable IRAs except for $\delta_{\\{\mathbb{C}\\}}$.

### Organization of the paper

In addition to this section, there are three other sections. We begin by
recalling some definitions and proving some preparatory results in Section 2,
which we put to use in later parts. In Section 3, after establishing a
singularity type result in the form of singularity, we proceed to prove
Thoerem A. In Section 4, we show that the push-forward of an IRS by the
natural subalgebra map gives rise to an IRA (see ex:IRAfromIRS). We also
provide an example of an IRA which is not induced from an IRS (see Example
4.3). After introducing the notion of normal closure for an IRA, we devote the
final subsections towards the proof of Theorem B.

### Acknowledgements

The authors thank Mehrdad Kalantar, Yongle Jiang and Chris Phillips for many
helpful discussions and for sharing their insights. They also thank Mehrdad
Kalantar, Ionut Chifan and Yongle Jiang for reading through a near complete
draft of the manuscript and for pointing out numerous typos and inaccuracies
which improved the exposition greatly.

This work has received funding from the European Research Council (ERC) under
the European Union’s Seventh Framework Programme (FP7-2007-2013) (Grant
agreement No. 101078193), and by the Israel Science Foundation (ISF 1175/18)

H.O. is partially supported by Early Stage Funding - Vice-rector for Research,
Universität Innsbruck, and the AIANI fellowship.

##  2\. Preliminaries

In this section, we briefly recall the notions of Effros-Meréchal topology,
amenable von Neumann algebras and that of the Furstenberg boundary along with
some other elementary observations for our later use.

###  2.1. Topology on the subalgebras of a von Neumann algebra

Let $\mathcal{N}$ be a von Neumann algebra with separable predual. We consider
the set of all sub-von Neumann algebras $\text{SA}(\mathcal{N})$ of
$\mathcal{N}$ and equip it with the Effros-Meréchal topology. We denote by
$so^{*}$ the strong-* operator topology on $\mathcal{N}$ and by $wo$ the weak-
operator-topology.

######  Definition 2.1.

[AHW16, Definition 2.2] Let $\mathcal{M}_{n}\in\text{SA}(\mathcal{N})$ for
$n\in\mathbb{N}$. We set

$\liminf\limits_{n\to\infty}\mathcal{M}_{n}:=\\{x\in\mathcal{N}\ |\
\exists(x_{n})_{n\in\mathbb{N}}\in
l^{\infty}(\mathbb{N},\mathcal{M}_{n})\,:\,\text{so}^{*}\text{-}\lim\limits_{n\to\infty}x_{n}=x\\},$

and

$\limsup\limits_{n\to\infty}\mathcal{M}_{n}:=\langle\\{x\in\mathcal{N}\ |\
\exists(x_{n})_{n\in\mathbb{N}}\in
l^{\infty}(\mathbb{N},\mathcal{M}_{n})\,:\,\text{wo}\text{-}\lim\limits_{n\to\infty}x_{n}=x\\}\rangle$

where $\langle\cdot\rangle$ means the von Neumann algebra generated by the
set. The Effros-Maréchal topology is defined such that

$\mathcal{M}_{n}\to\mathcal{M}\ \ \text{ iff }\ \
\liminf\limits_{n\to\infty}\mathcal{M}_{n}=\limsup\limits_{n\to\infty}\mathcal{M}_{n}=\mathcal{M}.$

It is well known that this topology gives a standard Borel structure on
$\text{SA}(\mathcal{N})$ (see [Eff65] and [Mar73]).
Let us state the following fact [HW98, Corollary 2.12], which we will need
later. Let $\mathcal{M}_{n},\mathcal{M}\in\text{SA}(\mathcal{N})$,
$n\in\mathbb{N}$, for a finite, separable von Neumann algebra $\mathcal{N}$.
If $\mathcal{M}_{n}\to\mathcal{M}$ in Effros-Maréchal topology, then

(1)
$\mathbb{E}_{\mathcal{M}_{n}}(x)\xrightarrow[]{\text{so*}}\mathbb{E}_{\mathcal{M}}(x),\
\forall x\in\mathcal{N},$

where $\mathbb{E}_{\mathcal{M}}:\mathcal{N}\longrightarrow\mathcal{M}$ denotes
the canonical conditional expectation.

###  2.2. Amenable von Neumann algebras

Let $(\mathcal{M},\tau)\subset\mathbb{B}(\mathcal{H})$ be a tracial von
Neumann algebra on a separable Hilbert space $\mathcal{H}$.

######  Definition 2.2.

A $(\mathcal{M},\tau)$-hypertrace is a state $\phi$ on
$\mathbb{B}(\mathcal{H})$ such that on $\phi|_{\mathcal{M}}=\tau$ and
$\phi(Tm)=\phi(mT),\ \forall m\in\mathcal{M},\ \forall
T\in\mathbb{B}(\mathcal{H})$.

We denote by $\text{Hyp}_{\tau}(\mathcal{M})$ the set of all
$(\mathcal{M},\tau)$ hypertraces.

######  Definition 2.3.

[ADPon, Proposition 10.2.5] A tracial von Neumann algebra $(\mathcal{M},\tau)$
is called amenable (also referred to as injective) if
$\text{Hyp}_{\tau}(\mathcal{M})\neq\emptyset$.

From [HW98, Theorem 5.2], it follows that the collection of all amenable von
Neumann sub-algebras of $\mathcal{N}$, denoted by
$\text{SA}_{am}(\mathcal{N})$, is a $G_{\delta}$-set in the Effros-Maréchal-
topology, thus in particular Borel measurable w.r.t. this topology. The
following proposition holds the key for Theorem B. We denote by
$S(\mathcal{N})$ the collection of all states on $\mathcal{N}$.

######  Proposition 2.4.

keylink Let $(\mathcal{M},\tau)$ be a finite vNa with a separable predual. Let
$\mathcal{N}\leq\mathcal{M}$ be an amenable sub-von Neumann algebra of
$\mathcal{N}$. Then, there exists an $\mathcal{N}$-hyperstate $\varphi\in
S(\mathbb{B}(L^{2}(\mathcal{M},\tau)))$ such that
$\varphi|_{\mathcal{M}}=\tau$.

###### Proof.

Let $\mathcal{N}\leq\mathcal{M}$ be an amenable von Neumann algebra. Since
$\mathcal{N}$ is amenable, $\mathcal{N}$ is approximately finite dimensional
(see [Con76, Theorem 6]). Hence, we can find an increasing sequence of unital
finite dimensional von-Neumann subalgebras $\\{Q_{n}\\}$ of $\mathcal{N}$ such
that $\mathcal{N}=\left(\cup_{n}Q_{n}\right)^{\prime\prime}$. Now, let us
consider

$\mathcal{C}=\left\\{\varphi\in
S(\mathbb{B}(L^{2}(\mathcal{M},\tau)):\varphi|_{\mathcal{M}}=\tau\right\\}.$

Clearly, $\mathcal{C}$ is weak∗-compact. Let us note that the unitary group
$\mathcal{U}(Q_{n})\curvearrowright\mathcal{C}$ by conjugation action. Indeed,
for $u\in\mathcal{U}(Q_{n})$ and $\varphi\in\mathcal{C}$, $u.\varphi\in
S(\mathbb{B}(L^{2}(\mathcal{M},\tau)))$. Moreover, for $x\in\mathcal{M}$,

$u.\varphi(x)=\varphi(uxu^{*})=\tau(uxu^{*})=\tau(x).$

Since $Q_{n}$ is finite-dimensional, the action
$\mathcal{U}(Q_{n})\curvearrowright\mathcal{C}$ is continuous and being
amenable, we will get a $\mathcal{U}(Q_{n})$-fixed point in $\mathcal{C}$,
which we call $\varphi_{n}$. Note that $\varphi_{n}|_{\mathcal{M}}=\tau$. And,
let $\psi=\lim_{n\to\omega}\varphi_{n}$,
$\omega\in\beta\mathbb{N}\setminus\mathbb{N}$. Then, clearly,
$\psi|_{\mathcal{M}}=\tau$. We now show that $\psi$ is $\mathcal{N}$-central,
i.e., $\psi(xT)=\psi(Tx)$ for all $x\in\mathcal{N}$ and
$T\in\mathbb{B}(L^{2}(\mathcal{M},\tau))$. Fix $0\neq
T\in\mathbb{B}(L^{2}(\mathcal{M},\tau))$. Let us observe that
$\varphi_{n}(uT)=\varphi_{n}(Tu)$ for all $u\in\mathcal{U}(Q_{n})$. Letting
$n\to\omega$, we see that $\psi(uT)=\psi(Tu)$ for all $u\in\mathcal{U}(Q_{n})$
and for every $n\in\mathbb{N}$. Since the linear span of unitary elements are
norm dense inside a finite-dimensional algebra, it follows that

(2) $\psi(xT)=\psi(Tx),\forall x\in\cup_{n}Q_{n}.$

Now, fix an arbitrary element $y\in\mathcal{N}$ and let $\epsilon>0$ be given.
We can find an element $x\in\cup_{n}Q_{n}$ such that
$\|y-x\|_{\tau}<\frac{\epsilon}{2\|T\|}$. Now, using the triangle inequality
along with equation (2), we see that

