# Horofunctions and metric compactification of
noncompact Hermitian symmetric spaces111The authors gratefully acknowledge the
support by the EPSRC (UK) (grant EP/R044228/1)

Cho-Ho Chu Email<EMAIL_ADDRESS>School of Mathematical Sciences, Queen
Mary, University of London,
London E1 4NS, UK  María Cueto-Avellaneda Email<EMAIL_ADDRESS>School of Mathematics, Statistics & Actuarial Science, University of Kent,
Canterbury CT2 7NX, UK  Bas Lemmens Email<EMAIL_ADDRESS>School of
Mathematics, Statistics & Actuarial Science, University of Kent,
Canterbury CT2 7NX, UK

###### Abstract

Given a Hermitian symmetric space $M$ of noncompact type, we give a complete
description of the horofunctions in the metric compactification of $M$ with
respect to the Carathéodory distance, via the realisation of $M$ as the open
unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan
structure, called a $\mathrm{JB}^{*}$-triple. The Carathéodory distance $\rho$
on $D$ has a Finsler structure. It is the integrated distance of the
Carathéodory differential metric, and the norm $\|\cdot\|$ in the realisation
is the Carathéodory norm with respect to the origin $0\in D$. We also identify
the horofunctions of the metric compactification of $(V,\|\cdot\|)$ and relate
its geometry and global topology to the closed dual unit ball (i.e., the polar
of $D$). Moreover, we show that the exponential map $\exp_{0}\colon
V\longrightarrow D$ at $0\in D$ extends to a homeomorphism between the metric
compactifications of $(V,\|\cdot\|)$ and $(D,\rho)$, preserving the geometric
structure. Consequently, the metric compactification of $M$ admits a concrete
realisation as the closed dual unit ball of $(V,\|\cdot\|)$.

Keywords: Hermitian symmetric space, bounded symmetric domain, horofunction,
metric compactification, $\mathrm{JB}^{*}$-triple, Carathéodory distance.

Subject Classification: 32M15, 17C65, 46L70, 53C60.

###### Contents

  1. 1 Introduction
  2. 2 Horofunctions
  3. 3 Jordan algebraic structures
  4. 4 Horofunctions of Hermitian symmetric spaces
  5. 5 Horofunctions of finite dimensional $\mathrm{JB}^{*}$-triples
  6. 6 Homeomorphism onto the dual unit ball
  7. 7 Geometry of $V\cup V(\infty)$
  8. 8 Extension of the exponential map

##  1 Introduction

Compactifications of symmetric spaces is a particularly rich subject, which
has been studied extensively [4, 16]. A variety of compactifications of
symmetric spaces have been introduced with different applications in mind. For
instance, Satake [33] introduced his compactifications in his study of
automorphic forms, and the Martin and Furstenberg compactifications [13, 29]
were introduced to analyse harmonic functions.

Recently, metric (or horofunction) compactifications with respect to invariant
Finsler metrics have been used to investigate different types of
compactifications of symmetric spaces. In particular, it was shown in [17, 34]
that generalised Satake compactifications and Martin compactifications of
symmetric spaces can be realised as metric compactifications under suitable
invariant Finsler metrics. In [22] an explicit invariant Finsler metric was
constructed on symmetric spaces whose metric compactification gives the
maximal Satake compactification, and in [14] the minimal Satake
compactification of $\mathrm{SL}_{n}(\mathbb{R})/\mathrm{SO}_{n}$ was realised
as a metric compactification.

However, for relatively few noncompact type symmetric spaces, with invariant
Finsler metrics, is the metric compactification well understood. In this paper
we completely determine the metric compactifcation of noncompact type
Hermitian symmetric spaces under the Carathéodory distance and provide a
detailed analysis of its geometry and global topology. The Carathéodory
distance plays an important role in the geometry and analysis of Hermitian
symmetric spaces, hence it is natural to consider its metric compactification.

Given a Hermitian symmetric space $M$ of noncompact type, the Harish-Chandra
embedding,

$M\approx\mathcal{D}\subset\mathfrak{p}^{+}\hookrightarrow M_{c},$

identifies $M$ bihomorphically with a bounded symmetric domain $\mathcal{D}$
in a complex Euclidean space $\mathfrak{p}^{+}$, which is biholomorphic to an
open dense subset of the compact dual $M_{c}$ of $M$. By the seminal works of
Loos [28] and Kaup [23], one can equip $\mathfrak{p}^{+}$ with a Jordan
algebraic structure and a norm $\|\cdot\|$ so that $\mathcal{D}$ is
biholomorphic to the open unit ball $D$ of $(\mathfrak{p}^{+},\|\cdot\|)$,
which is called a $\mathrm{JB}^{*}$-triple. Since the Carathéodory distance is
invariant under biholomorphisms, one can transfer from $M$ to the open unit
ball $D$ in the $\mathrm{JB}^{*}$-triple $V=(\mathfrak{p}^{+},\|\cdot\|)$ to
study the metric compactification and exploit Jordan theory and functional
analysis. This is our task in the paper.

For symmetric spaces of noncompact type, the metric compactification with
respect to the Riemannian distance can be identified with its geodesic
compactification ([2] and [19, Proposition 12.6]), and is known to be
homeomorphic to a Euclidean ball, see [5, Chapter II.8]. It has also been
observed for various classes of finite dimensional normed spaces that the
geometry and global topology of the metric compactification is closely related
to the closed dual unit ball of the norm. In [20, 21] this connection was
established for normed spaces with polyhedral unit balls, see also [10]. At
present, however, it is unknown [22, Question 6.18] if this duality phenomenon
holds for general finite dimensional normed spaces.

In [24, 26] it was shown that the duality phenomenon does not only appear in
finite dimensional normed spaces, but also occurs in metric compactifications
of certain symmetric spaces with invariant Finsler norms. More precisely, it
was shown in [24, 26] for symmetric cones equipped with the Thompson and
Hilbert distances that the geometry and global topology of the metric
compactification coincides with the geometry of the closed dual unit ball of
the Finsler norm in the tangent space at the base point of the metric
compactification.

In these symmetric spaces $N$ the connection between the geometry of the
metric compactification and the dual unit ball manifests itself in the
following way. The horofunction boundary $N(\infty)$ in the metric
compactification of the symmetric space $N$ carries an equivalence relation
where two horofunctions $g$ and $h$ are equivalent if $\sup_{x\in
N}|g(x)-h(x)|<\infty$. This relation yields a natural partition of the
horofunction boundary. On the other hand, the boundary of the closed dual unit
ball $B^{*}$ of the Finsler norm on the tangent space $T_{b}N$ at the base
point $b\in N$, is partitioned by the relative interiors of its boundary
faces. For the symmetric spaces $N$ considered in [24, 26] it was shown that
there exists a homeomorphism $\varphi$ from the metric compactification $N\cup
N(\infty)$ onto $B^{*}$, which maps each equivalence class of $N(\infty)$ onto
the relative interior of a boundary face of $B^{*}$. In this sense the dual
ball captures the geometry of the metric compactification.

Moreover, in [24] it was observed that the metric compactification of $N$ is
closely related to the metric compactification of the normed space
$(T_{b}N,\|\cdot\|_{b})$, where $\|\cdot\|_{b}$ is the Finsler norm on the
tangent space $T_{b}N$ at the base point $b\in N$. More explicitly, it was
shown that the exponential map $\exp_{b}\colon T_{b}N\longrightarrow N$
extends to a homeomorphism between the metric compactifications of the normed
space $(T_{b}N,\|\cdot\|_{b})$ and $N$ under the invariant Finsler distance.
Furthermore, the extension preserves the equivalence classes in the
horofunction boundaries. A case in point is the Hilbert distance on
$\mathrm{SL}_{n}(\mathbb{C})/\mathrm{SU}_{n}$, which can be realised as the
manifold of positive-definite Hermitian $n\times n$ matrices with determinant
$1$. The tangent space at the identity $I$ is the real vector space of the
$n\times n$ Hermitian matrices with trace $0$ and has Finsler norm,
$|A|_{I}=\max\sigma(A)-\min\sigma(A)$, where $\sigma(A)$ is the set of
eigenvalues of $A$.

For a noncompact type Hermitian symmetric space $M$ identified as the open
unit ball $D$ of a JB*-triple $V=(\mathfrak{p}^{+},\|\cdot\|)$, with
Carathéodory distance $\rho$, the $\mathrm{JB}^{*}$-triple norm $\|\cdot\|$ is
the corresponding Finsler norm [23, (4.5)]. In this paper we show that the
analogues of the results in [24, 26] hold for the metric compactification of
$(D,\rho)$.

More specifically, the following results are established. We provide a
complete description of the horofunctions of $(D,\rho)$ in Theorem 4.4. We
also determine the horofunctions of the $\mathrm{JB}^{*}$-triple
$(V,\|\cdot\|)$ in Theorem 5.2, and establish in Theorem 6.5 an explicit
homeomorphism $\varphi$ between the horofunction compactification of
$(V,\|\cdot\|)$ and its closed dual unit ball. Further, we show in Section 7
that the homeomorphism $\varphi$ maps each equivalence class in the
horofunction boundary of $(V,\|\cdot\|)$ onto the relative interior of a
boundary face of the closed dual unit ball. Finally we prove in Section 8 that
the exponential map $\operatorname{\mathrm{exp}_{0}}\colon V\longrightarrow D$
extends to a homeomorphism between the metric compactifications $V\cup
V(\infty)$ and $D\cup D(\infty)$, which maps equivalence classes onto
equivalence classes in the horofunction boundaries, see Theorem 8.1. Combining
the results we see that the geometry and global topology of the metric
compactification of $(D,\rho)$ coincides with the geometry of the closed dual
unit ball of $(V,\|\cdot\|)$.

We start by recalling the essential background on metric compactifications and
the theory of $\mathrm{JB}^{*}$-triples in Sections 2 and 3, respectively.

##  2 Horofunctions

The origins of the idea of the metric (or horofunction) compactification go
back to Gromov [2]. It has proven to be a valuable tool in numerous fields
such as, geometric group theory, complex and real dynamics, Riemannian
geometry, and geometric analysis. It captures asymptotic geometric properties
of metric spaces and provides ways to analyse mappings or groups acting on
them. Here we mostly follow the set up as in [31].

Let $(X,d)$ be a metric space and let $\mathbb{R}^{X}$ be the space of all
real functions on $X$ equipped with the topology of pointwise convergence. Fix
a point $b\in X$, called the base point, and write $\mathrm{Lip}^{1}_{b}(X)$
to denote the set of all functions $h\in\mathbb{R}^{X}$ such that $h(b)=0$ and
$h$ is $1$-Lipschitz, i.e., $|h(x)-h(y)|\leq d(x,y)$ for all $x,y\in X$.

The set $\mathrm{Lip}^{1}_{b}(X)$ is a compact subset of $\mathbb{R}^{X}$.
Indeed, it is easy to verify that the complement of $\mathrm{Lip}^{1}_{b}(X)$
is open. Moreover, as $|h(x)|=|h(x)-h(b)|\leq d(x,b)$ for all $x\in X$ and
$h\in\mathrm{Lip}^{1}_{b}(X)$, we have that
$\mathrm{Lip}_{b}^{1}(X)\subseteq[-d(x,b),d(x,b)]^{X}$. The set
$[-d(x,b),d(x,b)]^{X}$ is compact by Tychonoff’s theorem, and hence
$\mathrm{Lip}^{1}_{b}(X)$ is a compact subset of $\mathbb{R}^{X}$.

For $y\in X$ define the real valued function,

$h_{y}(z)=d(z,y)-d(b,y)\qquad(z\in X).$ (2.1)

Note that $h_{y}(b)=0$ and $|h_{y}(z)-h_{y}(w)|=|d(z,y)-d(w,y)|\leq d(z,w)$
for all $z,w\in X$, and hence $h_{y}\in\mathrm{Lip}_{b}^{1}(X)$.

######  Definition 2.1.

Denote the closure of $\\{h_{y}\colon y\in X\\}$ in $\mathbb{R}^{X}$ by
$\overline{X}$, which is compact. The set
$X(\infty)=\overline{X}\setminus\\{h_{y}\colon y\in X\\}$ is called the
horofunction boundary of $(X,d)$. The elements of $X(\infty)$ are called
horofunctions, and the set $X\cup X(\infty)$ is called the metric (or
horofunction) compactification of $(X,d)$. The elements of $\overline{X}$ are
called metric functionals, and the metric functionals $h_{y}$ in (2.1) are
called internal points.

The topology of pointwise convergence on $\mathrm{Lip}_{b}^{1}(X)$ coincides
with the topology of uniform convergence on compact sets, see [30, p. 291]. In
general the topology of pointwise convergence on $\mathrm{Lip}_{b}^{1}(X)$ is
not metrizable, and hence horofunctions are limits of nets rather than
sequences. However, if the metric space is separable, then the topology is
metrizable and each horofunction is the limit of a sequence. In fact, one can
verify that given a countable dense subset $\\{y_{k}\colon k\in\mathbb{N}\\}$
of $(X,d)$, the function $\sigma$ on
$\mathrm{Lip}_{b}^{1}(X)\times\mathrm{Lip}_{b}^{1}(X)$ given by,

$\sigma(f,g)=\sum_{k}2^{-k}\min\\{1,|f(y_{k})-g(y_{k})|\\}\mbox{\quad for
$f,g\in\mathrm{Lip}_{b}^{1}(X)$,}$

is a metric whose topology coincides with the pointwise convergence topology
on $\mathrm{Lip}_{b}^{1}(X)$, [30, p. 289, Ex. 10*].

The metric compactification may not be a compactification in the usual
topological sense, as the embedding $\iota\colon y\in X\mapsto
h_{y}\in\mathrm{Lip}_{b}^{1}(X)$ may fail to have the necessary properties.
However, the embedding $\iota\colon X\longrightarrow\iota(X)$ is always a
continuous bijection. Indeed, if $x\in X$ and we consider a neighbourhood
$U=\\{h\in\mathrm{Lip}_{b}^{1}(X)\colon|h(y)-h_{x}(y)|<\varepsilon,\,y\in
X\\}$ of $h_{x}$ for $\varepsilon>0$, then for $z\in X$ with
$d(x,z)<\varepsilon/2$ we have that
$|h_{z}(y)-h_{x}(y)|\leq|d(y,z)-d(y,x)|+|d(b,z)-d(b,x)|\leq
2d(z,x)<\varepsilon,$ and hence $\iota(z)\in U$, which shows that $\iota$ is
continuous. Moreover, if $x,z\in X$ and $h_{x}=h_{z}$, then
$0=(h_{z}(x)-h_{x}(x))+(h_{x}(z)-h_{z}(z))=2d(x,z)$, which gives injectivity.

It can happen that $\iota\colon X\longrightarrow\iota(X)$ does not have a
continuous inverse. If, however, $(X,d)$ is proper and geodesic, then the
metric compactification will be a compactification in the usual topological
sense. We provide some details of this fact below. Recall that a metric space
$(X,d)$ is proper if all its closed balls are compact. Note that a proper
metric space is separable, as each closed ball is compact and hence separable.