$\displaystyle\left|\psi(yT)-\psi(Ty)\right|$
$\displaystyle\leq\left|\psi(yT)-\psi(xT)\right|+\left|\psi(xT)-\psi(Tx)\right|+\left|\psi(Tx)-\psi(Ty)\right|$
$\displaystyle=\left|\psi(yT)-\psi(xT)\right|+\left|\psi(Tx)-\psi(Ty)\right|$

Appealing to the Cauchy-Schwartz inequality, we see that

$\left|\psi(yT)-\psi(xT)\right|\leq\|y-x\|_{\tau}\|T\|<\frac{\epsilon}{2}.$

Similarly, $\left|\psi(Tx)-\psi(Ty)\right|<\frac{\epsilon}{2}$. Putting these
together, we see that $\left|\psi(yT)-\psi(Ty)\right|<\epsilon$. Since
$\epsilon>0$ is arbitrary, the claim follows. ∎

###  2.3. Furstenberg boundary

Furstenberg [Fur73] introduced the notion of topological boundary. One purpose
is to relate lattices of semisimple lie groups with their ambient groups
(e.g., ${SL}_{n}(\mathbb{Z})$ in ${SL}_{n}(\mathbb{R})$). We make this notion
precise below.
Let $\Gamma$ be a discrete countable group and $X$ be a $\Gamma$-space, i.e.,
$X$ is a compact Hausdorff space and $\Gamma\curvearrowright X$ by
homeomorphisms. The action $\Gamma\curvearrowright X$ is called a boundary
action if $\left\\{\delta_{x}:x\in
X\right\\}\subset\overline{\Gamma\nu}^{\text{weak}^{*}}$ for every
$\nu\in\text{Prob}(X)$. If $\Gamma\curvearrowright X$ is a boundary action,
then $X$ is called a $\Gamma$-boundary. Note that this action is always
minimal.

######  Proposition 2.5.

[Fur73, Fur63] The Furstenberg boundary of $\Gamma$, $\partial_{F}\Gamma$ is a
$\Gamma$ boundary which is universal in the sense that every other
$\Gamma$-boundary $Y$ is a $\Gamma$-equivariant continuous image of
$\partial_{F}\Gamma$.

The fact that such a space exists follows from a standard product argument
involving the representatives of all boundaries (see, e.g., [Fur73, P. 199]).
Moreover, $\partial_{F}\Gamma$ is unique up to $\Gamma$-equivariant
homeomorphism. Over the years, this notion has been used to study the
properties of the groups. For example, Kalantar and Kennedy [KK17] gave a
dynamical characterization of $C^{*}$-simplicity in terms of the action
$\Gamma\curvearrowright\partial_{F}\Gamma$ on the Furstenberg boundary
$\partial_{F}\Gamma$. It is also known that the group $\Gamma$ is amenable if
and only if the associated Furstenberg boundary $\partial_{F}\Gamma$ is
trivial (see, e.g., [Gla76, Theorem 3.1, Chapter 3]). More generally, Furman
[Fur03, Proposition 7] showed that $\text{Rad}(\Gamma)$ is exactly the kernel
of the action $\Gamma\curvearrowright\partial_{F}\Gamma$ (also see [BKKO17,
Proposition 2.8]). It is this later criterion which we exploit for our
purposes.

##  3\. Maximal amenable $\Gamma$-invariant subalgebra

Let $\Gamma$ be a discrete countable group. It follows from Day’s result that
there is a unique maximal normal subgroup of $\Gamma$, called the amenable
radical. We denote it by $\text{Rad}(\Gamma)$. By $L(\Gamma)$ we denote the
group von Neumann algebra, i.e.
$L(\Gamma)=\overline{\operatorname{span}(\lambda(\Gamma))}^{so}$, where
$\lambda$ denotes the left regular representation. It then immediately follows
that $L(\text{Rad}(\Gamma))$ is a $\Gamma$-invariant amenable von Neumann
subalgebra of $L(\Gamma)$. We show that $L(\text{Rad}(\Gamma))$ is also the
largest $\Gamma$-invariant amenable subalgebra of $L(\Gamma)$.

######  Proposition 3.1.

maximalamenable Let $\Gamma$ be a discrete countable group. Let
$\mathcal{M}\leq L(\Gamma)$ be a $\Gamma$-invariant amenable subalgebra. Then,
$\mathcal{M}\subset L(\text{Rad}(\Gamma))$. In particular,
$L(\text{Rad}(\Gamma))$ is the largest amenable $\Gamma$-invariant subalgebra
of $L(\Gamma)$.

The classical argument (see for example [Nev94]) in the setting of groups does
not have an obvious modification to encompass the von Neumann setup. To prove
our result, we resort to the dynamical characterization of the amenable
radical in terms of the action $\Gamma\curvearrowright\partial_{F}\Gamma$. Let
$\tau_{0}$ denote the canonical trace on $L(\Gamma)$.

######  Definition 3.2.

invstate Let $\Gamma$ be a discrete countable group. Let
$\mathcal{A}\subset\mathbb{B}(\ell^{2}(\Gamma))$ be a unital $C^{*}$-algebra
and $\mathcal{M}\leq L(\Gamma)$ be a von Neumann subalgebra. A state
$\varphi\in S(\mathcal{A})$ is called $(\mathcal{M},\tau_{0})$-invariant if
there exists an $\mathcal{M}$-hyperstate $\psi\in
S(\mathbb{B}(\ell^{2}(\Gamma)))$ such that $\psi|_{\mathcal{A}}=\varphi$ and
$\psi|_{L(\Gamma)}=\tau_{0}$. We denote the collection of such states by
$S^{\mathcal{M}}_{\tau_{0}}(\mathcal{A})$.

Let $X$ be a minimal $\Gamma$-space. We can view $C(X)$ as multiplication
operators on $\mathbb{B}(\ell^{2}(\Gamma))$. Fix $x_{0}\in X$. For $f\in
C(X)$, the map $M(f):\ell^{2}(\Gamma)\to\ell^{2}(\Gamma)$ defined by
$M(f)(\delta_{t})=f(t.x_{0})\delta_{t}$ is linear and bounded. Since the
action $\Gamma\curvearrowright X$ is minimal, we see that
$\|M(f)\|=\|f\|_{\infty}$ for every $f\in C(X)$. Therefore, we can identity
$C(X)$ with its image $M(C(X))$ inside $\mathbb{B}(\ell^{2}(\Gamma))$. We note
that this embedding is not canonical. However, it is enough to fix one
embedding for our purposes.

######  Lemma 3.3.

singularity Let $\Gamma$ be a discrete group and $\mathcal{M}\leq L(\Gamma)$
be an amenable von Neumann subalgebra. Let $s\in\Gamma\setminus\\{e\\}$.
Assume that there exists a minimal $\Gamma$-space $X$ and an embedding of
$C(X)\leq\mathbb{B}(\ell^{2}(\Gamma))$ (possibly depending on s) such that the
following hold: There exists $x\in X$ such that

  1. (1)

$sx\neq x$, and

  2. (2)

$\delta_{x}\in S^{\mathcal{M}}_{\tau_{0}}(C(X))$.

Then, $\tau_{0}(a\lambda(s)^{*})=0$ for all $a\in\mathcal{M}$.

###### Proof.

Let $\mathcal{M}\leq L(\Gamma)$ be an amenable von Neumann algebra. Let
$s\in\Gamma\setminus\\{e\\}$. Assume a minimal $\Gamma$-space $X$ satisfies
the above conditions and let $x\in X$ be such that $sx\neq x$. Since
$\delta_{x}\in S^{\mathcal{M}}_{\tau_{0}}(C(X))$, there exists a
$\mathcal{M}$-hyperstate $\varphi$ such that $\varphi|_{C(X)}=\delta_{x}$ and
$\varphi|_{L(\Gamma)}=\tau_{0}$. Choose $f\in C(X)$ such that $f(x)=1$ and
$f(sx)=0$ with $0\leq f\leq 1$. One can view $C(X)$ as a subset of
$\mathbb{B}(\ell^{2}(\Gamma))$. Now, since $C(X)$ falls in the multiplicative
domain of $\varphi$, we see that for any $a\in\mathcal{M}$,

$\displaystyle\tau_{0}(a\lambda(s)^{*})$
$\displaystyle=\varphi(a\lambda(s)^{*})$
$\displaystyle=\varphi\left(a\lambda(s)^{*}f\right)$
$\displaystyle=\varphi\left(a(s^{-1}.f)\lambda(s)^{*}\right)$
$\displaystyle=\varphi\left(a\sqrt{(s^{-1}.f)}\sqrt{(s^{-1}.f)}\lambda(s)^{*}\right).$

Now, using Cauchy-Schwartz inequality, we see that

$\displaystyle|\varphi(a\lambda(s)^{*})|$
$\displaystyle=\left|\varphi\left(a\sqrt{(s^{-1}.f)}\sqrt{(s^{-1}.f)}\lambda(s)^{*}\right)\right|$
$\displaystyle\leq\sqrt{\varphi(a(s^{-1}.f)a^{*})}\sqrt{\varphi(\lambda(s)(s^{-1}.f)\lambda(s)^{*})}$
$\displaystyle=\sqrt{\varphi((s^{-1}.f)a^{*}a)}\sqrt{\varphi(f)}$ ($\varphi$
is a $\mathcal{M}$-hyperstate)
$\displaystyle=\sqrt{(s^{-1}.f)(x)\varphi(a^{*}a)}\sqrt{\varphi(f)}$
($\varphi|_{C(X)}=\delta_{x}$)
$\displaystyle=\sqrt{f(sx)\varphi(a^{*}a)}\sqrt{\varphi(f)}$ $\displaystyle=0$