A map $\gamma$ from a, possibly unbounded, interval $I\subseteq\mathbb{R}$
into a metric space $(X,d)$ is called a geodesic path if

$d(\gamma(s),\gamma(t))=|s-t|\mbox{\quad for all }s,t\in I.$

The image, $\gamma(I)$, is called a geodesic. A metric space $(X,d)$ is said
to be geodesic if for each $x,y\in X$ there exists a geodesic path
$\gamma\colon[a,b]\longrightarrow X$ connecting $x$ and $y$, i.e,
$\gamma(a)=x$ and $\gamma(b)=y$.

Hermitian symmetric spaces with Carathéodory distance are proper geodesic
metric spaces (cf. [15]). In the discussion below we will focus on the metric
compactification of such metric spaces.

The horofunctions of a proper geodesic metric space $(X,d)$ are precisely the
limits of converging sequences $(h_{x_{k}})$ such that
$d(b,x_{k})\longrightarrow\infty$. A slightly stronger assertion was shown in
[31, Theorem 4.7], but for our purposes the following statement will suffice,
see also [25, Lemma 2.1].

######  Lemma 2.2.

If $(X,d)$ is a proper geodesic metric space, then $h\in X(\infty)$ if and
only if there exists a sequence $(x_{k})$ in $X$ with
$d(b,x_{k})\longrightarrow\infty$ such that $(h_{x_{k}})$ converges to
$h\in\overline{X}$ as $k\longrightarrow\infty$.

We use this lemma to show that, in this case, the embedding $\iota\colon
X\longrightarrow\iota(X)$ has a continuous inverse.

######  Lemma 2.3.

If $(X,d)$ is a proper geodesic metric space, then $\iota\colon
X\longrightarrow\iota(X)$ is a homeomorphism.

###### Proof.

From the previous observations it remains to show that $\iota\colon
X\longrightarrow\iota(X)$ has a continuous inverse. Let
$h_{z_{0}}=\iota(z_{0})$ where $z_{0}\in X$. Note that as $(X,d)$ is proper it
is also separable. So, to prove continuity of $\iota^{-1}$ at $h_{z_{0}}$, we
can use sequences, as the topology of pointwise convergence on
$\mathrm{Lip}_{b}^{1}(X)$ is metrizable. Let $(z_{k})$ be a sequence in $X$
with $h_{z_{k}}\longrightarrow h_{z_{0}}$. By Lemma 2.2 we know that $(z_{k})$
is bounded, and hence after taking a subsequence we may assume that
$z_{k}\longrightarrow z$. It follows that $h_{z}=h_{z_{0}}$, and hence the
injectivity of $\iota$ implies that $z=z_{0}$, which completes the proof. ∎

Thus, the metric compactification is a compactification in the usual
topological sense if $(X,d)$ is a proper geodesic space.

Special horofunctions come from so-called almost geodesics sequences. They
were introduced by Rieffel [31] and further developed by Walsh and co-workers
in [1, 27, 35, 36]. A sequence $(x_{k})$ in $(X,d)$ is called an almost
geodesic if for each $\varepsilon>0$ there exists an $N\geq 0$ such that

$d(x_{n},x_{m})+d(x_{m},x_{0})-d(x_{n},x_{0})<\varepsilon\mbox{\quad for all
}n\geq m\geq N.$

In particular, every unbounded almost geodesic sequence yields a horofunction
for a proper geodesic metric space, see [31].

######  Lemma 2.4.

Let $(X,d)$ be a proper geodesic metric space. If $(x_{k})$ is an unbounded
almost geodesic in $(X,d)$, then

$h(z)=\lim_{k}d(z,x_{k})-d(b,x_{k})$

exists for all $z\in X$ and $h\in X(\infty)$.

Given a proper geodesic metric space $(X,d)$, a horofunction $h$ is called a
Busemann point if there exists an almost geodesic $(x_{k})$ in $X$ such that
$h(z)=\lim_{k}d(z,x_{k})-d(b,x_{k})$ for all $z\in X$. The collection of all
Busemann points is denoted by $\mathcal{B}_{X}$.

The set of Busemann points can be equipped with a metric known as the detour
distance, which was introduced in [1], and is defined as follows. Suppose
$(X,d)$ is a proper geodesic metric space and $h,h^{\prime}\in X(\infty)$ are
horofunctions. Let $W_{h}$ be the collection of neighbourhoods of $h$ in
$\overline{X}$. Then the detour cost is given by

$H(h,h^{\prime})=\sup_{W\in W_{h}}\left(\inf_{x\colon\iota(x)\in
W}d(b,x)+h^{\prime}(x)\right),$

and the detour distance is defined by

$\delta(h,h^{\prime})=H(h,h^{\prime})+H(h^{\prime},h).$

It is known [27, 36] that if $(x_{k})$ is an almost geodesic converging to a
horofunction $h$, then

$H(h,h^{\prime})=\lim_{k}d(b,x_{k})+h^{\prime}(x_{k})$ (2.2)

for all horofunctions $h^{\prime}$. Moreover, on the set of Busemann points
$\mathcal{B}_{X}$ the detour distance is a metric where points can be at
infinite distance from each other, see [27, 36].

The detour distance induces a partition of $\mathcal{B}_{X}$ into equivalence
classes, called parts, where $h$ and $h^{\prime}$ in $\mathcal{B}_{X}$ are
equivalent if $\delta(h,h^{\prime})<\infty$. In particular, if all
horofunctions are Busemann points, so $\mathcal{B}_{M}=X(\infty)$, then
$X(\infty)$ is the disjoint union of parts, each of which is a metric space
under the detour distance.

The horofunction boundary $X(\infty)$ has a natural partition induced by the
equivalence relation:

$h\sim h^{\prime}\mbox{\quad if \quad}\sup_{x\in
X}|h(x)-h^{\prime}(x)|<\infty.$

It was shown in [36, Proposition 4.5] that two Busemann points $h$ and
$h^{\prime}$ in $X(\infty)$ have finite detour distance if, and only if,
$h\sim h^{\prime}$. Thus, if all horofunctions are Busemann points, then the
partition of $X(\infty)$ into parts coincides with the partition into
equivalence classes $X(\infty)/\sim$.

We also like to note that each (surjective) isometry $\psi\colon
X\longrightarrow X$ extends as a homeomorphism to $X(\infty)$ by

$\psi(h)(x)=h(\psi^{-1}(x))-h(\psi^{-1}(b))\qquad(x\in X,h\in X(\infty)).$

It is known that on the Busemann points in $X(\infty)$ the extension $\psi$ is
an isometry under the detour distance, see e.g., [27]. Also if $h\sim g$ in
$X(\infty)$, then $\psi(h)\sim\psi(g)$.

##  3 Jordan algebraic structures

In this section, we recall some necessary definitions and results concerning
Jordan algebraic structures associated with Hermitian symmetric spaces.
Throughout, $M$ will denote an arbitrary Hermitian symmetric space of
noncompact type. To determine the horofunctions of $M$ and analyse the
geometry and global topology of its metric compactification with respect to
the Carathéodory distance (and base point $b\in M$), we make use of the fact
that $M$ can be realised as the open unit ball in a $\mathrm{JB}^{*}$-triple.
Indeed, there exists a biholomorphism,

$\psi\colon M\leavevmode\nobreak\ \leavevmode\nobreak\
^{\Large\underrightarrow{\psi_{1}}}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{D}\leavevmode\nobreak\ \leavevmode\nobreak\
^{\Large\underrightarrow{\psi_{2}}}\leavevmode\nobreak\ \leavevmode\nobreak\
D\subset V,$ (3.1)

onto the open unit ball $D$ of a finite dimensional $\mathrm{JB}^{*}$-triple
$V$, with $\psi(b)=0$, where $\psi_{1}$ is the Harish-Chandra embedding and
$\psi_{2}$ is the Kaup Riemann mapping, which is unique up to a linear
isometry [23, Theorem 4.9].

Since the biholomorphism $\psi$ preserves the Carathéodory distance, it
induces a homeomorphism between the metric compactifications of $M$ and $D$.
Hence we can, and will, work in the setting

$D\subset V,\mbox{ where $D$ is the open unit ball of a finite dimensional
$\mathrm{JB}^{*}$-triple $V$.}$

The results can be transferred to the corresponding ones for $M$ via $\psi$.

Let us now present some of the essential background on Jordan triple systems,
where we focus on the finite dimensional case. A $\mathrm{JB}^{*}$-triple is a
complex Banach space $V$ equipped with a continuous triple product

$\\{\cdot,\cdot,\cdot\\}\colon V\times V\times V\longrightarrow V,$

called a Jordan triple product, which is linear and symmetric in the outer
variables, and conjugate linear in the middle variable, and satisfies the
following axioms:

  1. (i)

$\\{a,b,\\{x,y,z\\}\\}=\\{\\{a,b,x\\},y,z\\}-\\{x,\\{b,a,y\\},z\\}+\\{x,y,\\{a,b,z\\}\\}$,
(Jordan triple identity)

  2. (ii)

$a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern
4.0pt\vrule width=0.4pt}\hrule height=0.4pt}}\,}\,a$ is Hermitian, that is,
$\|\exp it(a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a)\|=1$ for all $t\in\mathbb{R}$;

  3. (iii)

$a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern
4.0pt\vrule width=0.4pt}\hrule height=0.4pt}}\,}\,a$ has nonnegative spectrum
$\sigma(a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a)$;

  4. (iv)

$\|a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a\|=\|a\|^{2},\qquad$

for $a,b,x,y,z\in V$, where $a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,b:V\longrightarrow V$ is a bounded linear map, called a box
operator, defined by

$a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern
4.0pt\vrule width=0.4pt}\hrule height=0.4pt}}\,}\,b(x)=\\{a,b,x\\}\qquad(x\in
V)$ (3.2)

and condition (iv) can be replaced by

$\|\\{a,a,a\\}\|=\|a\|^{3}\qquad(a\in V).$ (3.3)

We note that the box operator in (3.2) satisfies $\|a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,b\|\leq\|a\|\|b\|$, as the triple
product is a contractive mapping, that is,

$\|\\{a,b,c\\}\|\leq\|a\|\|b\|\|c\|\quad{\rm for\leavevmode\nobreak\
all}\leavevmode\nobreak\ a,b,c\in V.$

######  Remark 3.1.

By definition, a Hermitian operator $T\colon V\longrightarrow V$ has real
numerical range, which is the closed convex hull of its spectrum $\sigma(T)$
and $\|T\|=\sup\\{|\lambda|\colon\lambda\in\sigma(T)\\}$ [3, pp. 46–54]. In
particular, given $a,b$ in a $\mathrm{JB}^{*}$-triple, (i) and (ii) above
implies

$\|a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a\|=\sup\\{\lambda\colon\lambda\in\sigma(a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a)\\}$

and $a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,b+b{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a$ is Hermitian. Further, if $\|a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,b+b{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a\|\leq 1$, then we have
$\sigma(a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,b+b{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a)\subset[-1,1]$.

A prime example of a JB*-triple is the space of $p\times q$ complex matrices
$M_{p,q}(\mathbb{C})$ with Jordan triple product,

$\\{A,B,C\\}=\frac{1}{2}(AB^{*}C+CB^{*}A)\qquad(A,B,C\in
M_{p,q}(\mathbb{C})),$

which has open unit ball $D=\\{A\in M_{p,q}(\mathbb{C})\colon I-AA^{*}\mbox{
positive definite}\\}$. In particular if $q=1$, we get the complex Euclidean
space $\mathbb{C}^{p}$ with Jordan triple product

$\\{x,y,z\\}=\frac{1}{2}(\langle x,y\rangle z+\langle z,y\rangle
x)\qquad(x,y,z\in\mathbb{C}^{p})$

and $D=\\{z\in\mathbb{C}^{p}\colon\langle z,z\rangle<1\\}$ is the Euclidean
ball.

Given $\mathrm{JB}^{*}$-triples $V_{1},\ldots,V_{d}$, the direct sum
$V_{1}\oplus\cdots\oplus V_{d}$, with the $\ell_{\infty}$-norm,

$\|(a_{1},\ldots,a_{d})\|_{\infty}=\max\\{\|a_{i}\|\colon
i=1,\ldots,d\\},\quad(a_{i}\in V_{i})$

is a $\mathrm{JB}^{*}$-triple with the coordinatewise triple product.

In fact, a finite dimensional $\mathrm{JB}^{*}$-triple $V$ decomposes into a
finite direct sum $V_{1}\oplus\cdots\oplus V_{d}$ of so-called Cartan factors
$V_{j}$ ($j=1,\dots,d$) with $\ell_{\infty}$-norm. There are six different
types of (finite dimensional) Cartan factors:

(1) $M_{p,q}(\mathbb{C})$ (2) $S_{q}(\mathbb{C})$ (3) $H_{q}(\mathbb{C})$ (4)
$Sp_{n}(\mathbb{C})$ (5) $M_{1,2}(\mathcal{O})$ (6) $H_{3}(\mathcal{O})$,
where $S_{q}(\mathbb{C})$ and $H_{q}(\mathbb{C})$ are norm closed subspaces of
$M_{p,q}(\mathbb{C})$ consisting of $p\times q$ skew-symmetric and symmetric
matrices, respectively; and $Sp_{n}(\mathbb{C})$ is a spin factor of dimension
$n>2$. The Cartan factors of types 5 and 6 are _exceptional_ Cartan factors
(cf. [6, Theorem 2.5.9]).

There are various operators that play an important role in the theory of
$\mathrm{JB}^{*}$-triples. Besides the box operators, we will use the Bergman
operator $B(b,c)\colon V\longrightarrow V$ and the Möbius transformation
$g_{a}\colon D\longrightarrow D$, where $a\in D$ and $b,c\in V$, which are
defined as follows:

$B(b,c)(x)=x-2(b{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c)(x)+\\{b,\\{c,x,c\\},b\\}\qquad(x\in V),$ (3.4)
$g_{a}(x)=a+B(a,a)^{1/2}(I+x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a)^{-1}(x)\qquad(x\in D).$ (3.5)

Here $I$ denotes the identity operator on $V$, and the inverse
$(I+x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a)^{-1}\colon V\longrightarrow V$ exists, as
$\|x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a\|\leq\|x\|\|a\|<1$.

We note that $B(a,b)$ is invertible for $\|a\|\|b\|<1$. The proof of the
following two identities can be found in [6, Proposition 3.2.13. Lemma
3.2.17].