The claim follows. ∎

###### Proof of maximalamenable.

Suppose that $\mathcal{M}$ is an amenable $\Gamma$-invariant subalgebra of
$L(\Gamma)$. Let us consider $\text{Hype}_{\tau_{0}}(\mathcal{M})$, the
collection of all $(\mathcal{M},\tau_{0})$ -hyperstates whose restrictions to
$L(\Gamma)$ is the canonical trace $\tau_{0}$. Due to Proposition
LABEL:keylink, this is a non-empty set. Let $\partial_{F}\Gamma$ denote the
Furstenberg boundary of the group $\Gamma$ and fix an embedding of
$C(\partial_{F}\Gamma)$ inside $\mathbb{B}(\ell^{2}(\Gamma))$. The group
$\Gamma$ acts on $S(\mathbb{B}(\ell^{2}(\Gamma)))$ by
$(\gamma.\phi)(T)=\phi(\lambda(\gamma)T\lambda(\gamma^{-1}))$ for $\phi\in
S(\mathbb{B}(\ell^{2}(\Gamma)))$, $\gamma\in\Gamma$,
$T\in\mathbb{B}(\ell^{2}(\Gamma))$. We note that
$\text{Hype}_{\tau_{0}}(\mathcal{M})|_{C(\partial_{F}\Gamma)}$ is a
$\Gamma$-invariant weak∗-closed convex subset of
$\text{Prob}(\partial_{F}\Gamma)$. Using the irreducibility of
$\text{Prob}(\partial_{F}\Gamma)$, we obtain that
$S_{\tau_{0}}^{\mathcal{M}}(C(\partial_{F}\Gamma))=\text{Prob}(\partial_{F}\Gamma)$.
This, in particular, says that every probability measure $\nu$ on
$\partial_{F}\Gamma$ can be realized as $\varphi|_{C(\partial_{F}\Gamma)}$ for
some $\varphi\in\text{Hype}_{\tau_{0}}(\mathcal{M})$. We now proceed to show
that $\mathcal{M}\subset L(\text{Rad}(\Gamma))$. To do so, it is enough to
show that $\tau_{0}(a\lambda(s)^{*})=0$ for all $a\in\mathcal{M}$ and
$s\in\Gamma\setminus(\text{Rad}(\Gamma))$. Let $a\in\mathcal{M}$ and choose
$s\in\Gamma\setminus\text{Rad}(\Gamma)$. Since
$\text{Rad}(\Gamma)=\text{Ker}\left(\Gamma\curvearrowright\partial_{F}\Gamma\right)$,
there exists a point $x\in\partial_{F}\Gamma$ such that $sx\neq x$. It now
follows from singularity that $\tau_{0}(a\lambda(s)^{*})=0$. Since
$s\in\Gamma\setminus\text{Rad}(\Gamma)$ is arbitrary, we see that
$\mathcal{M}\subset L\left(\text{Rad}(\Gamma)\right)$. ∎

######  Corollary 3.4.

subfactors Let $\Gamma$ be a countable group with trivial amenable radical.
Then every $\Gamma$-invariant von Neumann subalgebra $\mathcal{M}\leq
L(\Gamma)$ is a subfactor.

###### Proof.

Let $\mathcal{M}\leq L(\Gamma)$ be a $\Gamma$-invariant subalgebra. Then
$\mathcal{Z}(\mathcal{M})$, the center of $\mathcal{M}$, is a
$\Gamma$-invariant, amenable subalgebra. It now follows from maximalamenable
that $\mathcal{Z}(\mathcal{M})\leq L(\text{Rad}(\Gamma))=\mathbb{C}$. ∎

##  4\. Amenable Invariant Random Algebras

Using the following proposition, we see IRAs of $L(\Gamma)$ naturally extend
IRSs.

######  Proposition 4.1.

ex:IRAfromIRS Let $\Gamma$ be a countable discrete group and
$\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. Let
$\lambda\in\text{IRS}(\Gamma)$. Consider the map
$L:\text{Sub}(\Gamma)\to\text{SA}(L(\Gamma))$ defined by $H\mapsto
L(H),\leavevmode\nobreak\ H\in\text{Sub}(\Gamma)$. $L$ is continuous with
respect to the Chabauty topology on $\text{Sub}(\Gamma)$ and Effros-Maréchal
topology on $\text{SA}(L(\Gamma))$.

###### Proof.

Let $H_{n}\in\text{Sub}(\Gamma)$ be such that $H_{n}\to H$ in the Chabauty
topology. We show $L(H_{n})\to L(H)$ in the Effros-Maréchal topology. Let
$x\in L(H)$ be given and choose $\epsilon>0$. Then, we can find
$h_{1},h_{2},\ldots,h_{m}\in H$ and $c_{1},c_{2},\ldots,c_{m}\in\mathbb{C}$
such that $\|x-\sum_{i=1}^{m}c_{i}\lambda(h_{i})\|_{2}<\epsilon$. We remark
that the $\|.\|_{2}$-norm is with respect to the canonical trace $\tau_{0}$ on
$L(\Gamma)$. Since $H_{n}\to H$, there exists a $n_{0}\in\mathbb{N}$ such that
for all $n\geq n_{0}$, $h_{i}\in H_{n}$ for all $i=1,2,\ldots,m$.
Consequently, it follows that $\sum_{i=1}^{m}c_{i}\lambda(h_{i})\in L(H_{n})$
for all $n\geq n_{0}$. Hence, for all $n\geq n_{0}$, we can find
$x_{n}=\sum_{i=1}^{n}c_{i}\lambda(h_{i})\in L(H_{n})$ such that
$x_{n}\xrightarrow[]{\text{so}}x$. Using the Uniform boundedness principle
(or, Kaplansky density theorem) we can assume $\|x_{n}\|\leq\|x\|$. On the
ball $B(0,\|x\|)$, the so- and $\text{so}^{*}\text{-}$topology coincide. So,
$x_{n}\xrightarrow[]{\text{so}^{*}}x$. Therefore, $x\in\liminf_{n}L(H_{n})$.
Now, let $x\in\limsup_{n}L(H_{n})$ be such that we can find $x_{n}\in
L(H_{n})$ with $x_{n}\xrightarrow{\text{wo}}x$. Let $g\in\Gamma\setminus H$.
Since $H_{n}\to H$, there exists a subsequence $\\{n_{k}\\}_{k}$ such that
$g\not\in H_{n_{k}}$ for all $k\in\mathbb{N}$. Let $\tau_{0}$ denote the
canonical trace on $L(\Gamma)$. Since
$\tau_{0}(\cdot)=\langle(\cdot)\delta_{e},\delta_{e}\rangle$, $\tau_{0}$ is
wo-continuous (see [ADPon, Proposition 2.1.1]). Since
$x_{n_{k}}\lambda(g^{-1})\xrightarrow[]{\text{wo}}x\lambda(g^{-1})$, it
follows that
$\tau_{0}\left(x_{n_{k}}\lambda(g^{-1})\right)\xrightarrow{k\to\infty}\tau_{0}\left(x\lambda(g^{-1})\right)$.
Since $g\not\in H_{n_{k}}$ and $x_{n_{k}}\in L(H_{n_{k}})$, for every
$k\in\mathbb{N}$, we see that

$\tau_{0}\left(x_{n_{k}}\lambda(g^{-1})\right)=\tau_{0}\left(\mathbb{E}_{H_{n_{k}}}(x_{n_{k}}\lambda(g^{-1}))\right)=\tau_{0}\left((x_{n_{k}}\mathbb{E}_{H_{n_{k}}}(\lambda(g^{-1}))\right)=0,$

where $\mathbb{E}_{H}:L(\Gamma)\longrightarrow L(H)$ denotes the canonical
conditional expectation, confer [ADPon, Theorem 9.1.2]. This implies that
$\tau_{0}(x\lambda(g^{-1}))=0$ for all $g\in\Gamma\setminus H$. This implies
that $x\in L(H)$. Therefore,

$\displaystyle\limsup_{n}L(H_{n})$ $\displaystyle=\left\langle\\{x\in L(H)\ |\
\exists(x_{n})_{n\in\mathbb{N}}\in
l^{\infty}(\mathbb{N},L(H_{n}))\,:\,\text{wo}\text{-}\lim\limits_{n\to\infty}x_{n}=x\\}\right\rangle$
$\displaystyle\leq L(H).$

Consequently, we see that

$L(H)\leq\liminf_{n}L(H_{n})\leq\limsup_{n}L(H_{n})\leq L(H).$

It now follows from Definition 2.1 that $\lim_{n}L(H_{n})=L(H)$ in the Effros-
Marechal topology, which in turn implies that the map $L$ is continuous. This
completes the proof. ∎

######  Example 4.2 (IRAs coming from IRSs).

Let $\lambda\in IRS(\Gamma)$. It follows from ex:IRAfromIRS that the map
$L:\text{Sub}(\Gamma)\to\text{SA}(L(\Gamma))$ is continuous, hence measurable.
Since $L$ is equivariant, $L_{*}\lambda\in\text{IRA}(L(\Gamma))$.