$\|B(z,z)^{-1/2}\|=\frac{1}{1-\|z\|^{2}}\qquad(\|z\|<1),$ (3.6)
$1-\|g_{-y}(z)\|^{2}=\frac{1}{\|B(z,z)^{-1/2}B(z,y)B(y,y)^{-1/2}\|}\qquad(y,z\in
D).$ (3.7)

For the Euclidean ball $D\subset\mathbb{C}^{d}$ with inner product
$\langle\cdot,\cdot\rangle$, we have from [7, Example 3.2.29] the formula

$1-\|g_{-y}(z)\|^{2}=\frac{(1-\|y\|^{2})(1-\|z\|^{2})}{|1-\langle
y,z\rangle|^{2}}\qquad(y,z\in D).$ (3.8)

Given $a\in V$, the quadratic operator $Q_{a}\colon V\longrightarrow V$ is
defined by

$Q_{a}(x)=\\{a,x,a\\}\qquad(x\in V).$

An element $e$ in a $\mathrm{JB}^{*}$-triple $V$ is called a tripotent if
$\\{e,e,e\\}=e$. Although $0$ is a tripotent in a $\mathrm{JB}^{*}$-triple, we
are only interested in the nonzero ones, of which the norm is always $1$.
Tripotents in C*-algebras are exactly the partial isometries.

Any tripotent $e$ in $V$ induces an eigenspace decomposition of $V$, called
the Peirce decomposition associated with $e$. The eigenvalues of the box
operator $e{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e\colon V\longrightarrow V$ are in the set $\\{0,1/2,1\\}$.
Let

$V_{k}(e)=\\{x\in V\colon(e{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e)(x)=\frac{k}{2}x\\}\qquad(k=0,1,2)$

be the corresponding eigenspaces, called the Peirce $k$-space of $e$. We have
the algebraic direct sum

$V=V_{0}(e)\oplus V_{1}(e)\oplus V_{2}(e).$

where the Peirce $k$-spaces satisfy

$\\{V_{i}(e),V_{j}(e),V_{k}(e)\\}\subseteq V_{i-j+k}(e)$ (3.9)

if $i-j+k$ belongs to the set $\\{0,1,2\\},$ and
$\\{V_{i}(e),V_{j}(e),V_{k}(e)\\}=\\{0\\}$ otherwise. Further, we have

$\\{V_{2}(e),V_{0}(e),V\\}=\\{V_{0}(e),V_{2}(e),V\\}=0.$ (3.10)

The Peirce $k$-space $V_{k}(e)$ is the range of the Peirce $k$-projection
$P_{k}(e)\colon V\longrightarrow V$, which are contractive and given by

$P_{2}(e)=Q_{e}^{2},\quad P_{1}(e)=2(e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e-Q_{e}^{2}),\quad P_{0}(e)=B(e,e).$

A tripotent $e$ in a JB*-triple $V$ is called minimal if
$Q_{e}(V)=\mathbb{C}e$, or equivalently, $V_{2}(e)=\mathbb{C}e$. It is called
maximal if $V_{0}(e)=\\{0\\}$. In fact, the maximal tripotents in $V$ coincide
with the extreme points of the closed unit ball $V$ (cf. [6, Theorem 3.2.3]).

We note that, with the inherited norm from $V$, the Peirce 2-space $V_{2}(e)$
is a $\mathrm{JB}^{*}$-algebra with identity $e$, Jordan product and
involution

$x\circ y=\\{x,e,y\\},\quad x^{*}=\\{e,x,e\\}=Q_{e}x,\quad(x,y\in V_{2}(e))$
(3.11)

respectively [7, Example 2.4.18]. In particular, we have

$\|x\|=\|x^{*}\|=\|Q_{e}x\|\qquad(x\in V_{2}(e)).$ (3.12)

We refer to [7, Definition 2.4.16] for the definition of a JB*-algebra, which
are examples of JB∗-triples [6, Lemma 3.1.6].

Let

$A(e)=\\{x\in V_{2}(e)\colon x^{*}=x\\}=\\{x\in
V_{2}(e)\colon\\{e,x,e\\}=x\\}$ (3.13)

be the self-adjoint part of $V_{2}(e)$. Then it is a closed real subalgebra of
$(V_{2}(e),\circ)$ satisfying

$\|a\|^{2}=\|a^{2}\|\leq\|a^{2}+b^{2}\|,\qquad(a,b\in A(e))$

where $a^{2}=a\circ a$, in other words, it is a so-called JB-algebra [18,
3.1.4].

There is a natural partial ordering $\leq$ on $A(e)$ defined by the closed
cone

$A(e)_{+}=\\{x^{2}:x\in A(e)\\}$

where $x\leq y$ if and only if $y-x\in A(e)_{+}$. We will make use of the fact
that $\\{a,A(e)_{+},a\\}\subset A(e)_{+}$ for all $a\in A(e)$, and

$\|a\|\leq 1\mbox{ if and only if }-e\leq a\leq e$ (3.14)

(cf. [18, Proposition 3.3.6; 3.1.5]). An element $a\in V_{2}(e)$ is called
invertible if there is a (unique) element $a^{-1}$, called the inverse of $a$,
such that $a\circ a^{-1}=e$ and $a^{2}\circ a^{-1}=a$. If $a\in A(e)$, then
$a^{-1}\in A(e)$.

Given $a,b\in V$, we say that $a$ is _orthogonal_ to $b$ if
$a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern
4.0pt\vrule width=0.4pt}\hrule height=0.4pt}}\,}\,b=0$. It is known that $a$
is orthogonal to $b$ if and only if $\\{a,a,b\\}=0$. Moreover, $a$ orthogonal
to $b$ implies $b$ orthogonal to $a$, in which case we have

$\|a+b\|=\max\\{\|a\|,\|b\|\\}$

from [6, Corollary 3.1.21].

A linear subspace $W\subset V$ of a JB∗-triple $V$ is called a JB∗-subtriple
if $x,y,z\in W$ implies $\\{x,y,z\\}\in W$, in the inherited Jordan triple
product.

The rank $r$ of a finite dimensional JB∗-triple $V$ is defined by

$r=\sup\\{\hbox{dim}V(a)\colon a\in V\\},$

where $V(a)$ denotes the smallest closed subtriple of $V$ containing $a\in V$.
It can be shown that $r$ is the maximal number of mutually orthogonal
tripotents in $V$ [7, Example 3.3.3].

Let $\\{e_{1},\dots,e_{n}\\}$ a family of mutually orthogonal tripotents in a
JB∗-triple $V$. For $i,j\in\\{0,1,\ldots,n\\}$, the joint Peirce space
$V_{ij}$ is defined by

$\displaystyle V_{ij}=V_{ij}(e_{1},\dots,e_{n})=\\{z\in V\colon
2\\{e_{k},e_{k},z\\}=(\delta_{ik}+\delta_{jk})z\mbox{ for }k=1,\ldots,n\\},$
(3.15)

where $\delta_{ij}$ is the Kronecker delta and $V_{ij}=V_{ji}$.

The decomposition

$\displaystyle V$ $\displaystyle=$ $\displaystyle\bigoplus_{0\leq i\leq j\leq
n}V_{ij}$

is called a joint Peirce decomposition.

The Peirce multiplication rules

$\displaystyle\\{V_{ij},V_{jk},V_{k\ell}\\}$ $\displaystyle\subset$
$\displaystyle V_{i\ell}\quad{\rm and}\quad V_{ij}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,V_{pq}=\\{0\\}\quad{\rm for}\quad
i,j\notin\\{p,q\\}$

hold. The contractive projection $P_{ij}(e_{1},\ldots,e_{n})$ from $V$ onto
$V_{ij}(e_{1},\ldots,e_{n})$ is called a joint Peirce projection which
satisfies

$P_{ij}(e_{1},\dots,e_{n})(e_{k})=\left\\{\begin{array}[]{ll}0&(i\neq j)\\\
\delta_{ik}e_{k}&(i=j).\end{array}\right.$ (3.16)

We shall simplify the notation $P_{ij}(e,\ldots,e_{n})$ to $P_{ij}$ if the
tripotents $e_{1},\ldots,e_{n}$ are understood.

Let $M=\\{0,1,\ldots,n\\}$ and $N\subset\\{1,\ldots,n\\}$. The Peirce
$k$-spaces of the tripotent $e_{N}=\sum_{i\in N}e_{i}$ are given by

$\displaystyle V_{2}(e_{N})$ $\displaystyle=$ $\displaystyle\bigoplus_{i,j\in
N}V_{ij},$ (3.17) $\displaystyle V_{1}(e_{N})$ $\displaystyle=$
$\displaystyle\bigoplus_{\begin{subarray}{c}i\in N\\\ j\in M\backslash
N\end{subarray}}V_{ij},$ $\displaystyle V_{0}(e_{N})$ $\displaystyle=$
$\displaystyle\bigoplus_{i,j\in M\backslash N}V_{ij}.$

The Peirce projections provide a very useful formulation of the Bergman
operators. Let $e_{1},\ldots,e_{n}$ be mutually orthogonal tripotents in a
JB*-triple $V$ and let $x=\sum_{i=1}^{n}\lambda_{i}e_{i}$ with
$\lambda_{i}\in\mathbb{C}$. Then the Bergmann operator $B(x,x)$ satisfies

$\displaystyle B(x,x)$ $\displaystyle=$ $\displaystyle\sum_{0\leq i\leq j\leq
n}(1-|\lambda_{i}|^{2})(1-|\lambda_{j}|^{2})P_{ij},$

where we set $\lambda_{0}=0$ and $P_{ij}=P_{ij}(e_{1},\ldots,e_{n})$. This
gives the following formulae for the square roots

$\displaystyle B(x,x)^{1/2}$ $\displaystyle=$ $\displaystyle\sum_{0\leq i\leq
j\leq
n}(1-|\lambda_{i}|^{2})^{1/2}(1-|\lambda_{j}|^{2})^{1/2}P_{ij}\quad(\|x\|<1),$
(3.18) $\displaystyle B(x,x)^{-1/2}$ $\displaystyle=$
$\displaystyle\sum_{0\leq i\leq j\leq
n}(1-|\lambda_{i}|^{2})^{-1/2}(1-|\lambda_{j}|^{2})^{-1/2}P_{ij}\quad(\|x\|<1).$
(3.19)

The following lemma will be useful later for computing the horofunctions in a
noncompact Hermitian symmetric space.

######  Lemma 3.2.

Let $D$ be the open unit ball of a JB*-triple $V$. Given a sequence $(y_{k})$
in $D$ such that $y_{k}\longrightarrow\xi\in\partial D$, we have

$\lim_{k}\|g_{-y_{k}}(z)\|=1\qquad(z\in D).$

###### Proof.

By (3.6) and (3.7), we have

$\displaystyle 0<1-\|g_{-y_{k}}(z)\|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\|B(z,z)^{-1/2}B(z,y_{n})B(y_{k},y_{k})^{-1/2}\|}$
$\displaystyle\leq$
$\displaystyle\frac{\|B(z,y_{k})^{-1}B(z,z)^{1/2}\|}{\|B(y_{k},y_{k})^{-1/2}\|}$
$\displaystyle=$
$\displaystyle\|B(z,y_{k})^{-1}B(z,z)^{1/2}\|(1-\|y_{k}\|^{2})\longrightarrow
0\qquad(z\in D)$

as $k\rightarrow\infty$, since
$\lim_{k}\|B(z,y_{k})^{-1}B(z,z)^{1/2}\|=\|B(z,\xi)^{-1}B(z,z)^{1/2}\|$. ∎

For more details of JB*-triples, we refer to [7] and the references therein.

##  4 Horofunctions of Hermitian symmetric spaces

Given the open unit ball $D$ in a finite dimensional $\mathrm{JB}^{*}$-triple
$V$, we now determine the horofunctions of $D$ under the Carathéodory distance
$\rho$, with the origin $0\in D$ as a base point . Recall that the
horofunctions of the corresponding Hermitian symmetric space $M$ with base
point $b\in M$ can be obtained via $\psi$ in (3.1), with $\psi(b)=0$, where
$\psi$ preserves the Carathéodory distance. In fact, we have

$M(\infty)=\\{h\circ\psi\colon h\in D(\infty)\\}.$

The Carathéodory distance $\rho$ on $D$ is given by

$\rho(x,y)=\sup\\{\omega(f(x),f(y))\colon f\in H(D,\mathbb{D})\\}\qquad(x,y\in
D),$

where $H(D,\mathbb{D})$ is the set of all holomorphic functions $f\colon
D\longrightarrow\mathbb{D}$ and $\omega$ is the Poincaré distance of the unit
disc $\mathbb{D}=\\{z\in\mathbb{C}\colon|z|<1\\}$. We will make use of the
formula (cf. [7, Theorem 3.5.9]):

$\rho(x,y)=\tanh^{-1}\|g_{-x}(y)\|.$

For each $y,z\in D$ we have

$h_{y}(z)=\rho(z,y)-\rho(0,y)=\frac{1}{2}\log\frac{1+\|g_{-y}(z)\|}{1-\|g_{-y}(z)\|}-\frac{1}{2}\log\frac{1+\|y\|}{1-\|y\|}=\frac{1}{2}\log\left(\frac{1-\|y\|^{2}}{1-\|g_{-y}(z)\|^{2}}\left(\frac{1+\|g_{-y}(z)\|}{1+\|y\|}\right)^{2}\right),$
(4.1)

which can also be written as

$h_{y}(z)=\frac{1}{2}\log\left(\frac{1-\|y\|^{2}}{1-\|g_{-z}(y)\|^{2}}\left(\frac{1+\|g_{-z}(y)\|}{1+\|y\|}\right)^{2}\right).$
(4.2)

######  Lemma 4.1.

Let $D$ be the open unit in a finite dimensional $\mathrm{JB}^{*}$-triple $V$,
and $\rho$ the Carathéodory distance on $D$. Then $h\in D(\infty)$ if and only
if there exists a sequence $(y_{k})$ in $D$ with
$y_{k}\longrightarrow\xi\in\partial D$ such that

$h(z)=\lim_{k}\frac{1}{2}\log\left(\frac{1-\|y_{k}\|^{2}}{1-\|g_{-y_{k}}(z)\|^{2}}\right)=\lim_{k}\frac{1}{2}\log\left(\frac{1-\|y_{k}\|^{2}}{1-\|g_{-z}(y_{k})\|^{2}}\right)$
(4.3)

for all $z\in D$.