It is not true that every IRA is of this form. We construct such an example
below. The above results are a proper generalization of [BDLW16] since there
are IRAs that are supported on collections of sub-von Neumann algebras that
are not of the form $LK$ for any subgroup $K$ of $\Gamma$.

######  Example 4.3 (IRAs not coming from IRSs).

Consider the lamplighter group

$\Gamma=\text{LL}(\mathbb{Z},\mathbb{Z}/3\mathbb{Z})=\mathbb{Z}\ltimes\text{Lamps}$

with $\text{Lamps}:=\\{f:\mathbb{Z}\longrightarrow\mathbb{Z}/3\mathbb{Z}\text{
with }|\operatorname{supp}(f)|<\infty\\}$. For any $S\subseteq\mathbb{Z}$ set

$\text{Lamps}_{S}:=\\{f\in\text{Lamps}\,:\,\operatorname{supp}(f)\subseteq
S\\}.$

Observe that $\Lambda_{S}:=\\{0\\}\ltimes\text{Lamps}_{S}$ is a group with
$(0,f)\cdot(0,h)=(0,f+h)$ for any $f,h\in\text{Lamps}_{S}$. Let
$\sigma:\mathbb{Z}/3\mathbb{Z}\longrightarrow\mathbb{Z}/3\mathbb{Z}$ be the
non-trivial homomorphism. Then, the set

$\mathcal{M}_{S}:=\overline{\\{\sum\limits_{f\in\text{Lamps}_{S}}a_{f}\lambda((0,f))\
:\ a_{f}=a_{\sigma\circ f},\ a_{f}\in\mathbb{C}\\}}^{\text{wo}}$

is a non-trivial von Neumann sub-algebra of $L(\Lambda_{S})$. Moreover, we
also observe that $\\{\mathcal{M}_{S}:S\subset\mathbb{Z}\\}$ is
$\Gamma$-invariant. Indeed, let

$x:=\sum\limits_{f\in\text{Lamps}_{S}}a_{f}\lambda((0,f))\in\mathcal{M}_{S},$

then

$x^{*}=\sum\limits_{f\in\text{Lamps}_{S}}\overline{a_{f}}\lambda((0,-f))\
\in\mathcal{M}_{S},$

because the coefficient at the position $(0,-f)$ is
$c_{-f}:=\overline{a_{f}}=\overline{a_{\sigma(f)}}=c_{-\sigma(f)}=c_{\sigma(-f)},$
since by assumption $a_{f}=a_{\sigma(f)}$. Further, for
$y=\sum\limits_{h\in\text{Lamps}_{S}}\overline{b_{h}}\lambda((0,h))\
\in\mathcal{M}_{S},$ we see that

$xy=\sum\limits_{h,f\in\text{Lamps}_{S}}a_{f}b_{h}\lambda((0,f+h))\
\in\mathcal{M}_{S},$

due to
$d_{f+h}:=a_{f}b_{h}=a_{\sigma(f)}b_{\sigma(h)}=d_{\sigma(f)+\sigma(h)}=d_{\sigma(f+h)}.$

Now, as pointed out by [AJ23, Example 3.5], for $\sigma\neq id$, there is no
sub-group $K$ of $\Gamma$ such that $\mathcal{M}_{S}$ can be written as
$L(K)$. Indeed, if we had $\mathcal{M}_{S}=L(K)$, then for any $(0,f)\in K$ we
would obtain e.g. $5\lambda(0,-f)+\lambda(0,f)\in\mathcal{M}_{S}$ but by
construction $a_{f}=a_{-f}$ which is a contradiction.
Moreover, $\mathcal{M}_{S}$ is clearly $\Gamma$-invariant.
Let us consider the map

$\psi:\\{0,1\\}^{\mathbb{Z}}\longrightarrow\text{SA}(L(\Gamma)),\
S\mapsto\mathcal{M}_{S}.$

Then, $\psi$ is Borel measurable w.r.t. Effros-Maréchal topology on
$\text{SA}(L(\Gamma))$: For $S\in\\{0,1\\}^{\mathbb{Z}}$, let
$H_{S}:=\\{(0,f)\ :\ \operatorname{supp}(f)\subseteq S\\}$. We first observe
that the map
$\text{Sub}:\\{0,1\\}^{\mathbb{Z}}\to\text{Sub}(\Gamma),\leavevmode\nobreak\
S\mapsto H_{S}$ is a continuous map with respect to the Chabauty topology.
Furthermore, it follows from Proposition LABEL:ex:IRAfromIRS above that the
map $L:\text{Sub}(\Gamma)\to\text{SA}(L(\Gamma)),\leavevmode\nobreak\
H_{S}\mapsto L(H_{S})$ is continuous. Let
$j:\text{SA}(L(\Gamma))\to\text{SA}(L(\Gamma))\times\text{SA}(L(\Gamma))$ be
given by $j(\mathcal{M})=\left(\mathcal{M},\mathcal{M}_{\mathbb{Z}}\right)$.
We equip $\text{SA}(L(\Gamma))\times\text{SA}(L(\Gamma))$ with the product
topology. $j$ is continuous with respect to the Effros-Marechal topology. It
follows from [Eff65, Corollary 2] that the map

$\text{Ints}:\text{SA}(L(\Gamma))\times\text{SA}(L(\Gamma))\longrightarrow\text{SA}(L(\Gamma)),(\mathcal{M},\mathcal{N})\mapsto\mathcal{M}\cap\mathcal{N}$

is Borel measurable w.r.t. the Effros-Maréchal topology. Therefore,

$\text{Ints}\circ j\circ
L\circ\text{Sub}:\\{0,1\\}^{\mathbb{Z}}\to\text{SA}(L(\Gamma)),\leavevmode\nobreak\
S\mapsto\mathcal{M}_{\mathbb{Z}}\cap L(H_{S})$

being a composition of measurable maps is measurable. Let us now observe that

$\displaystyle\mathcal{M}_{\mathbb{Z}}\cap L(H_{S})$
$\displaystyle=\overline{\\{\sum\limits_{f\in\text{Lamps}}a_{f}\lambda((0,f))\
:\ a_{f}=a_{\sigma(f)},\
a_{f}\in\mathbb{C}\\}\cap\operatorname{span}(\lambda(H_{S}))}^{\text{wo}}$
$\displaystyle=\mathcal{M}_{S}.$

Therefore, the map $\psi=\text{Ints}\circ j\circ L\circ\text{Sub}$ is
measurable.
Moreover, $\psi$ is $\Gamma$-equivariant, where $\Gamma$ acts by shifting on
$\\{0,1\\}^{\mathbb{Z}}$: The map $L$ is clearly $\Gamma$-equivariant. Since
$\mathcal{M}_{\mathbb{Z}}$ is a $\Gamma$-invariant subalgebra, the maps Ints
and $j$ are $\Gamma$-equivariant. Therefore, it is enough to show that the map
Sub is $\Gamma$-equivariant. Let $g=(m,h)\in\Gamma$, then $g\cdot S=(m,0)S$
shifts the position of the ones in the set $S$ to the left by
$m\in\mathbb{Z}$. So,

$H_{gS}=\\{(0,f)\ :\ \operatorname{supp}(f)\subseteq gS\\}=\\{(0,f)\ :\
\operatorname{supp}(g^{-1}f)\subseteq S\\}$ $=\\{(0,mf)\ :\
\operatorname{supp}(f)\subseteq S\\}=gH_{S}g^{-1}$

since $g(0,f)g^{-1}=(m-m,h+mf-h)=(0,mf)$ for $g=(m,h)$.
Therefore, we can push forward any $\Gamma$-invariant probability measure
$\kappa$ on $\\{0,1\\}^{\mathbb{Z}}$, e.g. the Bernoulli measure, to obtain an
IRA on $\text{SA}(L(\Gamma))$ via the map $\psi$. Observe that by the above,
$\psi_{*}\kappa$ cannot be purely supported on von Neumann sub-algebras of the
form $L(K)$ for some subgroup $K$ of $\Gamma$ since it lives on sub-algebras
of the form $\mathcal{M}_{S}$ for $S\in\\{0,1\\}^{\mathbb{Z}}$.

###  4.1. Normal closure of IRAs

In [HT16] the so-called normal closure of an IRSs was introduced. We mimic
this concept in the von Neumann algebra setting for IRAs.
Let $\mathcal{M}$ be a von Neumann algebra with separable Hilbert space on
which $\Gamma$ acts.

######  Lemma 4.4.

normalclosure Let $\mu\in\text{IRA}(\mathcal{M})$. Then there exists a unique
minimal sub-von Neumann algebra $\mathcal{N}\in\text{SA}(\mathcal{M})$ such
that $\mu(\text{SA}(\mathcal{M}))=1$. Moreover, this von Neumann algebra
$\mathcal{N}$ is $\Gamma$-invariant. We denote this von Neumann algebra
$\mathcal{N}$ by $\overline{\langle\mu\rangle}$ and call it the _normal
closure_ of $\mu$.