###### Proof.

Suppose that $h\in D(\infty)$. Then by Lemma 2.2 we know that there exists a
sequence $(y_{k})$ in $D$ with $\rho(0,y_{k})\longrightarrow\infty$ and
$\lim_{k}h_{y_{k}}(z)$ exists for all $z\in D$. By taking a subsequence we may
assume that $y_{k}\longrightarrow\xi\in\partial D$. The implication now
follows from (4.1) and Lemma 3.2. On the other hand, if there exists a
sequence $(y_{k})$ in $D$ with $y_{k}\longrightarrow\xi\in\partial D$ such
that (4.3) holds for all $z\in D$, then
$\rho(0,y_{k})=\tanh^{-1}\|y_{k}\|\longrightarrow\infty$, as
$\|y_{k}\|\longrightarrow 1$. So, we deduce from (4.1) and Lemma 2.2 that
$h\in D(\infty)$. ∎

######  Remark 4.2.

The notion of a horofunction $h$ on $D$ is essentially the same as the
function $F$ introduced in [9, Lemma 4.1]. Indeed, the formula (4.3) in Lemma
4.1 for $h$ is related to $F$ by

$h(x)=\frac{1}{2}\log F(x)\qquad(x\in D).$

In case $D$ is the open Euclidean ball in $\mathbb{C}^{d}$ and
$y_{k}\longrightarrow\xi$ with $\|\xi\|_{2}=1$, we find that

$h(z)=\lim_{k}\frac{1}{2}\log\left(\frac{1-\|y_{k}\|^{2}}{1-\|g_{-y_{k}}(z)\|^{2}}\right)$

is a horofunction, and from (3.8) we have

$h(z)=\lim_{k}\frac{1}{2}\log\left(\frac{|1-\langle
z,y_{k}\rangle|^{2}}{1-\|z\|^{2}}\right)=\frac{1}{2}\log\left(\frac{|1-\langle
z,\xi\rangle|^{2}}{1-\|z\|^{2}}\right).$

In particular, for the disc $\mathbb{D}\subset\mathbb{C}$ we find that
$h\colon\mathbb{D}\longrightarrow\mathbb{R}$ is given by the well known
expression,

$h(z)=\frac{1}{2}\log\frac{|1-z\overline{\xi}|^{2}}{1-|z|^{2}}=\frac{1}{2}\log\frac{|\xi-z|^{2}}{1-|z|^{2}}\qquad(z\in\mathbb{D}).$

We now compute the limit in (4.3) for general $D$. By [7, Lemma 3.2.28], one
can express the limit $h=\lim_{k}h_{y_{k}}$ in (4.3) in terms of the Bergman
operators as

$h(z)=\lim_{k\rightarrow\infty}\frac{1}{2}\log(1-\|y_{k}\|^{2})\|B(z,z)^{-1/2}B(z,y_{k})B(y_{k},y_{k})^{-1/2}\|\qquad(z\in
D).$

Let $r$ be the rank of $V$. Each $y_{k}\in D$ has a spectral decomposition

$y_{k}=\alpha_{1k}e_{1k}+\cdots+\beta_{dk}e_{rk}$ (4.4)

where $e_{1k},\ldots,e_{rk}$ are mutually orthogonal minimal tripotents in $V$
and

$1>\|y_{k}\|=\alpha_{1k}\geq\alpha_{2k}\geq\cdots\geq\alpha_{rk}\geq 0.$

Choosing a subsequence, we may assume for each $i$ that the sequence
$(\alpha_{ik})$ converges to some $\alpha_{i}\in[0,1]$ and the minimal
tripotents $e_{ik}$ converge to minimal tripotent $e_{i}$, as the set of
minimal tripotents is a closed subset of $\partial D$. Note that
$\alpha_{1}=\lim_{k}\|y_{k}\|=1$, as $\|y_{k}\|\longrightarrow\|\xi\|=1$.

By [9, Lemma 5.8], we have

$\xi=\lim_{k}y_{k}=\alpha_{1}e_{1}+\cdots+\alpha_{r}e_{r}$

and there exists $r_{0}\in\\{1,\ldots,r\\}$ such that

  1. (i)

$\alpha_{s}>0$ for each $1\leq s\leq r_{0}$,

  2. (ii)

$\alpha_{s}=0$ for $s>r_{0}$,

  3. (iii)

$\\{e_{1},\ldots,e_{r_{0}}\\}$ is family of mutually orthogonal minimal
tripotents.

######  Remark 4.3.

The minimal tripotents $e_{1k},\ldots,e_{rk}$ in the spectral decomposition of
$y_{k}$ induce a joint Peirce decomposition of $V$, with joint Peirce
projections $P^{k}_{ij}=P_{ij}(e_{1k},\ldots,e_{rk})$ and $0\leq i\leq j\leq
r$. By (3.19), the Bergman operator $B(y_{k},y_{k})^{-1/2}$ is of the form

$B(y_{k},y_{k})^{-1/2}=\sum_{0\leq i\leq j\leq
r}(1-\alpha_{ik}^{2})^{-1/2}(1-\alpha_{jk}^{2})^{-1/2}P_{ij}^{k}\quad(\alpha_{0k}=0).$

As $(e_{sk})_{k}$ converges to a minimal tripotent $e_{s}$ for $1\leq s\leq
r_{0}$, with $r_{0}$ as above, and the $e_{s}$’s are pairwise orthogonal, we
have the following norm convergence,

$\lim_{k\rightarrow\infty}P_{ij}(e_{1k},\ldots,e_{r_{0}k})=P_{ij}(e_{1},\ldots,e_{r_{0}})$

of Peirce projections (cf. [9, Remark 5.9]).

Furthermore, if $e_{1},\ldots,e_{m}$ are mutually orthogonal tripotents and
$1\leq q<m$, then $P_{ij}(e_{1},\ldots,e_{q})=P_{ij}(e_{1},\ldots,e_{m})$ for
all $1\leq i\leq j\leq q$, see [9, Lemma 2.1(i)].

We will use the observations in the previous remark to prove the following
theorem.

######  Theorem 4.4.

Let $D$ be the open unit ball of a finite dimensional $\mathrm{JB}^{*}$-triple
$V$, with rank $r$. Then the horofunction functions in $D(\infty)$ are exactly
the functions of the form

$h(z)=\frac{1}{2}\log\left\|\sum_{1\leq i\leq j\leq
p}\lambda_{i}\lambda_{j}B(z,z)^{-1/2}B(z,e)P_{ij}\right\|\qquad(z\in D),$

where $p\in\\{1,\ldots,r\\}$, $\lambda_{i}\in(0,1]\leavevmode\nobreak\
(i=1,\ldots,p)$ with $\max_{i}\lambda_{i}=1$,
$e=e_{1}+e_{2}+\cdots+e_{p}\in\partial D$, and $P_{ij}\colon V\rightarrow V$
are the Peirce projections induced by the mutually orthogonal minimal
tripotents $e_{1},\ldots,e_{p}$.

###### Proof.

Suppose that $h$ is a horofunction. Then by Lemma 4.1 there exists a sequence
$(y_{k})$ in $D$ converging to $\xi\in\partial D$ such that
$h(z)=\lim_{k\rightarrow\infty}h_{y_{k}}(z)$ for all $z\in D$.

Let

$y_{k}=\alpha_{1k}e_{1k}+\cdots+\alpha_{rk}e_{rk}$

be the spectral decomposition. From Remark 4.3 we know there is an
$r_{0}\in\\{1,\ldots,r\\}$ such that

$\xi=\lim_{k}y_{k}=\alpha_{1}e_{1}+\cdots+\alpha_{r}e_{r}$

with $\alpha_{1}=1$, $\alpha_{s}>0$ for $1\leq s\leq r_{0}$ and $\alpha_{s}=0$
for $s>r_{0}$. Moreover,

$B(y_{k},y_{k})^{-1/2}=\sum_{0\leq i\leq j\leq
r}(1-\alpha_{ik}^{2})^{-1/2}(1-\alpha_{jk}^{2})^{-1/2}P_{ij}^{k}\quad(\alpha_{{}_{0k}}=0).$

Since $0<{1-\alpha_{1k}^{2}}\leq 1-\alpha_{ik}^{2}$ for
$i\in\\{1,\ldots,r\\}$, we may assume, by choosing subsequence if necessary,
for each $i$ that

$\frac{1-\alpha_{1k}^{2}}{1-\alpha_{ik}^{2}}\leavevmode\nobreak\ {\rm
converges\leavevmode\nobreak\ to\leavevmode\nobreak\ some\leavevmode\nobreak\
}\lambda_{i}\in[0,1].$

Note that $\lambda_{1}=1$ and $\lambda_{i}=0$ for $i>r_{0}$. Combining this
with Remark 4.3 we get that

$\displaystyle\lim_{k\rightarrow\infty}(1-\|y_{k}\|^{2})B(y_{k},y_{k})^{-1/2}$
$\displaystyle=$ $\displaystyle\lim_{k\rightarrow\infty}\sum_{0\leq i\leq
j\leq
r}\sqrt{\frac{1-\alpha_{1k}^{2}}{1-\alpha_{ik}^{2}}}\sqrt{\frac{1-\alpha_{1k}^{2}}{1-\alpha_{jk}^{2}}}\,P_{ij}^{k}$
$\displaystyle=$ $\displaystyle\lim_{k\rightarrow\infty}\sum_{0\leq i\leq
j\leq
r_{0}}\sqrt{\frac{1-\alpha_{1k}^{2}}{1-\alpha_{ik}^{2}}}\sqrt{\frac{1-\alpha_{1k}^{2}}{1-\alpha_{jk}^{2}}}\,P_{ij}(e_{1k},\ldots,e_{r_{0}k})$
$\displaystyle=$ $\displaystyle\sum_{0\leq i\leq j\leq
r_{0}}\lambda_{i}\lambda_{j}P_{ij}$

where $P_{ij}$ are the Peirce projections induced by the orthogonal minimal
tripotents $e_{1},\ldots,e_{r_{0}}$. So,

$\displaystyle h(z)$ $\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}\frac{1}{2}\log(1-\|y_{k}\|^{2})\|B(z,z)^{-1/2}B(z,y_{k})B(y_{k},y_{k})^{-1/2}\|$
(4.5) $\displaystyle=$ $\displaystyle\frac{1}{2}\log\left\|\sum_{1\leq i\leq
j\leq
r_{0}}\lambda_{i}\lambda_{j}B(z,z)^{-1/2}B(z,\xi)P_{ij}(e_{1},\ldots,e_{r_{0}})\right\|\qquad(z\in
D).$

Let $p\in\\{1,\ldots,r_{0}\\}$ be such that $\alpha_{i}=1$ for $i\leq p$, and
$\alpha_{i}<1$ otherwise. Since $\lambda_{i}=0$ when $\alpha_{i}<1$, the
horofunction $h$ in (4.5) reduces to

$h(z)=\frac{1}{2}\log\left\|\sum_{1\leq i\leq j\leq
p}\lambda_{i}\lambda_{j}B(z,z)^{-1/2}B(z,\xi)P_{ij}(e_{1},\ldots,e_{p})\right\|,$
(4.6)

as $P_{ij}(e_{1},\ldots,e_{p})=P_{ij}(e_{1},\ldots,e_{r_{0}})$ for $1\leq
i\leq j\leq p$ by [9, Lemma 2.1].

Let $e=e_{1}+\cdots+e_{p}$. For $k\not\in\\{i,j\\}$ and $w\in
V_{ij}(e_{1},\ldots,e_{r_{0}})$ we have that $(e_{k}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,w)(V)=\\{0\\}$ by the Peirce
multiplication rules, as $e_{k}\in V_{kk}(e_{1},\ldots,e_{r_{0}})$. Therefore,
for $1\leq i\leq j\leq p$,

$\xi{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{ij}(e_{1},\ldots,e_{p})(\cdot)=\xi{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{ij}(e_{1},\ldots,e_{r_{0}})(\cdot)=e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{ij}(e_{1},\ldots,e_{r_{0}})(\cdot)=e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,P_{ij}(e_{1},\ldots,e_{p})(\cdot),$

and likewise

$\displaystyle Q_{\xi}P_{ij}(e_{1},\dots,e_{p})(\cdot)$ $\displaystyle=$
$\displaystyle\\{e_{1}+\alpha_{2}e_{2}+\cdots+\alpha_{r_{0}}e_{r_{0}},\,P_{ij}(e_{1},\ldots,e_{p})(\cdot),\,e_{1}+\alpha_{2}e_{2}+\cdots+\alpha_{r_{0}}e_{r_{0}}\\}$
$\displaystyle=$ $\displaystyle\\{e,P_{ij}(e_{1},\ldots,e_{p})(\cdot),e\\}.$

Thus, $Q_{\xi}P_{ij}(e_{1},\dots,e_{p})=Q_{e}P_{ij}(e_{1},\dots,e_{p})$. It
now follows for $1\leq i\leq j\leq p$ that

$B(z,\xi)P_{ij}(e_{1},\ldots,e_{p})=P_{ij}(e_{1},\ldots,e_{p})-2(z{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,e)P_{ij}(e_{1},\ldots,e_{p})+Q_{z}Q_{e}P_{ij}(e_{1},\ldots,e_{p})\\\
=B(z,e)P_{ij}(e_{1},\ldots,e_{p}).$

Thus, the horofunction $h$ in (4.6) can be expressed as

$h(z)=\frac{1}{2}\log\left\|\sum_{1\leq i\leq j\leq
p}\lambda_{i}\lambda_{j}B(z,z)^{-1/2}B(z,e)P_{ij}(e_{1},\ldots,e_{p})\right\|\qquad(z\in
D).$

To prove that each function of the form (4.4) is a horofunction we let

$h(z)=\frac{1}{2}\log\left\|\sum_{1\leq i\leq j\leq
p}\lambda_{i}\lambda_{j}B(z,z)^{-1/2}B(z,e)P_{ij}\right\|\qquad(z\in D)$

where $p\in\\{1,\ldots,r\\}$, $\lambda_{i}\in(0,1]\leavevmode\nobreak\
(i=1,\ldots,p)$ with $\max_{i}\lambda_{i}=1$, and
$e=e_{1}+e_{2}+\cdots+e_{p}\in\partial D$, with Peirce projections
$P_{ij}\colon V\rightarrow V$ induced by the mutually orthogonal minimal
tripotents $e_{1},\ldots,e_{p}$.

For all $k\in\mathbb{N}$ sufficiently large (depends on $\min_{i}\lambda_{i}$)
we can define

$\alpha_{ik}=\sqrt{1-\frac{2k-1}{k^{2}\lambda_{i}^{2}}}$

for $1\leq i\leq p$. For those $k$ set
$y_{k}=(1-1/k)e_{1}+\alpha_{2k}e_{2}+\cdots+\alpha_{pk}e_{p}$ and note that
$y_{k}\in D$.

Then the sequence $(y_{k})$ norm converges to $e$ and

$\lim_{k\rightarrow\infty}(1-\|y_{k}\|^{2})B(y_{k},y_{k})^{-1/2}=\lim_{k\rightarrow\infty}\sum_{0\leq
i\leq j\leq
p}\sqrt{\frac{1-(1-1/k)^{2}}{1-\alpha_{ik}^{2}}}\sqrt{\frac{1-(1-1/k)^{2}}{1-\alpha_{jk}^{2}}}\,P_{ij}=\sum_{0\leq
i\leq j\leq p}\lambda_{i}\lambda_{j}P_{ij}.$

Hence

$\displaystyle\lim_{k\rightarrow\infty}h_{y_{k}}(z)$ $\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}\frac{1}{2}\log\left(\frac{1-\|y_{k}\|^{2}}{1-\|g_{-y_{k}}(z)\|^{2}}\right)$
$\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}\frac{1}{2}\log(1-\|y_{k}\|^{2})\|B(z,z)^{-1/2}B(z,y_{k})B(y_{k},y_{k})^{-1/2}\|$
$\displaystyle=$ $\displaystyle\frac{1}{2}\log\left\|\sum_{1\leq i\leq j\leq
p}\lambda_{i}\lambda_{j}B(z,z)^{-1/2}B(z,e)P_{ij}\right\|,$

which completes the proof. ∎

We see from the proof of Theorem 4.4 that it can happen that two sequence
$(y_{k})$ and $(z_{k})$ in $D$ converging to distinct points in $\partial D$
can give the same horofunction. Indeed, if we let $y_{k}$ be as in the proof
Theorem 4.4 and set $z_{k}=y_{k}+(1-1/\sqrt{k})(e_{p+1}+\cdots+e_{r})$, then
both $h_{y_{k}}$ and $h_{z_{k}}$ converge to $h$ given by (4.4).

We also note that the horofunction $h$ given by (4.4) can be obtained by
taking the limit of an appropriate sequence in the flat
$\mathbb{R}e_{1}\oplus\cdots\oplus\mathbb{R}e_{p}$. This is consistent with
the observation in [17, Lemma 4.4]. Later in Lemma 8.3 we shall show that one
can obtain the horofunctions in $D(\infty)$ by taking limits along geodesics
in the flats.

##  5 Horofunctions of finite dimensional $\mathrm{JB}^{*}$-triples

We now determine the horofunctions of finite dimensional
$\mathrm{JB}^{*}$-triples $(V,\|\cdot\|)$ as normed spaces, with base point
$0$. Throughout we let $r$ be the rank of $V$. As in (4.4), each element $a\in
V$ has a spectral decomposisiton,

$a=\lambda_{1}e_{1}+\lambda_{2}e_{2}+\cdots+\lambda_{r}e_{r},\quad(\|a\|=\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}\geq
0),$

where $e_{1},\ldots,e_{r}$ are mutually orthogonal minimal tripotents in $V$.

Given a sequence $(a_{k})$ in $V$ with $h_{a_{k}}\longrightarrow h\in
V(\infty)$, we have $r_{k}=\|a_{k}\|\longrightarrow\infty$ (by Lemma 2.2) and

$h_{a_{k}}(x)=\|x-a_{k}\|-\|a_{k}\|=\frac{\|x-a_{k}\|^{2}-\|a_{k}\|^{2}}{\|x-a_{k}\|+\|a_{k}\|}=\frac{(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})}{2^{-1}(\|r_{k}^{-1}(x-a_{k})\|+1)}.$

As the denominator goes to $1$ when $k\longrightarrow\infty$, we need to
analyse

$\lim_{k\longrightarrow\infty}(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2}).$

First we note that, for each $y\in V$, the spectrum
$\sigma(y{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,y)$ is the set of eigenvalues in $[0,\infty)$ since $\dim
V<\infty$, and by Remark 3.1, $\|y{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,y\|=\sup\sigma(y{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,y).$ On the other hand, $V$ is a finite
dimensional Hilbert space with inner-product

$\langle x,y\rangle=\mathrm{Tr}(x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,y)\qquad(x,y\in V).$ (5.1)

For each self-adjoint operator $T$ on $(V,\langle\cdot,\cdot\rangle)$, we use
the notation

$\Lambda(T)=\sup\\{\langle Tz,z\rangle\colon z\in V,\langle z,z\rangle=1\\}$

to denote the maximum eigenvalue of $T$. In particular, we have

$\Lambda(y{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,y)=\sup\sigma(y{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,y),$

as $y{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,y$ is a positive self-adjoint operator on the Hilbert space
$V$ (cf. [6, Lemma 1.2.22]).

Since $h_{a_{k}}(x)=\|x-a_{k}\|-\|a_{k}\|\geq-\|x\|$ for all $x\in V$, we have

$(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})\geq-2\|x\|$ (5.2)

for sufficiently large $k$ , which will be useful later.

The start we prove the following technical lemma.

######  Lemma 5.1.

Let $(a_{k})$ be a sequence in $V$ such that
$r_{k}=\|a_{k}\|\longrightarrow\infty$. Let
$a_{k}=\sum_{i=1}^{r}\lambda_{ik}e_{ik}$ be a spectral decomposition of
$a_{k}$, with $\lambda_{1k}\geq\lambda_{2k}\geq\ldots\geq\lambda_{rk}\geq 0$
and mutually orthogonal minimal tripotents $e_{1k},\ldots,e_{rk}$.

If $\alpha_{ik}=r_{k}-\lambda_{ik}\longrightarrow\alpha_{i}\in[0,\infty]$ and
$e_{ik}\longrightarrow e_{i}$ for all $i$, then

$\displaystyle\lim_{k\longrightarrow\infty}(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})$ (5.3)
$\displaystyle=\sup_{u\in V_{2}(e_{I})\colon\langle
u,u\rangle=1}\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,u\rangle,$

where $I=\\{i\colon\alpha_{i}<\infty\\}$ and $e_{I}=\sum_{i\in I}e_{i}$.

###### Proof.

We will show that every subsequence of
$((2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2}))_{k}$ has a convergent subsequence
whose limit is the right-hand side of (5.3). Let
$((2r_{m})^{-1}(\|(x-a_{m}){\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{m})\|-r_{m}^{2}))_{m}$ be a subsequence. Note that

$(2r_{m})^{-1}(\|(x-a_{m}){\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{m})\|-r_{m}^{2})=\Lambda\left(\frac{1}{2r_{m}}x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,x-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)\right),$

where $I\colon V\longrightarrow V$ is the identity operator.

There exists $w^{m}\in V$ with $\langle w^{m},w^{m}\rangle=1$ such that

$\Lambda\left(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)\right)\\\
=\left\langle\left(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)\right)w^{m},\,w^{m}\right\rangle.$

After taking a further subsequence, we may assume that $w^{m}\longrightarrow
w$ and $\lambda_{im}/r_{m}\longrightarrow\mu_{i}\in[0,1]$ for all $i$. So,
$a_{m}/r_{m}\longrightarrow a=\sum_{i=1}^{r}\mu_{i}e_{i}$. Also note that
$\mu_{i}=1$ for all $i\in I$, as

$\mu_{i}=\lim_{m}\frac{\lambda_{im}}{r_{m}}=\lim_{m}\frac{r_{m}-\alpha_{im}}{r_{m}}=1\mbox{\qquad
for all $i\in I$.}$

Consider the Peirce decomposition $V=\bigoplus_{0\leq s\leq t\leq
r}V_{st}^{m}$ with respect to $e_{1m},\ldots,e_{rm}$ and write

$w^{m}=\sum_{0\leq s\leq t\leq r}w^{m}_{st}.$

Set $\lambda_{0m}=0$ and $\alpha_{0m}=r_{m}$ for all $m$. From (3.15), we have

$(e_{im}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{im})w_{st}^{m}=\left\\{\begin{array}[]{ll}w_{st}^{m}&(i=s=t)\\\
\frac{1}{2}w_{st}^{m}&(i=s\neq t\,{\rm or}\,i=t\neq s)\\\ 0&\mbox{
otherwise.}\end{array}\right.$

Therefore

$\displaystyle\langle\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)w^{m},\,w^{m}\rangle$
$\displaystyle=\langle\sum_{i=1}^{r}\frac{1}{2r_{m}}(\lambda_{im}^{2}(e_{im}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{im})w_{m}-r^{2}_{m}w^{m}),\,w^{m}\rangle$
$\displaystyle=\sum_{0\leq s\leq
r}\frac{\lambda_{sm}^{2}-r_{m}^{2}}{2r_{m}}\langle
w^{m}_{ss},\,w^{m}_{ss}\rangle+\sum_{0\leq s<t\leq
r}\frac{(\lambda_{sm}^{2}+\lambda_{tm}^{2})/2-r_{m}^{2}}{2r_{m}}\langle
w^{m}_{st},\,w^{m}_{st}\rangle$ $\displaystyle=-\sum_{0\leq s\leq
r}\frac{\alpha_{sm}(2r_{m}-\alpha_{sm})}{2r_{m}}\langle
w^{m}_{ss},w^{m}_{ss}\rangle$ $\displaystyle\qquad-\sum_{0\leq s<t\leq
r}\Big{(}\frac{\alpha_{sm}(2r_{m}-\alpha_{sm})}{4r_{m}}+\frac{\alpha_{tm}(2r_{m}-\alpha_{tm})}{4r_{m}}\Big{)}\langle
w^{m}_{st},w^{m}_{st}\rangle.$ (5.4)

Note that, as the of set minimal tripotents is compact and
$e_{1k},\ldots,e_{rk}$ are mutually othogonal tripotents, $e_{1},\ldots,e_{r}$
are mutually orthogonal minimal tripotents. Let $V=\bigoplus_{0\leq s\leq
t\leq r}V_{st}$ and $w=\sum_{0\leq s\leq t\leq r}w_{st}$ be the Peirce
decompositions with respect to $e_{1},\ldots,e_{r}$.

Then $w_{st}=0$ if $\\{s,t\\}\not\subset I$. Indeed,
$w^{m}_{st}\longrightarrow w_{st}$ for all $0\leq s\leq t\leq r$, and if
$w_{st}\neq 0$ for some $\\{s,t\\}\not\subset I$, then the right-hand side of
(5.4) goes to $-\infty$ as $m\longrightarrow\infty$, since

$\frac{\alpha_{sm}(2r_{m}-\alpha_{sm})}{2r_{m}}\geq\frac{\alpha_{sm}}{2}\longrightarrow\infty\mbox{\quad
or\quad}\frac{\alpha_{tm}(2r_{m}-\alpha_{tm})}{2r_{m}}\geq\frac{\alpha_{tm}}{2}\longrightarrow\infty.$

As $\langle\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}w^{m},w^{m}\rangle\longrightarrow 0$ and

$\langle-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})w^{m},w^{m}\rangle\longrightarrow\langle-\frac{1}{2}(a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a)w,w\rangle,$

we find that $(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})\longrightarrow-\infty$, which
contradicts (5.2). Hence

$w\in\bigoplus_{s,t\in I\colon s\leq t}V_{st}=V_{2}(e_{I}).$

Now using the Peirce decomposition $V=V_{2}(e_{I})\oplus V_{1}(e_{I})\oplus
V_{0}(e_{I})$ with respect to the tripotent $e_{I}$ and the Peirce
multiplication rules, we find for each $z\in V_{2}(e_{I})$ (and in particular
for $w$) that

$\langle(a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x)z,z\rangle=\langle\\{e_{I},x,z\\}+\\{\sum_{i\not\in
I}\mu_{i}e_{I},x,z\\},z\rangle=\langle(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x)z,z\rangle$

and

$\langle(x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a)z,z\rangle=\langle\\{x,e_{I},z\\}+\\{x,\sum_{i\not\in
I}\mu_{i}e_{I},z\\},z\rangle=\langle(x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})z,z\rangle.$

Hence we deduce from (5.4) that

$\displaystyle\limsup_{m\longrightarrow\infty}\,(2r_{m})^{-1}(\|(x-a_{m}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,(x-a_{m})\|-r_{m}^{2})$
$\displaystyle\leq\langle-\frac{1}{2}(a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a)w,w\rangle-\sum_{s\in
I}\alpha_{s}\langle w_{ss},w_{ss}\rangle$ $\displaystyle\qquad-\sum_{s,t\in
I\colon s<t}\frac{1}{2}(\alpha_{s}+\alpha_{t})\langle w_{st},w_{st}\rangle$
$\displaystyle=\langle-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})w,w\rangle-\sum_{s\in
I}\alpha_{s}\langle(e_{s}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{s})w,w\rangle.$ (5.5)

Next we show

$\liminf_{m\longrightarrow\infty}\,(2r_{m})^{-1}(\|(x-a_{m}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{m})\|-r_{m}^{2})\geq\langle-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})w,w\rangle-\sum_{s\in
I}\alpha_{s}\langle(e_{s}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{s})w,w\rangle.$ (5.6)

Let $e_{I}^{m}=\sum_{i\in I}e_{im}$ and set $u^{m}=P_{2}(e_{I}^{m})w^{m}$.
Then $u^{m}\longrightarrow P_{2}(e_{I})w=w$ and

$\Lambda\left(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)\right)\\\
\geq\left\langle\left(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)\right)u^{m},u^{m}\right\rangle\langle
u^{m},u^{m}\rangle^{-1}$

for large $m$, as $\langle u^{m},u^{m}\rangle\longrightarrow\langle
w,w\rangle=1$. Since

$\langle(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}}))u^{m},u^{m}\rangle\longrightarrow\langle-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})w,w\rangle$

and

$\lim_{m}\langle\frac{1}{2r_{m}}((a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,a_{m})u^{m}-r^{2}_{m}u^{m}),u^{m}\rangle=-\sum_{s\in
I}\alpha_{s}\langle(e_{s}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{s})w,w\rangle,$

the inequality follows.

From (5.5) and (5.6), we now obtain

$\lim_{m}(2r_{m})^{-1}(\|(x-a_{m}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{m})\|-r_{m}^{2})=\langle-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})w,w\rangle-\sum_{s\in
I}\alpha_{s}\langle(e_{s}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{s})w,w\rangle,$

which implies that the left-hand side of (5.3) does not exceed the right-hand
side.

To prove equality of the two sides, pick $v\in V_{2}(e_{I})$ with $\langle
v,v\rangle=1$ such that

$\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{I})-\sum_{i\in I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{i}))v,\,v\rangle=\sup_{u\in
V_{2}(e_{I})\colon\langle
u,u\rangle=1}\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,\,u\rangle.$

Let $v^{m}=P_{2}(e^{m}_{I})v$. Again by definition of $w^{m}$, we have for
large $m$,

$\langle(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I))w^{m},w^{m}\rangle\\\
\geq\langle(\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})+\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I))v^{m},v^{m}\rangle\langle
v^{m},v^{m}\rangle^{-1},$

where $e^{m}_{I}=\sum_{i\in I}e_{im}$ and
$v^{m}=P_{2}(e^{m}_{I})v\longrightarrow P_{2}(e_{I})v=v$.

Write $v^{m}=\displaystyle\sum_{s,t\in I\colon s\leq
t}v^{m}_{st}\in\bigoplus_{0\leq s\leq t\leq r}V_{st}^{m}$. Then
$\langle\frac{x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,x}{2r_{m}}v^{m},v^{m}\rangle\longrightarrow 0$ and

$\langle-\frac{1}{2}(\frac{a_{m}}{r_{m}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,\frac{a_{m}}{r_{m}})v^{m},v^{m}\rangle\longrightarrow\langle-\frac{1}{2}(a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a)v,v\rangle.$

As before,

$\displaystyle\langle\frac{1}{2r_{m}}(a_{m}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a_{m}-r^{2}_{m}I)v^{m},v^{m}\rangle$
$\displaystyle=-\sum_{s\in
I}\frac{\alpha_{sm}(2r_{m}-\alpha_{sm})}{2r_{m}}\langle
v^{m}_{ss},v^{m}_{ss}\rangle$ $\displaystyle\qquad\qquad-\sum_{s,t\in I\colon
s<t}\Big{(}\frac{\alpha_{sm}(2r_{m}-\alpha_{sm})}{4r_{m}}+\frac{\alpha_{tm}(2r_{m}-\alpha_{tm})}{4r_{m}}\Big{)}\langle
v^{m}_{st},v^{m}_{st}\rangle$ $\displaystyle\longrightarrow-\sum_{s\in
I}\alpha_{s}\langle(e_{s}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{s})v,v\rangle.$

We conclude that

$\lim_{m\longrightarrow\infty}(2r_{m})^{-1}(\|(x-a_{m}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{m})\|-r_{m}^{2})\geq\langle-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})v-\sum_{s\in
I}\alpha_{s}(e_{s}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{s})v,v\rangle,$

which completes the proof. ∎

######  Remark.

In (5.3), we have

$\sup_{u\in V_{2}(e_{I})\colon\langle
u,u\rangle=1}\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,u\rangle\\\ =\sup_{u\in V_{2}(e_{I})\colon\langle
u,u\rangle=1}\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(e_{I})x+P_{2}(e_{I})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,u\rangle\\\
=\Lambda_{V_{2}(e_{I})}(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(e_{I})x+P_{2}(e_{I})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i})),$ (5.7)

where the latter denotes the maximum eigenvalue of the operator

$-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(e_{I})x+P_{2}(e_{I})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i})$

restricted to the subspace $V_{2}(e_{I})$, which it leaves invariant.