###### Proof.

We follow the argument outlined in [GH23, Lemma 2.3] and adapt it to the von
Neumann setting. Let us denote

$\mathcal{W}=\left\\{\tilde{\mathcal{M}}\in\text{Sub}(\mathcal{M}):\mu(\text{SA}(\tilde{\mathcal{M}}))=1\right\\}.$

Let

$\mathcal{N}:=\bigcap\limits_{\tilde{\mathcal{M}}\in\mathcal{W}}\tilde{\mathcal{M}}.$

Clearly, $\mathcal{N}$ is a von Neumann algebra again and it is
$\Gamma$-invariant since $\mu$ is $\Gamma$-invariant by assumption. It is
unique since any algebra $\tilde{\mathcal{M}}$ fulfilling
$\mu(\text{SA}(\tilde{\mathcal{M}}))=1$ contains $\mathcal{N}$, and we want a
minimal one. It is left to show that $\mu(\text{SA}(\mathcal{N}))=1$. We now
observe that

$\text{SA}(\mathcal{N})=\cap_{\tilde{\mathcal{M}}\in\mathcal{W}}\text{SA}(\tilde{\mathcal{M}}).$

Now, let
$\mathcal{K}\in\text{SA}(\mathcal{M})\setminus\text{SA}(\mathcal{N})$. This
means that there exists $\mathcal{M}_{\mathcal{K}}\in\mathcal{W}$ such that
$\mathcal{K}\not\in\text{SA}(\mathcal{M}_{\mathcal{K}})$. Since
$\text{SA}(\mathcal{M}_{\mathcal{K}})$ is closed, we can find
$\epsilon_{\mathcal{K}}>0$ such that

(3)
$B(\mathcal{K},\epsilon_{\mathcal{K}})\cap\text{SA}(\mathcal{M}_{\mathcal{K}})=\emptyset.$

In particular, we see that

$\text{SA}(\mathcal{M})\setminus\text{SA}(\mathcal{N})=\bigcup_{\mathcal{K}\in\text{SA}(\mathcal{M})\setminus\text{SA}(\mathcal{N})}B(\mathcal{K},\epsilon_{\mathcal{K}})$

Since $\text{SA}(\mathcal{M})$ is a separable metric space, hence, second
countable, and therefore, hereditarily Lindelöf. Since
$\text{SA}(\mathcal{N})$ is closed in Effros-Maréchal topology ([HW98, Theorem
2.8]), $\text{SA}(\mathcal{M})\setminus\text{SA}(\mathcal{N})$ is open. In a
hereditary Lindelöf space, every open set is Lindelöf. Therefore, we can find
$\mathcal{K}_{n}\in\text{SA}(\mathcal{M})\setminus\text{SA}(\mathcal{N})$ and
$\epsilon_{\mathcal{K}_{n}}>0$ such that

(4)
$\text{SA}(\mathcal{M})\setminus\text{SA}(\mathcal{N})=\bigcup_{n}B(\mathcal{K}_{n},\epsilon_{\mathcal{K}_{n}})$

Moreover, using equation (3), we see that
$\text{SA}(\mathcal{M}_{\mathcal{K}_{n}})\subseteq\left(B(\mathcal{K},\epsilon_{\mathcal{K}_{n}})\right)^{c}$
for each $n\in\mathbb{N}$. Therefore,

(5)
$\bigcap_{n}\text{SA}(\mathcal{M}_{\mathcal{K}_{n}})\subseteq\bigcap_{n}\left(B(\mathcal{K},\epsilon_{\mathcal{K}_{n}})\right)^{c}$

Putting together equations (4) and (5), we see that

$\bigcap_{n}\text{SA}(\mathcal{M}_{\mathcal{K}_{n}})\subseteq\bigcap_{n}\left(B(\mathcal{K},\epsilon_{\mathcal{K}_{n}})\right)^{c}\subseteq\text{SA}(\mathcal{N})$

Therefore,

$\mu(\text{SA}(\mathcal{N}))\geq\lim\limits_{m\to\infty}\mu(\bigcap_{n=1}^{m}\text{SA}(\mathcal{M}_{\mathcal{K}_{n}}))=1,$

since $\mu\left(\text{SA}(\mathcal{M}_{\mathcal{K}_{n}})^{c}\right)=0$ by
construction. ∎

######  Remark 4.5.

Let $\mathcal{Z}=L\Gamma$ and $\mu\in IRA(\mathcal{Z})$. Let

$H_{0}=\\{g\in\Gamma\ :\ \mu(\\{\mathcal{M}\ni\lambda(g)\\})>0\\}.$

If $H$ is the subgroup of $\Gamma$ generated by $H_{0}$, then,
$\overline{\langle\mu\rangle}\supseteq LH$. Indeed, let us write for
simplicity $\\{\mathcal{M}\ni
m\\}:=\\{\mathcal{M}\in\text{SA}(L\Gamma)\,:\,m\in\mathcal{M}\\}$ for $m\in
L\Gamma$ and first verify that this is a measurable set. It is closed in the
Effros-Maréchal topology: If $\mathcal{M}_{\alpha}\in\\{\mathcal{M}\ni m\\}$
with $\mathcal{M}_{\alpha}\to\mathcal{M}$, then by Definition 2.1 the elements
in $\mathcal{M}$ are precisely the $\text{so}^{*}\text{-}$limit points of
elements in $\mathcal{M}_{\alpha}$, in particular the constant sequence
$x_{\alpha}=m$ converges and thus $m\in\mathcal{M}$. Assume there exists some
$g\in\Gamma$ such that $\lambda(g)\notin\overline{\langle\mu\rangle}$. Then
clearly,
$\text{SA}(\overline{\langle\mu\rangle})\cap\\{\mathcal{M}\ni\lambda(g)\\}=\emptyset$.
Since $\mu(\text{SA}(\overline{\langle\mu\rangle}))=1$ we thus obtain
$\mu(\\{\mathcal{M}\ni\lambda(g)\\})=0$. Therefore $g\notin H_{0}$. Thus
$LH\subseteq\overline{\langle\mu\rangle}$.

######  Remark 4.6.

Let us recall that $\Gamma$ is said to have Invariant Subgroup Rigidity
property (ISR-property) if every $\Gamma$-invariant subalgebra
$\mathcal{M}\leq L(\Gamma)$ is of the form $L(N)$ for some normal subgroup
$N\triangleleft\Gamma$. This property was studied in the context of higher
rank lattices in [AB21, KP22] and in the context of hyperbolic groups in
[AJ23, CDS22].

If $\Gamma$ has the (ISR)-property, then, the von Neumann algebra
$\overline{\langle\mu\rangle}$ is of the form $LN$ for some normal subgroup
$N$ of $\Gamma$ with $N\geq H$ as in Remark 4.5 above.

###  4.2. Closedness of amenable subalgebras

######  Proposition 4.7.

closedamenablecollection Let $(\mathcal{M},\tau)$ be a finite von Neumann
algebra with a separable predual. Then, the collection of amenable subalgebras
of $\mathcal{M}$ is closed in the Effros-Maréchal topology.

###### Proof.

Let $\mathcal{M}_{n}$ be a sequence of amenable subalgebras of $\mathcal{M}$
such that $\mathcal{M}_{n}\to\tilde{\mathcal{M}}$ and
$\varphi_{\mathcal{M}_{n}}$the $\mathcal{M}_{n}$-hyperstates on
$\mathbb{B}(L^{2}(\mathcal{M},\tau))$ such that
$\varphi_{\mathcal{M}_{n}}|_{\mathcal{M}}=\tau$. The existence is guaranteed
by keylink. We shall denote $\varphi_{\mathcal{M}_{n}}$ by $\varphi_{n}$ for
ease of notation. Then, there exists a sub-sequence $(n_{k})$ and $\varphi\in
S(\mathbb{B}(L^{2}(\mathcal{M},\tau)))$ such that

$\varphi_{n_{k}}\xrightarrow[]{\text{weak}^{*}}\varphi.$

We shall show that $\varphi\in\text{Hype}_{\tau}(\tilde{\mathcal{M}})$ and
this shall complete the proof. Clearly, $\varphi|_{\mathcal{M}}=\tau$. Let
$x\in\tilde{\mathcal{M}}$ be given and $0\neq
T\in\mathbb{B}(L^{2}(\mathcal{M},\tau))$ be fixed. It follows from Definition
2.1 that there exist $x_{n}\in\mathcal{M}_{n}$ such that
$x_{n}\xrightarrow[n\to\infty]{\text{so}^{*}}x$. In particular,
$\|x_{n_{k}}-x\|_{\tau}\xrightarrow{k\to\infty}0$. Let $\epsilon>0$ be given.
We can find $k_{0}\in\mathbb{N}$ such that for all $k,j\geq k_{0}$,

$\|x_{n_{k}}-x\|_{\tau}<\frac{\epsilon}{6\|T\|}\text{ and
}\|x_{n_{k}}-x_{n_{j}}\|_{\tau}<\frac{\epsilon}{6\|T\|}.$

Since $\varphi_{n_{k}}\to\varphi$ in the weak∗-topology, we can find $j\geq
k_{0}$ such that