We are now ready to describe the horofunction boundary $V(\infty)$.

######  Theorem 5.2.

Let $h$ be a horofunction in ${V}(\infty)$. Then there exist
$I\subset\\{1,\ldots,r\\}$ nonempty, mutually orthogonal minimal tripotents
$e_{i}\in V$ and $\alpha_{i}\geq 0$ for $i\in I$, with $\min_{i\in
I}\alpha_{i}=0$, such that

$h(x)=\sup\\{\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,u\rangle\colon u\in V_{2}(e_{I}),\,\langle
u,u\rangle=1\\}\\\
=\Lambda_{V_{2}(e_{I})}(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(e_{I})x+P_{2}(e_{I})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))\qquad(x\in V),$ (5.8)

where $e_{I}=\sum_{i\in I}e_{i}$ is a tripotent with Peirce $2$-space
$V_{2}(e_{I})$ and Peirce projection $P_{2}(e_{I})$.

Conversely, each function $h\colon V\longrightarrow\mathbb{R}$ of the form in
(5.8) is a horofunction of $V$.

###### Proof.

Let $h=\lim_{k}h_{a_{k}}$ be a horofunction, where $a_{k}\in V$ and
$r_{k}=\|a_{k}\|\longrightarrow\infty$. For each $k$, let
$a_{k}=\sum_{i=1}^{r}\lambda_{ik}e_{ik}$ be a spectral decomposition, with
pairwise orthogonal minimal tripotents $e_{ik}$ and
$\|a_{k}\|=\lambda_{1k}\geq\ldots\geq\lambda_{rk}\geq 0$.

After taking a subsequence, we may assume
$\alpha_{ik}=r_{k}-\lambda_{ik}\longrightarrow\alpha_{i}\in[0,\infty]$ and
$e_{ik}\longrightarrow e_{i}$ for all $i=1,\ldots,r$, where
$e_{1},\ldots,e_{r}$ are mutually orthogonal minimal tripotents. Let
$I=\\{i\colon\alpha_{i}<\infty\\}$ and $e_{I}=\sum_{i\in I}e_{i}$, and note
that $I\neq\emptyset$, since $\alpha_{1}=0$.

We have

$h_{a_{k}}(x)=\frac{\|x-a_{k}\|^{2}-\|a_{k}\|^{2}}{\|x-a_{k}\|+\|a_{k}\|}=\frac{(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})}{2^{-1}(\|r_{k}^{-1}(x-a_{k})\|+1)}$

and $2^{-1}(\|r_{k}^{-1}(x-a_{k})\|+1)\longrightarrow 1$. Hence (5.8) follows
readily from Lemma 5.1 and the preceding remark.

Conversely, let $h\colon V\longrightarrow\mathbb{R}$ be of the form (5.8),
where $I$ is a nonempty subset of $\\{1,\ldots,r\\}$, $\min_{i\in
I}\alpha_{i}=0$ and $e_{I}=\sum_{i\in I}e_{i}$ is the sum of mutually
orthogonal minimal tripotents $e_{i}$.

We show $h\in V(\infty)$. For $k=1,2,\ldots,$ define

$a_{k}=ke_{I}-\sum_{i\in I}\alpha_{i}e_{i}.$

For $k\geq\max_{i\in I}\alpha_{i}$, we have $r_{k}=\|a_{k}\|=k$ and, in the
notation of Lemma 5.1,

$\alpha_{ik}=\left\\{\begin{matrix}\alpha_{i}&(i\in I)\\\
k&(\rm{otherwise})\end{matrix}\right.$

It follows from Lemma 5.1 that

$\lim_{k\longrightarrow\infty}(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})=\sup_{u\in
V_{2}(e_{I})\colon\langle
u,u\rangle=1}\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,u\rangle$

and

$\lim_{k\longrightarrow\infty}h_{a_{k}}(x)=\lim_{k\longrightarrow\infty}\frac{(2r_{k})^{-1}(\|(x-a_{k}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(x-a_{k})\|-r_{k}^{2})}{2^{-1}(\|r_{k}^{-1}(x-a_{k})\|+1)}\\\
=\sup_{u\in V_{2}(e_{I})\colon\langle
u,u\rangle=1}\langle(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,x+x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))u,u\rangle=h(x)$

for all $x\in V$. Hence $h\in V(\infty)$. ∎

If $I$ is a singleton in the preceding theorem, $e_{I}=e$, where $e$ is a
minimal tripotent. In that case the last term of (5.8) vanishes and $h$ is a
real continuous linear functional of $V$. Indeed,
$P_{2}(e)(V)=V_{2}(e)=\mathbb{C}e$ implies $P_{2}(e)x=\ell(x)e$ for some
functional $\ell\in V^{*}$ and

$h(x)=\langle-\frac{1}{2}(e{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(e)x+P_{2}(e)x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e)e,e\rangle\langle
e,e\rangle^{-1}=-\frac{1}{2}(\langle e,(\ell(x)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,e)e\rangle+\langle(\ell(x)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e)e,e\rangle)\langle e,e\rangle^{-1}\\\
=-\frac{1}{2}(\overline{\ell(x)}+\ell(x))=-\mathrm{Re}\,\ell(x).$

######  Remark 5.3.

In the course of proving the preceding theorem, we observe that each
horofunction $h\in V(\infty)$ can actually be constructed from a sequence
$(a_{k})$ going to infinity along a straight line, which is a geodesic in the
normed space $V$. In fact, the sequence $a_{k}=ke_{I}-\sum_{i\in
I}\alpha_{i}e_{i}$ used in the proof lies on the straight line, $t\mapsto
te_{I}-\sum_{i\in I}\alpha_{i}e_{i}$ in the flat $\oplus_{i\in
I}\mathbb{R}e_{i}$. Also note that if $k\geq m$ with $k\geq\max_{i\in
I}\alpha_{i}$, then $\|a_{k}\|=k$ and
$h_{a_{k}}(a_{m})=\|a_{k}-a_{m}\|-\|a_{k}\|=(k-m)-k=-m$, so that $h(a_{m})=-m$
for all $m$.

By the remark we have the following corollary.

######  Corollary 5.4.

Each horofunction in ${V}(\infty)$ is a Busemann point.

For general finite dimensional normed vector spaces it need not be true that
all horofunctions are Busemann points, see [35].

##  6 Homeomorphism onto the dual unit ball

In this section we give a homeomorphism of the metric compactification of $V$
onto the closed dual unit ball $B^{*}$ of $(V,\|\cdot\|)$. We subsequently
show in the next section that this homeomorphism maps each equivalence class
in $V(\infty)/\sim$ onto the relative interior of a boundary face of $B^{*}$.
So, the dual ball $B^{*}$ captures the geometry of the metric compactification
of $V$.

To prove these results we need a lemma concerning the partial ordering on the
set of tripotents of a JB∗-triple. Recall that, given two tripotents $c$ and
$e$ in $V$, we shall write $c\leq e$ if $e-c$ is a tripotent in $V$ orthogonal
to $c$, or equivalently, if and only if $P_{2}(c)e=c$ ([12, Corollary 1.7]).
We also have $c\leq e$ if and only if $c$ is a projection in the JB∗-algebra
$V_{2}(e)$. Particularly,

$c=\\{e,c,e\\}$ (6.1)

(cf. [6, pp.34-36]).

######  Lemma 6.1.

Given two tripotents $c,e\in V$, the following conditions are equivalent:

  1. (i)

$c\leq e$,

  2. (ii)

$\\{c,P_{2}(c)e,c\\}+\\{c,c,P_{2}(c)e\\}=2c$,

  3. (iii)

$\\{c,e,c\\}+\\{c,c,e\\}=2c$,

  4. (iv)

$B(e,c)w=0$ for all $w\in V_{2}(c)$.

###### Proof.

(i) $\Rightarrow$ (ii). If $c\leq e$, then $P_{2}(c)e=c$ and (ii) follows
directly.

(ii) $\Rightarrow$ (i). We note that $\\{c,P_{2}(c)e,c\\}\in V_{2}(c)$ by
(3.9), and $\\{c,c,P_{2}(c)e\\}=P_{2}(c)e\in V_{2}(c)$.

Observe that $c$ is a maximal tripotent in the JB*-triple $V_{2}(c)$, and
hence $c$ is an extreme point of the closed unit ball of $V_{2}(c)$ ([6,
Theorem 3.2.3]). As $\|\\{c,P_{2}(c)e,c\\}\|,\|P_{2}(c)e\|\leq 1$, we conclude
that $P_{2}(c)e=c$, that is, $c\leq e$.

(i) $\Rightarrow$ (iii). Suppose $c\leq e$. The orthogonality between $e$ and
$e-c$ gives clearly (iii).

(iii) $\Rightarrow$ (i). The arguments used in the implication (ii)
$\Rightarrow$ (i) apply, by observing that $\\{c,e,c\\}=\\{c,P_{2}(c)e,c\\}\in
V_{2}(c)$ from (3.9) and (3.10), which gives $\\{c,c,e\\}=2c-\\{c,e,c\\}\in
V_{2}(c)$.

(i) $\Rightarrow$ (iv). Let $w\in V_{2}(c)$. Using orthogonality, (6.1) and
the Jordan triple identity, we deduce

$\displaystyle B(e,c)w$ $\displaystyle=$ $\displaystyle
w-2\\{e,c,w\\}+\\{e,\\{c,w,c\\},e\\}$ $\displaystyle=$ $\displaystyle
w-2\\{e,c,w\\}+2\\{e,c,\\{e,c,w\\}\\}-\\{w,c,\\{e,c,e\\}\\}$ $\displaystyle=$
$\displaystyle w-2\\{e,c,w\\}+2\\{e,c,\\{e,c,w\\}\\}-\\{w,c,c\\}$
$\displaystyle=$ $\displaystyle w-2\\{e,c,w\\}+2\\{e,c,\\{e,c,w\\}\\}-w$
$\displaystyle=$ $\displaystyle w-2w+2w-w=0.$

(iv) $\Rightarrow$ (i). Let $a=P_{2}(c)e\in V_{2}(c)$. We show $a=c$. Using
the identities (JP1), (JP3) and (JP2) in [28, Appendix], one deduces that

$Q(c)B(a,c)w=Q(c)B(e,c)w=0,\qquad(w\in V_{2}(c)).$

Now take $w=c-\\{c,a,c\\}\in V_{2}(c)$. Then we have

$\displaystyle B(a,c)w$ $\displaystyle=$ $\displaystyle
B(a,c)c-B(a,c)\\{c,a,c\\}$ $\displaystyle=$ $\displaystyle
c-2a+\\{a,c,a\\}-(\\{c,a,c\\}-2\\{a,c,\\{c,a,c\\}\\}+\\{a,\\{c,\\{c,a,c\\},c\\},a\\})$
$\displaystyle=$ $\displaystyle
c-2a+\\{a,c,a\\}-\\{c,a,c\\}+2\\{c,a,a\\}-\\{a,a,a\\}$ $\displaystyle=$
$\displaystyle\\{c-a,c-a,c-a\\},$

where
$\\{c,a,a\\}=\\{c,a,\\{c,c,a\\}\\}=\\{\\{c,a,c\\},c,a\\}-\\{c,\\{a,c,c\\},a\\}+\\{c,c,\\{c,a,a\\}\\}=\\{\\{c,a,c\\},c,a\\}$.
As $\\{c-a,c-a,c-a\\}\in V_{2}(c)$, it follows from (3.12) that

$\|\\{c-a,c-a,c-a\\}\|=\|Q(c)\\{c-a,c-a,c-a\\}\|=\|Q(c)B(a,c)w\|=0,$

so that $c-a=0$ by (3.3). ∎

######  Remark 6.2.

Given tripotents $c,e\in V$ and identity map $I\colon V\rightarrow V$, we have
$\|c{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c)e+P_{2}(c)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c\|\leq 2$ and
$2I-(c{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c)e+P_{2}(c)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c)$ is a positive operator on the
Hilbert space $(V_{2}(c),\langle\cdot,\cdot\rangle)$ by Remark 3.1. Hence
$\langle(2I-(c{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c)e+P_{2}(c)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c))v,v\rangle\geq 0$ for all $v\in
V_{2}(c)$. Moreover, $\langle(2I-(c{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,P_{2}(c)e+P_{2}(c)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c))v,v\rangle=0$ for all $v\in V_{2}(c)$
is equivalent to $2I=c{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c)e+P_{2}(c)e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c$ on $V_{2}(c)$.

The previous lemma has the following useful corollary.

######  Corollary 6.3.

Suppose that $h$ and $h^{\prime}$ are horofunctions given, respectively, by
(5.8) and

$h^{\prime}(x)=\Lambda_{V_{2}(c_{J})}(-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})x+P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})-\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j})).$

Let $a=\sum_{i\in I}\alpha_{i}e_{i}$ and $b=\sum_{j\in J}\beta_{j}c_{j}$. Then
$h=h^{\prime}$ if and only if $e_{I}=c_{J}$ and $a=b$.

###### Proof.

The sufficiency follows from the observation that $e_{I}=c_{J}$ and $a=b$
implies that

$\sum_{i\in I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i})=(\sum_{i\in I}\alpha_{i}e_{i}){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I}=(\sum_{j\in
J}\beta_{j}c_{j}){\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{J}=\sum_{j\in J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{j}).$

Conversely, let $\lim_{k}h_{a_{k}}=h=h^{\prime}=\lim_{k}h_{b_{k}}$, where we
can choose $a_{k}=ke_{I}-a$ and $b_{k}=kc_{J}-b$ by Remark 5.3. As
$h(a_{k})=-k$, we have $k+h^{\prime}(a_{k})=0$ and

$k+h^{\prime}(a_{k})=k+\sup_{v\in V_{2}(c_{J})\colon\langle
v,v\rangle=1}\langle(-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,(ke_{I}-a)+(ke_{I}-a){\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J}-\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))v,v\rangle\\\ =k+\sup_{v\in
V_{2}(c_{J})\colon\langle
v,v\rangle=1}(\langle-\frac{k}{2}((c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})e_{I})+(P_{2}(c_{J})e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J}))v,v\rangle\\\
+\langle(\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a+a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{J})-\sum_{j\in J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{j}))v,v\rangle)\\\ =k(\sup_{v\in
V_{2}(c_{J})\colon\langle
v,v\rangle=1}(\langle(I-\frac{1}{2}((c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})e_{I})+(P_{2}(c_{J})e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})))v,v\rangle\\\
+\langle(\frac{1}{2k}(c_{J}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a+a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{J})-\frac{1}{k}\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))v,v\rangle))$

for $k=1,2,\ldots$. So we find that
$\langle(I-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})e_{I}+P_{2}(c_{J})e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J}))v,v\rangle=0$ for all $v\in
V_{2}(c_{J})$, and hence $2I=c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})e_{I}+P_{2}(c_{J})e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J}$ by Remark 6.2 on $V_{2}(c_{J})$.
Moreover,

$\sup_{v\in V_{2}(c_{J})\colon\langle
v,v\rangle=1}\langle(\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a+a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})-\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))v,v\rangle=0.$ (6.2)

In particular, we have $2c_{J}=(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})e_{I})(c_{J})+(P_{2}(c_{J})e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})(c_{J})$, so $c_{J}\leq e_{I}$ by
Lemma 6.1.

Analogously, $0=k+h^{\prime}(b_{k})=k+h(b_{k})$ implies $e_{I}\leq c_{J}$ and

$\sup_{v\in V_{2}(e_{I})\colon\langle
v,v\rangle=1}\langle(\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,b+b{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{j\in
J}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))v,v\rangle=0.$ (6.3)

We conclude that $e_{I}=c_{J}$ and note that this implies that

$\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,a+a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{J})=\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,a+a{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})=\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i})$

and

$\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,b+b{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{I})=\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,b+b{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})=\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}).$

It now follows from (6.2) and (6.3) that $\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i})=\sum_{j\in J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{j})$ on $V_{2}(e_{I})=V_{2}(c_{J})$,
as both operators are self-adjoint. In particular,

$a=\sum_{i\in I}\alpha_{i}e_{i}=\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i})(e_{I})=\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j})(c_{J})=\sum_{j\in J}\beta_{j}c_{j}=b.$

∎

To show that the metric compactification $V\cup V(\infty)$ is homeomorphic to
closed dual unit ball of $(V,\|\cdot\|)$, we identify $V$ with its (algebraic)
dual space $V^{*}$ by using an inner-product on $V$, which we can do as $V$ is
finite dimensional. For convenience we adjust the inner-product
$\langle\cdot,\cdot\rangle$ on $V$ to define a new inner-product
$[\cdot,\cdot]$ such that $[c,c]=1$ for each minimal tripotent $c\in V$.

To realise this we note that a finite dimensional $\mathrm{JB}^{*}$-algebra
$V$ decomposes into a finite $\ell_{\infty}$-sum, $V=V_{1}\oplus\cdots\oplus
V_{d}$, of Cartan factors, each of which contains a minimal tripotent (cf. [7,
Theorem 3.3.5, Theorem 3.8.17]. In a finite dimensional Cartan factor $V_{j}$,
the tripotents form a compact submanifold $M_{j}$ of $V_{j}$, in which the
minimal tripotents form a connected component $N_{j}$ [28, §5]. As shown in
[8, Proposition 2.2], the tangent space $T_{e}(N_{j})$ at each $e\in N_{j}$
identifies with $\mathrm{i}A(e)\oplus V_{1}(e)$, where $A(e)$ is defined in
(3.13).

In particular, $\dim(\mathrm{i}A(e)\oplus V_{1}(e))=1+\dim V_{1}(e)$ is
constant for all $e\in N_{j}$. On $V_{j}$ we define, for a fixed $e\in N_{j}$,
a normalised trace form

$\langle x,y\rangle_{j}=\frac{1}{1+\dim V_{1}(e)/2}{\rm
Trace}\,(x{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,y)\qquad(x,y\in V_{j}),$

so that $\langle c,c\rangle_{j}=1$ for all $c\in N_{j}$, and
$(V_{j},\langle\cdot,\cdot\rangle_{j})$ is a Hilbert space.

Henceforth, we denote by $[\cdot,\cdot]$ the inner-product of the Hilbert
space direct sum $V=V_{1}\oplus\cdots\oplus V_{d}$, with
$(V_{j},\langle\cdot,\cdot\rangle_{j})$. Then each minimal tripotent $c\in V$
lies in some $V_{j}$ and $[c,c]=1$. Moreover, the inner-product is
associative, i.e.,

$[\\{a,b,y\\},z]=[y,\\{b,a,z\\}]\qquad(a,b,y,z\in V)$

(cf. [6, (2.31)]). In particular, we have $[P_{2}(c)y,z]=[y,P_{2}(c)z]$ for
each tripotent $c\in V$. We note that if $a,b\in V$ are triple orthogonal,
i.e., $a{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,b=0$, then $[a,b]=0$.

By the Riesz representation theorem, the map

$x\in V\mapsto\widetilde{x}=[\cdot,x]\in V^{*}$ (6.4)

is a conjugate linear isomorphism.

Let $D^{*}=\\{\widetilde{x}\in V^{*}\colon\|x\|_{*}\leq 1\\}$ be the closed
dual unit ball in dual space of the $\mathrm{JB}^{*}$-triple $(V,\|\cdot\|)$.
Then for each $\widetilde{x}=[\cdot,x]\in V^{*}$, where $x$ has spectral
decomposition $x=\sum_{j=1}^{r}\lambda_{j}c_{j}\in V$, we have
$\|\widetilde{x}\|_{*}=\sum_{j=1}^{r}\lambda_{j}$. Indeed,

$\|\widetilde{x}\|_{*}=\sup_{\|y\|=1}|[y,x]|=\sup_{\|y\|=1}\sum_{j=1}^{r}\lambda_{j}|[y,c_{j}]|=\sup_{\|y\|=1}\sum_{j=1}^{r}\lambda_{j}|[P_{2}(c_{j})y,c_{j}]|\leq\sup_{\|y\|=1}\sum_{j=1}^{r}\lambda_{j}[c_{j},c_{j}]=\sum_{j=1}^{r}\lambda_{j},$

as $P_{2}(c_{j})y=\mu c_{j}$ for some $\mu\in\mathbb{C}$ and
$|\mu|=\|P_{2}(c_{j})y\|\leq\|y\|=1$. On the other hand,
$y=\sum_{j=1}^{r}c_{j}$ satisfies $\|y\|\leq 1$ and hence
$\|\widetilde{x}\|_{*}\geq|\sum_{j=1}^{r}[c_{j},x]|=\sum_{j=1}^{r}\lambda_{j}$.

We define

$D^{\circ}=\\{x\in V\colon x=\sum_{j=1}^{r}\lambda_{j}c_{j}\mbox{ spectral
decomposition with }\sum_{j=1}^{r}\lambda_{j}\leq 1\\}.$ (6.5)

It follows from the previous observations that the conjugate linear
isomorphism $x\in V\mapsto\widetilde{x}\in V^{*}$ in (6.4) maps $D^{\circ}$
onto $D^{*}$, and hence $D^{\circ}$ is the closed unit ball of a norm on $V$.

Thus, to prove that $V\cup V(\infty)$ is homeomorphic to $D^{*}$, it suffices
to show that there exists a homeomorphism $\varphi$ from $V\cup V(\infty)$
onto $D^{\circ}$, which is what we will do.

######  Remark 6.4.

Given $x\in V$ with spectral decomposition $x=\sum_{i=1}^{r}\lambda_{i}e_{i}$,
so $r$ is the rank of $V$, we have that the eigenvalues
$\lambda_{1}\geq\ldots\geq\lambda_{r}\geq 0$ are unique, but the pairwise
orthogonal minimal tripotents $e_{i}$ need not be unique. We can however
collect terms with equal non-zero eigenvalue in the sum and write
$x=\sum_{i=1}^{s}\mu_{i}c_{i}$, where $\mu_{1}>\ldots>\mu_{s}>0$ and the
$c_{i}$’s are (not necessarily minimal) pairwise orthogonal tripotents. In
this case both the $\mu_{i}$’s and $c_{i}$’s are unique, see [28, Corollary
3.12]. For clarity we refer to this decomposition of $x$, as the unique
spectral decomposition.

We define the map $\varphi\colon V\cup V(\infty)\longrightarrow{D}^{\circ}$ by

$\varphi(x)=\frac{\sum_{i=1}^{r}(\exp\lambda_{i}-\exp(-\lambda_{i}))c_{i}}{\sum_{i=1}^{r}(\exp\lambda_{i}+\exp(-\lambda_{i}))}$
(6.6)

for each $x\in V$ with spectral decomposition
$x=\sum_{i=1}^{r}\lambda_{i}c_{i}\in V$, and

$\varphi(h)=\frac{\sum_{i\in I}\exp(-\alpha_{i})e_{i}}{\sum_{i\in
I}\exp(-\alpha_{i})}$ (6.7)

for $h\in{V}(\infty)$ of the form,

$h(x)=\Lambda_{V_{2}(e_{I})}(-\frac{1}{2}(e_{I}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(e_{I})x+P_{2}(e_{I})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{I})-\sum_{i\in
I}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,e_{i}))\qquad(x\in V).$

To see that $\varphi(x)$ is well-defined note that the right-hand side of
(6.6) is the spectral decomposition of $\varphi(x)$. Moreover, if
$\lambda_{i}=0$, then the correspond coefficient in $\varphi(x)$ is also $0$.
So by switching to the unique spectral decomposition we find that $\varphi(x)$
is well-defined by Remark 6.4. Also $\varphi(h)$ is well defined. Indeed, if
$h$ was expressed as

$h(x)=\Lambda_{V_{2}(c_{J})}(-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})x+P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})-\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))\qquad(x\in V),$

then $e_{I}=c_{J}$ and $a=\sum_{i\in I}\alpha_{i}e_{i}=\sum_{j\in
J}\beta_{j}c_{j}=b$ by Corollary 6.3. Viewing $a$ and $b$ in the
$\mathrm{JB}^{*}$-subtriple $V_{2}(e_{I})=V_{2}(c_{J})$ we may assume after
relabelling that $I=J$ and $\alpha_{i}=\beta_{i}$ for all $i\in I=J$. Now
using the unique spectral decomposition and the fact that $e_{I}=c_{J}$, we
find that $\varphi(h)$ is well-defined.

######  Theorem 6.5.

The map $\varphi\colon V\cup V(\infty)\longrightarrow{D}^{\circ}$ is a
homeomorphism.

The proof of the theorem will be split in to several lemmas. Note that the
interior of $D^{\circ}$, denoted $\mathrm{int}\,{D}^{\circ}$, consists of
those $x=\sum_{j=1}^{r}\lambda_{j}c_{j}$ with $\sum_{j=1}^{r}\lambda_{j}<1$,
and the boundary, $\partial D^{\circ}$, of $D^{\circ}$ are precisely those
$x=\sum_{j=1}^{r}\lambda_{j}c_{j}$ with $\sum_{j=1}^{r}\lambda_{j}=1$. This
follows from the fact that $x\mapsto\widetilde{x}=[\cdot,x]$ is a conjugate
linear isomorphism mapping $D^{\circ}$ onto $B^{*}$.

######  Lemma 6.6.

We have $\varphi(V)\subset\mathrm{int}\,{D}^{\circ}$ and
$\varphi(V(\infty))\subset\partial{D}^{\circ}$.

###### Proof.

The right-hand side of (6.6) is the spectral decomposition of $\varphi(x)$ and

$0\leq(\sum_{i=1}^{r}e^{\lambda_{i}}+e^{-\lambda_{i}})^{-1}(\sum_{i=1}^{r}e^{\lambda_{i}}-e^{-\lambda_{i}})<1,$

so $\varphi(x)\in\mathrm{int}\,D^{\circ}$. Clearly, $\varphi(h)\in\partial
D^{\circ}$. ∎

######  Lemma 6.7.

The map $\varphi\colon V\cup V(\infty)\longrightarrow{D}^{\circ}$ is
continuous on $V$.

###### Proof.

Let $(v_{k})$ be a sequence in $V$ converging to $v\in V$. To show
$\varphi(v_{k})\rightarrow\varphi(v)$ as $k\rightarrow\infty$, we show that
each subsequence of $(v_{k})$ contains a subsequence $(v_{n})$ satisfying
$\varphi(v_{n})\rightarrow\varphi(v)$ as $n\rightarrow\infty$.

For each $k$, let $v_{k}=\sum_{i=1}^{r}\mu_{ik}c_{ik}$ be a spectral
decomposition. By convergence, $(v_{k})$ is bounded. Hence each subsequence of
$(v_{k})$ contains a subsequence

$v_{n}=\sum_{i=1}^{r}\mu_{in}c_{in}$

such that $\mu_{in}\rightarrow\mu_{i}\geq 0$ and $c_{in}\rightarrow c_{i}$ as
$n\rightarrow\infty$, where $c_{1},\ldots,c_{r}$ are mutually orthogonal
minimal tripotents.

It follows that $v=\lim_{n}v_{n}=\sum_{i=1}^{r}\mu_{i}c_{i}$ and

$\varphi(v_{n})=\frac{\sum_{i=1}^{r}(e^{\mu_{in}}-e^{-\mu_{in}})c_{in}}{\sum_{i=1}^{r}e^{\mu_{in}}+e^{-\mu_{in}}}\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\
\frac{\sum_{i=1}^{r}(e^{\mu_{i}}-e^{-\mu_{i}})c_{i}}{\sum_{i=1}^{r}e^{\mu_{i}}+e^{-\mu_{i}}}=\varphi(v)$

as $n\rightarrow\infty$, which completes the proof. ∎

######  Lemma 6.8.

We have $\varphi(V)=\mathrm{int}\,{D}^{\circ}$.

###### Proof.

As $\varphi$ is continuous on $V$ and maps $V$ into
$\mathrm{int}\,{D}^{\circ}$, we know from the Brouwer invariance of domain
theorem that $\varphi(V)$ is open in $\mathrm{int}\,{D}^{\circ}$. Suppose, for
the sake of contradiction, that $\varphi(V)\neq\mathrm{int}\,{D}^{*}$. Then
there exists $w\in\partial\varphi(V)\cap\mathrm{int}\,{D}^{\circ}$. Let
$(v_{n})$ in $V$ be such that $\varphi(v_{n})\longrightarrow w$. As $\varphi$
is continuous on $V$, we must have that $\|v^{n}\|\longrightarrow\infty$.

Consider the spectral decomposition $v_{n}=\sum_{i=1}^{r}\lambda_{in}c_{in}$.
After taking a subsequence we may assume that (1) $c_{in}\longrightarrow
c_{i}$, and (2)
$\alpha_{in}=\lambda_{1n}-\lambda_{in}\longrightarrow\alpha_{i}\in[0,\infty]$,
for all $i=1,\ldots,r$. Note that
$\lambda_{1n}=\|v_{n}\|\longrightarrow\infty$.