(6)
$\displaystyle\left\|\varphi_{n_{j}}\left(x_{n_{k_{0}}}T\right)-\varphi\left(x_{n_{k_{0}}}T\right)\right\|<\frac{\epsilon}{6},$
$\displaystyle\left\|\varphi_{n_{j}}\left(Tx_{n_{k_{0}}}\right)-\varphi\left(Tx_{n_{k_{0}}}\right)\right\|<\frac{\epsilon}{6}.$

Now, using the triangle inequality, we see that

$\displaystyle\left\|\varphi(xT-Tx)\right\|$
$\displaystyle\leq\left\|\varphi(xT)-\varphi(x_{n_{k_{0}}}T)\right\|+\left\|\varphi(x_{n_{k_{0}}}T)-\varphi_{n_{j}}(x_{n_{k_{0}}}T)\right\|$
$\displaystyle+\left\|\varphi_{n_{j}}(x_{n_{k_{0}}}T)-\varphi_{n_{j}}(x_{n_{j}}T)\right\|$
$\displaystyle+\left\|\varphi_{n_{j}}(x_{n_{j}}T)-\varphi_{n_{j}}(Tx_{n_{j}})\right\|+\left\|\varphi_{n_{j}}(Tx_{n_{j}})-\varphi_{n_{j}}(Tx_{n_{k_{0}}})\right\|$
$\displaystyle+\left\|\varphi_{n_{j}}(Tx_{n_{k_{0}}})-\varphi(Tx_{n_{k_{0}}})\right\|+\left\|\varphi(Tx_{n_{k_{0}}})-\varphi(Tx)\right\|$

Using Cauchy-Schwartz inequality, we observe that

$\displaystyle\left\|\varphi(xT)-\varphi(x_{n_{k_{0}}}T)\right\|$
$\displaystyle=\left\|\varphi\left((x-x_{n_{k_{0}}})T\right)\right\|$
$\displaystyle\leq\sqrt{\varphi\left((x-x_{n_{k_{0}}})(x^{*}-x_{n_{k_{0}}}^{*})\right)}\sqrt{\varphi(T^{*}T)}$

Since $x,x_{n_{k_{0}}}\in\mathcal{M}$ and $\varphi|_{\mathcal{M}}=\tau$, we
see that

$\sqrt{\varphi\left((x-x_{n_{k_{0}}})(x^{*}-x_{n_{k_{0}}}^{*})\right)}=\|x-x_{n_{k_{0}}}\|_{\tau}.$

Moreover, $\sqrt{\varphi(T^{*}T)}\leq\|T\|$. Hence,

$\left\|\varphi(xT)-\varphi(x_{n_{k_{0}}}T)\right\|\leq\|x-x_{n_{k_{0}}}\|_{\tau}\|T\|<\frac{\epsilon}{6}.$

Similarly,

$\displaystyle\left\|\varphi(Tx)-\varphi(Tx_{n_{k_{0}}})\right\|$
$\displaystyle\leq\sqrt{\varphi\left((x^{*}-x_{n_{k_{0}}}^{*})(x-x_{n_{k_{0}}})\right)}\sqrt{\varphi(TT^{*})}$
$\displaystyle\leq\|x-x_{n_{k_{0}}}\|_{\tau}\|T\|<\frac{\epsilon}{6}.$

Also, since $\varphi_{n_{j}}|_{\mathcal{M}}=\tau$, we see that

$\displaystyle\left\|\varphi_{n_{j}}(x_{n_{k_{0}}}T)-\varphi_{n_{j}}(x_{n_{j}}T)\right\|$
$\displaystyle\leq\sqrt{\varphi_{n_{j}}\left((x_{n_{k_{0}}}-x_{n_{j}})(x_{n_{k_{0}}}^{*}-x_{n_{j}}^{*})\right)}\sqrt{\varphi_{n_{j}}(T^{*}T)}$
$\displaystyle\leq\|x_{n_{k_{0}}}-x_{n_{j}}\|_{\tau}\|T\|<\frac{\epsilon}{6}.$

Since $\varphi_{n_{j}}$ is a hyperstate for $\mathcal{M}_{n_{j}}$,
$\varphi_{n_{j}}(x_{n_{j}}T)=\varphi_{n_{j}}(Tx_{n_{j}})$. Putting all of
these together, we see that $\|\varphi(xT-Tx)\|<\epsilon$ for every
$\epsilon>0$. As a result,
$\varphi\in\text{Hype}_{\tau}(\tilde{\mathcal{M}})$. ∎

###  4.3. Uppersemicontinuity of the Hype-map

Let $\mathcal{M}\leq L(\Gamma)$ be an amenable von Neumann algebra. Then,
$\text{Hype}(\mathcal{M})$, (the collection of all $\mathcal{M}$-hyperstates
$\varphi_{\mathcal{M}}$ on $\mathbb{B}(\ell^{2}(\Gamma))$) is a non-empty
compact (weak*-sense) and convex subset. Let us now fix a metrizable and
minimal $\Gamma$-space $X$. Let

$\text{Hyp}_{X}=\left\\{\text{Hype}(\mathcal{M})|_{C(X)}:\mathcal{M}\text{ is
an amenable subalgebra of }L(\Gamma)\right\\}$

We want to give a convex structure to Hyp, and we do this by following
[BDLW16]. Since $X$ is metrizable and compact, $C(X)$ is separable. Let
$\\{f_{n}\\}_{n=1}^{\infty}$ be a dense subset of the unit ball of $C(X)$. For
each $n\in\mathbb{N}$, we define

$f_{n}^{+}(\text{Hype}(\mathcal{M})|_{C(X)})=\sup_{\varphi\in\text{Hype}(\mathcal{M})}\text{Real}(\varphi)(f_{n}).$

We can then define the map

(7) $f:\text{Hyp}_{X}\to\prod_{n}[0,1],\leavevmode\nobreak\
\text{Hype}(\mathcal{M})|_{C(X)}\to\left(f_{n}^{+}\left(\text{Hype}(\mathcal{M})|_{C(X)}\right)\right)_{n\in\mathbb{N}}$

Applying the Hahn-Banach separation theorem implies that the map $f$ is
injective. The topology on $\text{Hyp}_{X}$ is then induced from the product
topology on $\prod_{n}[0,1]$. Moreover, the convex structure of
$\prod_{n}[0,1]$ carries over to $\text{Hyp}_{X}$ as described in [BDLW16].

The action of $\Gamma$ on $\text{S}(\mathbb{B}(\ell^{2}(\Gamma))\Gamma)$ is
defined as

$(g\cdot\phi)(x):=\phi(\lambda(g^{-1})x\lambda(g)),\leavevmode\nobreak\
g\in\Gamma,\phi\in\text{S}(\mathbb{B}(\ell^{2}(\Gamma))\Gamma),x\in\mathbb{B}(\ell^{2}(\Gamma))\Gamma.$

Let $\text{Hype}_{\tau_{0}}(\mathcal{M})$ denote the collection of all those
hyperstates whose restriction to $L(\Gamma)$ is the canonical trace
$\tau_{0}$. Note that this set is non-empty thanks to keylink.

######  Lemma 4.8.

The map

$\text{Hyp}:\\{\mathcal{M}\in\text{SA}(L(\Gamma))\,:\,\text{
amenable}\\}\longrightarrow\left\\{\text{Hype}_{\tau_{0}}(\mathcal{M})\right\\}$

is $\Gamma$-equivariant.

###### Proof.

To verify $\Gamma$-equivariance, it suffices to show that

$g\cdot\text{Hyp}(\mathcal{M})\subseteq\text{Hyp}(g\cdot\mathcal{M}),\ \forall
g\in\Gamma,\ \forall\mathcal{M}\in\text{SA}(\mathcal{Z})\text{ amenable},$

because the reverse inclusion follows by replacing $\mathcal{M}$ with
$g^{-1}\cdot\mathcal{M}$. Let $g\in\Gamma$ and
$\mathcal{M}\in\text{SA}(\mathcal{Z})$ and $\phi\in\text{Hyp}(\mathcal{M})$.
We shall show that $g\cdot\phi\in\text{Hyp}(g\cdot\mathcal{M})$. Recall that
$g\cdot\phi(x):=\phi(\lambda(g^{-1})x\lambda(g))$ and
$g\cdot\mathcal{M}=\lambda(g)\mathcal{M}\lambda(g^{-1})$. For any
$m\in\mathcal{M}$ and $T\in\mathbb{B}(\ell^{2}(\Gamma))\Gamma$, we have

$\displaystyle(g\cdot\phi)(\lambda(g)m\lambda(g^{-1})T)$
$\displaystyle=\phi(\lambda(g^{-1})\lambda(g)m\lambda(g^{-1})T\lambda(g))$
$\displaystyle=\phi(m\lambda(g^{-1})T\lambda(g))$
$\displaystyle=\phi(\lambda(g^{-1})T\lambda(g)m)$
$\displaystyle=\phi(\lambda(g^{-1})T\lambda(g)m\lambda(g^{-1})\lambda(g))$
$\displaystyle=(g\cdot\phi)(T\lambda(g)m\lambda(g^{-1})).$

Note that the last equality follows since
$\lambda(g^{-1})T\lambda(g)\in\mathbb{B}(\ell^{2}(\Gamma))\Gamma$ and $\phi$
is a hyperstate for $\mathcal{M}$. Moreover,
$(g\cdot\phi)_{|_{g\cdot\mathcal{M}}}=\phi_{|_{\mathcal{M}}}=\tau_{0}.$
Therefore $g\cdot\phi$ is a hyperstate for $g\cdot\mathcal{M}$, which was to
show. ∎

Let $X$ be a metrizable minimal $\Gamma$-space.

######  Proposition 4.9.

measurable The map

$\text{Hyp}:\\{\mathcal{M}\in\text{SA}(L(\Gamma))\,:\,\text{
amenable}\\}\longrightarrow\left\\{\text{Hype}_{\tau_{0}}(\mathcal{M})|_{C(X)}\right\\}$

is upper semi-continuous.

###### Proof.

Let $\\{f_{n}\\}_{n}$ be a countable dense subset of $C(X)$. It is enough to
prove that $\mathcal{M}\to
f_{n}^{+}(\text{Hype}_{\tau_{0}}(\mathcal{M})|_{C(X)})$ is upper semi-
continuous for each $n\in\mathbb{N}$. Now, fix $n_{0}\in\mathbb{N}$. We can
find $\varphi_{\mathcal{M}}\in\text{Hype}_{\tau_{0}}(\mathcal{M})$ such that

$f_{n_{0}}^{+}(\text{Hype}_{\tau_{0}}(\mathcal{M})|_{C(X)})=\text{Re}(\varphi_{\mathcal{M}}|_{C(X)}(f_{n_{0}}))$

due to compactness of $\text{Hype}_{\tau_{0}}(\mathcal{M})$. Let
$\mathcal{M}_{n}$ be a sequence of amenable subalgebras of $L(\Gamma)$ such
that $\mathcal{M}_{n}\to\mathcal{M}$ and $\varphi_{\mathcal{M}_{n}}$ as above
for each $n\in\mathbb{N}$. We shall denote $\varphi_{\mathcal{M}_{n}}$ by
$\varphi_{n}$ for ease of notation. Then, there exists a sub-sequence
$(n_{k})$ and $\varphi\in S(\mathbb{B}(\ell^{2}(\Gamma))$ such that

$\varphi_{n_{k}}\xrightarrow[]{\text{weak}^{*}}\varphi.$

It follows from closedamenablecollection that
$\varphi\in\text{Hype}_{\tau_{0}}(\mathcal{M})$. Then, we see that

$f_{n_{0}}^{+}(\text{Hype}_{\tau_{0}}(\mathcal{M}))\geq\text{Re}(\varphi|_{C(X)}(f_{n_{0}}))=\lim_{k}\text{Re}(\varphi_{n_{k}}|_{C(X)}(f_{n_{0}}))$

Hence, it follows that

$f_{n_{0}}^{+}(\text{Hype}_{\tau_{0}}(\mathcal{M}))\geq\limsup_{n}f_{n_{0}}^{+}(\text{Hype}_{\tau_{0}}(\mathcal{M}_{n})),$

thereby implying that the map $\mathcal{M}\to
f_{n}^{+}(\text{Hype}_{\tau_{0}}(\mathcal{M})|_{C(X)})$ is upper semi-
continuous. ∎

###  4.4. Amenable IRAs and amenable Radical

###### Proof of Theorem B.

Let $\mu$ be an amenable IRA on $L(\Gamma)$. We shall show that
$\mathcal{M}\leq L(\text{Rad}(\Gamma))$ for $\mu$-a.e. $\mathcal{M}\leq
L(\Gamma)$. Let $s\in\Gamma\setminus\text{Rad}(\Gamma)$. Using [Fur03,
Proposition 7(ii)], we can find a metrizable $\Gamma$-boundary $X_{s}$ such
that $s\not\in\text{Ker}(\Gamma\curvearrowright X_{s})$. Consider the map

$\text{Hyp}:\\{\mathcal{M}\in\text{SA}(L(\Gamma))\,:\,\text{
amenable}\\}\longrightarrow\left\\{\text{Hype}_{\tau_{0}}(\mathcal{M})|_{C(X_{s})}\right\\}.$

By Lemma 4.8 and measurable, this map is $\Gamma$-equivariant and measurable.
We now push forward $\mu$ to obtain an invariant probability measure on a
subset of $\mathcal{CC}(\operatorname{Prob}(X_{s}))$, the space of all
compact, convex subsets of $\operatorname{Prob}(X_{s})\cong S(C(X_{s}))$.
Since $\operatorname{Prob}(X_{s})$ is irreducible, it follows from [BDLW16,
Lemma 2.3] that $(\text{Hyp})_{*}\mu=\delta_{\operatorname{Prob}(X_{s})}$.
Hence,

$\text{for }\mu\text{-a.e }\mathcal{M}\in\text{SA}(L(\Gamma))\ :\
\text{Hyp}_{\tau_{0}}(\mathcal{M})|_{C(X_{s})}=\operatorname{Prob}(X_{s}).$

In particular, every $\nu\in\text{Prob}(X_{s})$ is
$(\mathcal{M},\tau_{0})$-invariant for $\mu$-a.e $\mathcal{M}$. Let
$V_{s}\subset\text{Subalg}(L(\Gamma))$ be a co-null measurable set such that
$\nu\in\text{Prob}(X_{s})$ is $(\mathcal{M},\tau_{0})$-invariant for every
$\mathcal{M}\in V_{s}$, i.e.,

$\text{Prob}(X_{s})=S_{\tau_{0}}^{\mathcal{M}}(C(X_{s})),\leavevmode\nobreak\
\forall\mathcal{M}\in V_{s}.$

Fix $\mathcal{M}\in V_{s}$. Let $s\in\Gamma\setminus\text{Rad}(\Gamma)$. By
assumption, we can find an element $x_{s}\in X_{s}$ such that $sx_{s}\neq
x_{s}$. Using singularity, we see that

$\tau_{0}(a\lambda(s)^{*})=0,\forall a\in\mathcal{M},\leavevmode\nobreak\
\forall\mathcal{M}\in V_{s}.$

Let $V=\cap_{s\in\Gamma\setminus\text{Rad}(\Gamma)}{V_{s}}$ be the measurable
co-null set obtained by taking the countable intersection of the co-null
measurable sets $V_{s}$. Let $\mathcal{M}\in V$. We see that
$\tau_{0}(a\lambda(s)^{*})=0$ for all $a\in\mathcal{M}$ and for all
$s\in\Gamma\setminus\text{Rad}(\Gamma)$. Consequently, $\mathcal{M}\subset
L(\text{Rad}(\Gamma))$ for all $\mathcal{M}\in V$. Therefore,
$\overline{\langle\mu\rangle}\subset L(\text{Rad}(\Gamma))$. Since
$\left(L(\text{Rad}(\Gamma)),\tau_{0}|_{\text{Rad}(\Gamma)}\right)$ is a
tracial von Neumann algebra, $\overline{\langle\mu\rangle}$ is in the image of
a conditional expectation
$\mathbb{E}_{\overline{\langle\mu\rangle}}:L(\text{Rad}(\Gamma))\to\overline{\langle\mu\rangle}$
(see [ADPon, Theorem 9.1.2]). Since amenability passes to subalgebras in the
image of a conditional expectation, it follows that
$\overline{\langle\mu\rangle}$ is amenable. ∎

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# Robust Neural Particle Identification Models

Aziz Temirkhanov1 Artem Ryzhikov1 Denis Derkach1
Mikhail Hushchyn1 Nikita Kazeev1 and Sergei Mokhnenko1
on behalf of LHCb collaboration  1 HSE University, 20 Myasnitskaya st., Moscow
101000, Russia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The volume of data processed by the Large Hadron Collider experiments demands
sophisticated selection rules typically based on machine learning algorithms.
One of the shortcomings of these approaches is their profound sensitivity to
the biases in training samples. In the case of particle identification (PID),
this might lead to degradation of the efficiency for some decays not present
in the training dataset due to differences in input kinematic distributions.
In this talk, we propose a method based on the Common Specific Decomposition
that takes into account individual decays and possible misshapes in the
training data by disentangling common and decay specific components of the
input feature set. We show that the proposed approach reduces the rate of
efficiency degradation for the PID algorithms for the decays reconstructed in
the LHCb detector.

##  1 Introduction

Particle identification (PID) plays a crucial part in many high-energy physics
analysis. A higher performance PID algorithm leads to a better background
rejection and thus more precise results. In addition, the algorithm is
required to work with high efficiency in the entire available wide range of
signal topologies and kinematics and provide good discrimination for various
analyses. Machine learning (ML) algorithms have gradually become the baseline
approach for this task [1]. One large family of such algorithms is neural
networks.

The PID algorithms of the LHCb experiment relies on several sub-detector
systems [2]. Compared to other LHC experiments, LHCb has more information for
hadron identification, particularly the separation between charged hadrons,
provided by the Ring Imaging Cherenkov (RICH) subdetector. The muon
identification is provided by the muon chambers, while the responsibility of
calorimeters is mainly leptons and photons. The LHCb experiment employs
several machine learning solutions [1] to aggregate the information for the
PID. These solutions proved to have overall high efficiency over a large
volume of the phase space. In this paper, we address one of the outstanding
issues that can occur during the application of machine learning algorithms in
the real-life scenario: algorithm’s efficiency degradation in case the testing
scenario is significantly different from training.

The paper is organized as follows. The problem formulation and description of
current LHCb PID algorithms are given in Section 2. Section 3 provides a
description of the Common Specific Decomposition algorithm used for the robust
PID models. Finally, the list of training and testing samples, and the results
for the different PID models are presented in Section 4.

##  2 Problem statement

The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, described in detail in Ref. [2].
Identification of various final state particles is performed by combining
together the information from the LHCb subdetectors, namely from ring-imaging
Cherenkov detectors (RICH), the electromagnetic and hadronic calorimeters,
muon chambers and tracking system, as demonstrated in Fig. 1. Apart from the
physics motivated likelihood observables based on observable subdetector
responses [3], track geometry variables and different detector flags are also
used. In addition, the muon identification [4] and calorimeter information
about neutral clusters [5] are also used.

Figure 1: Illustration of different particle type responses in the LHCb
systems. Figure from [6]

The objective of the PID algorithm is to identify the charged particle type
associated with a given track. In the LHCb experiment, there are five relevant
particle species: electron, muon, pion, kaon, and proton. A sixth hypothesis
corresponds to ghost tracks, which do not correspond to a real particle that
passed through the detector.

Figure 2: Distributions of transverse momentum for different decays in the
simulated samples (left) and performance of the PID models on $D^{0}\to
K^{-}\pi^{+}$ decay for the pion-vs-all task (right).

The classification models for this task are trained on simulated events
generated for the LHCb data taking condition for 2015-2018 data taking period
with minimum bias trigger selection. In this study, we also use dedicated
simulated samples of correctly reconstructed decays as a source of correctly
reconstructed particles of a certain type. Particles in these decays might
have distribution different from the training sample as demonstrated in Fig.
2. Such differences may cause a quality reduction of the classification models
in this regions. The aim of this research is to make PID algorithms [1] robust
to differences in the kinematic distributions between the events used for
training and events the algorithm faces in real use. The research is focused
on neural networks only.

In this work, we compare our PID algorithm with the following three models:
ProbNN, CombDLL, and DLRun2. ProbNN is a neural network with a single hidden
layer. CombDLL is a combination of differential log-likelihoods from the
detector’s subsytems. DLRun2 is a deep neural network with 3 wide hidden
layers. Neural-based models are trained on minimum bias sample. The
performance of the neural-based solutions is approximately the same when
tested on the minimum bias sample.

We notice that quality of the algorithms might deteriorate in some regions.
The performance of the classifiers is tested with a sample of $D^{0}\to
K^{-}\pi^{+}$ signal candidates (Fig 2). As can be seen from the Figure,
DLRun2 algorithm’s quality is lower than its counterparts.

##  3 CSD algorithm

To address the quality degradation, we use domain adaptation techniques [7].
The idea laying behind the method is as follows: assume that there exist
common features, whose correlation with the target is preserved in all decays,
and decay-specific features, whose correlation differs from decays to decay
(Fig 3). A classifier that relies solely on common features is robust to
domain shifts.

To achieve that, CSD algorithm use a weighted combination of these three terms
in training objective: orthonormality regularizes and two cross-entropy losses
between labels and distributions computed from the common and domain-specific
features.

This approach allows to compute low-rank decomposition along with a training
networks parameters in a single layer. Hence, we replace final classification
layer of DLRun2 model by CSD layer [7] as shown in Fig. 3.

Figure 3: The CSD assumption.

##  4 Data and results

For the experiment we use simulated minimum bias sample and pions from several
decay channels: $D^{0}\to K^{-}\pi^{+}$, $D^{*}\to({D^{0}\to
K^{-}\pi^{+}})\pi^{-}$, $\Lambda_{c}\to p^{+}K^{-}\pi^{-}$, and
$\Lambda_{b}^{0}\to p^{+}\pi^{-}$. We train CSD model similarly to DLRun2
model on minimum bias sample. Then, we test every model on simulated run2
sample and each decay channel. For the CSD model an additional round of
training with mixture of minimum bias sample and subset of listed above decays
has been done in order to teach model to generalize in pion domain.

In addition, we design a nearest neighbors test in the following way. For a
correctly reconstructed particle in the decay chain we choose the
kinematically closest particle of minimum bias sample for the different
species. We thus mimic the misreconstruction real-life scenario.

Table 1 contains ROC AOC (i.e. Area over ROC curve) increment with respect to
DLRun2 (uncertainty does not exceed 0.0008 as obtained by a bootstrap
procedure). The result for nearest neighbors test is shown in Fig. 4.

Model | Ghost | Electron | Muon | Pion | Kaon | Proton
---|---|---|---|---|---|---
DLRun2 | 100% | 100% | 100% | 100% | 100% | 100%
ProbNN | $+41\%$ | $+14\%$ | $+418\%$ | $+51\%$ | $20\%$ | $+21\%$
CombDLL | – | $+71\%$ | $+389\%$ | $+979\%$ | $+89\%$ | $+98\%$
CSD | -11% | -42% | -47% | -0.21% | -10% | -11%
Table 1: ROC AOCs change for different PID models for paricle-vs-all task.
DLRun2 AOC is taken as 100%. Figure 4: ROC AUCs for nearest neighbours test,
the background sample is constructed of the best matching candidates as
described in the text.

The quality on those 4 decays itself rises significantly, as it demonstrated
in Table 2.

Model | $D^{*}\to({D^{0}\to K^{-}\pi^{+}})\pi^{-}$ | $\Lambda_{c}\to p^{+}K^{-}\pi^{-}$ | $\Lambda_{b}^{0}\to p^{+}\pi^{-}$ | $D^{0}\to K^{-}\pi^{+}$
---|---|---|---|---
DLRun2 | 100% | 100% | 100% | 100%
CSD | -38% | -71% | -50% | -15%
Table 2: ROC AOC for the pionic decays unseen by the CSD algorithm during
training. DLRun2 AOC is taken as 100%.

In addition to previous tests, we cross-check that the obtained solution has
some generalization power. In order to prove this, we take two decays
containing muons: $B^{+}\to{K^{+}(J/\psi\to{\mu^{+}\mu^{-}})}$ and
$B^{+}\to{\mu^{+}\mu^{-}(K^{*}\to{K^{+}\pi_{0})}}$ and compare CSD and DLRun2
quality. ROC AUC error rate for proposed method almost halved. Note, that
models has never seen muon decays outside a minimum bias sample before. Table
3 illustrates that model can generalize well over different domains.

Model | $B^{+}\to{K^{+}(J/\psi\to{\mu^{+}\mu^{-}})}$ | $B^{+}\to{\mu^{+}\mu^{-}(K^{*}\to{K^{+}\pi_{0})}}$
---|---|---
DLRun2 | 100% | 100%
CSD | -97% | -98%
Table 3: ROC AOC on muonic decays. DLRun2 AOC is taken as 100%.

##  5 Conclusion

The degradation of machine learning solutions quality for the PID problem in
the specific parts of the phase space is addressed. The CSD algorithm based on
the separation of common and domain-specific features has shown promising
results. The algorithm is able to select common features even for the decays
that are not present in the original domains. The obtained algorithm shows
higher stability with respect to previously presented thus giving substantial
increase of solution’s quality for particular case. The method thus can be
used in many high-energy physics applications.

## Acknowledgement

The research leading to these results has received funding from Russian
Science Foundation under grant agreement n° 17-72-20127. This research was
supported in part through computational resources of HPC facilities at HSE
University [8].

## References

  * [1] Derkach D et al.  “Machine-Learning-based global particle-identification algorithms at the LHCb experiment”, J. Phys.: Conf. Ser. 1085 042038, 2018
  * [2] The LHCb Collaboration  “The LHCb Detector at the LHC”, JINST 3 S08005, 2008
  * [3] The LHCb RICH group  “Performance of the LHCb RICH detector at the LHC”, Eur. Phys. J. C 73 2431, 2013
  * [4] Archilli F et al., JINST 8 P10020 (Preprint 1306.0249), 2013
  * [5] Deschamps O, Machefert F P, Schune M H, Pakhlova G and Belyaev I  “Photon and neutral pion reconstruction”, Tech. Rep. LHCb-2003-091 CERN Geneva https://cds.cern.ch/record/691634, 2003
  * [6] Denis Derkach, Mikhail Hushchyn and Nikita Kazeev  “Machine Learning based Global Particle Identification Algorithms at the LHCb Experiment”  In _European Physical Journal Web of Conferences_ 214, European Physical Journal Web of Conferences, 2019, pp. 06011  DOI: 10.1051/epjconf/201921406011
  * [7] Vihari Piratla, Praneeth Netrapalli, Sunita Sarawag  “Efficient Domain Generalization via Common-Specific Low-Rank Decomposition”  In _ICML_ , https://arxiv.org/abs/2003.12815, 2020
  * [8] P.S. Kostenetskiy, R.A. Chulkevich and V.I. Kozyrev  “HPC Resources of the Higher School of Economics”  In _Journal of Physics: Conference Series._ 1740.1, 2021, pp. 012050  DOI: 10.1088/1742-6596/1740/1/012050

python pytorch