Let $I=\\{i\colon\alpha_{i}<\infty\\}$. Then $1\in I$ and

$\varphi(v_{n})=\frac{\sum_{i=1}^{r}(e^{\lambda_{in}}-e^{-\lambda_{in}})c_{in}}{\sum_{i=1}^{r}e^{\lambda_{in}}+e^{-\lambda_{in}}}=\frac{\sum_{i=1}^{r}(e^{-\alpha_{in}}-e^{-\lambda_{1n}-\lambda_{in}})c_{in}}{\sum_{i=1}^{r}e^{-\alpha_{in}}+e^{-\lambda_{1n}-\lambda_{in}}}\longrightarrow\frac{\sum_{i\in
I}e^{-\alpha_{i}}c_{i}}{\sum_{i\in I}e^{-\alpha_{i}}}.$

This implies that $w=\lim_{n\longrightarrow\infty}\varphi(v_{n})\in\partial
D^{\circ}$, which contradicts the assumption that
$w\in\mathrm{int}\,{D}^{\circ}$. ∎

######  Lemma 6.9.

The map $\varphi$ satisfies $\varphi(V(\infty))=\partial{D}^{\circ}$.

###### Proof.

Let $x\in\partial{D}^{\circ}$. Then $x$ has spectral decomposition
$x=\sum_{i=1}^{r}\lambda_{i}e_{i}$ with $\sum_{i=1}^{r}\lambda_{i}=1$ and
there exists a $p\in\\{1,\ldots,r\\}$ such that
$\lambda_{1}\geq\ldots\geq\lambda_{p}>0$ and $\lambda_{s}=0$ for $p<s\leq r$.

For $i=1,\ldots,p$ put $\mu_{i}=-\log\lambda_{i}$ and
$\alpha_{i}=\mu_{i}-\mu_{1}$. Then $\alpha_{i}\geq 0$ and $\alpha_{1}=0$. Now
consider the horofunction $h\in V(\infty)$ given by

$h(x)=\Lambda_{V_{2}(e)}(-\frac{1}{2}(e{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,P_{2}(e)x+P_{2}(e)x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,e)-\sum_{i=1}^{p}\alpha_{i}(e_{i}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,e_{i}))\qquad(x\in V).$

Then we have

$\varphi(h)=\frac{\sum_{i=1}^{p}\exp(-\alpha_{i})e_{i}}{\sum_{i=1}^{p}\exp(-\alpha_{i})}=\frac{\sum_{i=1}^{p}\lambda_{i}e_{i}}{\sum_{i=1}^{p}\lambda_{i}}=x.$

∎

######  Lemma 6.10.

The map $\varphi$ is injective on $V\cup V(\infty)$.

###### Proof.

Suppose first that $\varphi(x)=\varphi(y)$. Let
$x=\sum_{i=1}^{r}\lambda_{i}c_{i}$ and $y=\sum_{i=1}^{r}\mu_{i}d_{i}$ be the
spectral decompositions. Then we have

$\varphi(x)=\frac{\sum_{i=1}^{r}(e^{\lambda_{i}}-e^{-\lambda_{i}})c_{i}}{\sum_{i=1}^{r}e^{\lambda_{i}}+e^{-\lambda_{i}}}=\frac{\sum_{i=1}^{r}(e^{\mu_{i}}-e^{-\mu_{i}})d_{i}}{\sum_{i=1}^{r}e^{\mu_{i}}+e^{-\mu_{i}}}=\varphi(y)$

where the coefficients of the minimal tripotents of both sides are decreasing
through the order of the indices. If we let for $j=1,\ldots r$,

$\alpha_{j}=(\sum_{i=1}^{r}e^{\lambda_{i}}+e^{-\lambda_{i}})^{-1}(e^{\lambda_{j}}-e^{-\lambda_{j}})\mbox{\quad
and\quad}\beta_{j}=(\sum_{i=1}^{r}e^{\mu_{i}}+e^{-\mu_{i}})^{-1}(e^{\mu_{j}}-e^{-\mu_{j}}),$

then $\alpha_{j}=\beta_{j}$ by Remark 6.4. It now follows from [26, Lemma 3.7]
that $\lambda_{j}=\mu_{j}$ for $j=1,\ldots,r$.

Note that $\alpha_{i}=0$ if and only if $\lambda_{i}=0$, and similarly
$\beta_{i}=0$ if and only if $\mu_{i}=0$. So by considering the unique
spectral decompositions of $\varphi(x)$ and $\varphi(y)$ and using Remark 6.4
we find that $x=y$.

Now suppose $h,h^{\prime}\in V(\infty)$ with $\varphi(h)=\varphi(h^{\prime})$.
Let $h$ be of the form (5.8) and $h^{\prime}$ of the form

$h^{\prime}(x)=\Lambda_{V_{2}(c_{J})}(-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})x+P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})-\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j})).$

Then we have

$\frac{\sum_{i\in I}\exp(-\alpha_{i})e_{i}}{\sum_{i\in
I}\exp(-\alpha_{i})}=\frac{\sum_{j\in J}\exp(-\beta_{j})c_{j}}{\sum_{j\in
J}\exp(-\beta_{j})}$

where the coefficients of the minimal tripotents on both sides are strictly
positive. By relabelling the indices, we may assume that $I=\\{1,\ldots,p\\}$
and $0=\alpha_{1}\leq\alpha_{2}\leq\ldots\leq\alpha_{p}$. Likewise we can
assume that $J=\\{1,\ldots,q\\}$ and
$0=\beta_{1}\leq\beta_{2}\leq\ldots\leq\beta_{q}$. Since the norm of both
sides above in $V$ are equal, we have

$\sum_{i\in I}\exp(-\alpha_{i})=\sum_{j\in J}\exp(-\beta_{j}).$

By Remark 6.4, we have $p=q$, $e^{-\alpha_{i}}=e^{-\beta_{i}}$,
$e_{I}=e_{1}+\cdots+e_{p}=c_{1}+\cdots+c_{q}=c_{J}$ and
$\sum_{i}\alpha_{i}e_{i}=\sum\beta_{i}c_{i}.$ Hence $h=h^{\prime}$ by
Corollary 6.3. ∎

######  Lemma 6.11.

If $(a_{k})$ in $V$ is such that $h_{a_{k}}\longrightarrow h\in V(\infty)$,
then $\varphi(a_{k})\longrightarrow\varphi(h)$.

###### Proof.

Let $h$ be given by (5.8). To show that
$\varphi(a_{k})\longrightarrow\varphi(h)$, we show that each subsequence of
$(\varphi(a_{k}))$ has a convergent subsequence with limit $\varphi(h)$. So
let $(\varphi(a_{m}))$ be a subsequence. Using the spectral decomposition we
write $a_{m}=\sum_{i=1}^{r}\mu_{im}c_{im}$ with
$\mu_{1m}\geq\ldots\geq\mu_{rm}\geq 0$. As $h$ is a horofunction,
$\mu_{1m}=\|a_{m}\|\longrightarrow\infty$ by Lemma 2.2.

After taking a subsequence we may assume that
$\beta_{im}=\mu_{1m}-\mu_{im}\longrightarrow\beta_{i}\in[0,\infty]$ and
$c_{im}\longrightarrow c_{i}$ for all $i$. Let
$J=\\{i\colon\beta_{i}<\infty\\}=\\{1,\ldots,q\\}$ and note that $q\geq 1$, as
$\beta_{1}=0$.

It follows from Lemma 5.1 that $h_{a_{m}}\longrightarrow h^{\prime}$ where

$h^{\prime}(x)=\Lambda_{V_{2}(c_{J})}(-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})x+P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})-\sum_{j\in
J}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))\qquad(x\in V),$

and $h^{\prime}\in V(\infty)$ by Theorem 5.2. As $h_{a_{k}}\longrightarrow h$,
we know that $h^{\prime}=h$, and hence $e_{I}=c_{J}$ and

$\sum_{i\in I}\alpha_{i}e_{i}=\sum_{j\in J}\beta_{j}c_{j}$

by Corollary 6.3.

We can relabel the $\alpha_{i}$’s such that $I=\\{1,\ldots,p\\}$ and
$0=\alpha_{1}\leq\alpha_{2}\leq\ldots\leq\alpha_{p}$. It follows from Remark
6.4 that $p=q$ and $\alpha_{i}=\beta_{i}$ for all $i\in\\{1,\ldots,p\\}$.
Moreover,
$\sum_{i=1}^{p}e^{-\alpha_{i}}e_{i}=\sum_{i=1}^{p}e^{-\beta_{i}}c_{i}$.

As

$\varphi(a_{m})=\frac{\sum_{i=1}^{r}(e^{\mu_{im}}-e^{-\mu_{im}})c_{im}}{\sum_{i=1}^{r}e^{\mu_{im}}-e^{-\mu_{im}}}=\frac{\sum_{i=1}^{r}(e^{-\beta_{im}}-e^{-\mu_{1m}-\mu_{im}})c_{im}}{\sum_{i=1}^{r}e^{-\beta_{im}}-e^{-\mu_{1m}-\mu_{im}}},$

we find that

$\lim_{m}\varphi(a_{m})=(\sum_{i=1}^{p}e^{-\beta_{i}})^{-1}\sum_{i=1}^{p}e^{-\beta_{i}}c_{i}=(\sum_{i=1}^{p}e^{-\alpha_{i}})^{-1}\sum_{i=1}^{p}e^{-\alpha_{i}}e_{i}.$

Thus, $\varphi(a_{m})\longrightarrow\varphi(h)$, which completes the proof. ∎

To prove continuity of $\varphi$ on the boundary $V(\infty)$, we need the
following technical lemma.

######  Lemma 6.12.

Let $J\subset\\{1,\ldots,r\\}$ be nonempty, and for each $n$, let
$\\{c_{jn}\colon j\in J\\}$ be a collection of mutually orthogonal minimal
tripotents in $V$ such that $c_{jn}\longrightarrow c_{j}$ for all $j\in J$.
For $j\in J$ let $(\beta_{jn})_{n}$ a sequence in $[0,\infty)$ converging to
$\beta_{j}\in[0,\infty]$, with $\min_{j\in J}\beta_{jn}=0$ for each $n$. If we
let $J^{\prime}=\\{j\in J\colon\beta_{j}<\infty\\}\neq\emptyset$ and consider
the horofunctions,

$h_{n}(x)=\Lambda_{V_{2}(c_{Jn})}(-\frac{1}{2}(c_{Jn}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jn})x+P_{2}(c_{Jn})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jn})-\sum_{j\in
J}\beta_{jn}(c_{jn}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jn}))\qquad(x\in V),$

where $c_{Jn}=\sum_{i\in J}c_{jn}$, then

$\lim_{n\longrightarrow\infty}h_{n}(x)=\Lambda_{V_{2}(c_{J^{\prime}})}(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))$ (6.8)

with $c_{J^{\prime}}=\sum_{j\in J^{\prime}}c_{j}$.

###### Proof.

We show that each subsequence of $(h_{n}(x))$ has a convergent subsequence
with limit the right-hand side of (6.8). Pick a subsequence $(h_{k}(x))$. As
$\\{u\in V\colon\langle u,u\rangle\ =1\\}$ is compact, there exists $w^{k}\in
V_{2}(c_{Jk})$ with $\langle w^{k},w^{k}\rangle=1$ and

$h_{k}(x)=\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})-\sum_{j\in
J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))w^{k},w^{k}\rangle.$

Taking a subsequence, we may assume that $w^{k}\longrightarrow w$. For each
$k$, let

$V=\bigoplus_{0\leq s\leq t;s,t\in J}V_{st}^{k}$ (6.9)

be the Peirce decomposition of $V$ with respect to the tripotents
$\\{c_{jk}\colon j\in J\\}$, and let

$V=\bigoplus_{0\leq s\leq t;s,t\in J}V_{st}$

be the decomposition with respect to the tripotents $\\{c_{j}\colon j\in
J\\}$. We show

$w\in\bigoplus_{s,t\in J^{\prime}\colon s\leq t}V_{st}=V_{2}(c_{J^{\prime}}).$
(6.10)

Let $w_{st}^{k}\in V_{st}^{k}$ be the $(s,t)$-component of $w^{k}$ in the
Peirce decomposition in (6.9). Then we have $w_{st}^{k}\longrightarrow
w_{st}\in V_{st}$ and

$-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{Jk})w_{st}^{k}\longrightarrow-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})x+P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})w_{st}.$

Moreover,

$\sum_{j\in J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk})w^{k}_{st}=\sum_{j\in
J}\left(\frac{\delta_{sj}\beta_{jk}+\delta_{tj}\beta_{jk}}{2}\right)w^{k}_{st}.$

Recall that if $s,t\in J$ with $\\{s,t\\}\not\subset J^{\prime}$, then
$\beta_{sk}\longrightarrow\infty$ or $\beta_{tk}\longrightarrow\infty$. As
$h_{k}(x)\geq-\|x\|$ for all $k$ and $\beta_{jk}\geq 0$ for all $j\in J$, we
find that $w^{k}_{st}\longrightarrow 0$ for all $s,t\in J$ with
$\\{s,t\\}\not\subset J^{\prime}$. This implies that (6.10) holds.

Next, we make use of the following fact. Let $e$ be a tripotent. If $v\in
V_{2}(e)\cup V_{0}(e)$, $y\in V$ and $z\in V_{2}(e)$, then
$P_{2}(e)\\{v,y,z\\}=\\{P_{2}(e)v,P_{2}(e)y,z\\}$ and
$P_{2}(e)\\{y,v,z\\}=\\{P_{2}(e)y,P_{2}(e)v,z\\}$. In particular,

$\displaystyle\langle-\frac{1}{2}(c_{J}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J})x+P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J})w,w\rangle$ $\displaystyle=$
$\displaystyle\langle-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})P_{2}(c_{J})x+P_{2}(c_{J^{\prime}})P_{2}(c_{J})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})w,w\rangle$
$\displaystyle=$
$\displaystyle\langle-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})w,w\rangle,$

as $P_{2}(c_{J^{\prime}})P_{2}(c_{J})=P_{2}(c_{J^{\prime}})$.

It follows that

$\displaystyle\limsup_{k\longrightarrow\infty}h_{k}(x)$ $\displaystyle=$
$\displaystyle\limsup_{k\longrightarrow\infty}\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})-\sum_{j\in
J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))w^{k},w^{k}\rangle$ $\displaystyle\leq$
$\displaystyle\lim_{k\longrightarrow\infty}\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})-\sum_{j\in
J^{\prime}}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))w^{k},w^{k}\rangle$ $\displaystyle=$
$\displaystyle\langle(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}}-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))w,w\rangle.$

On the other hand,

$\liminf_{k\longrightarrow\infty}h_{k}(x)\geq\liminf_{k\longrightarrow\infty}\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})\\\ -\sum_{j\in
J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))P_{2}(c_{J^{\prime}k})w^{k},P_{2}(c_{J^{\prime}k})w^{k}\rangle\langle
P_{2}(c_{J^{\prime}k})w^{k},P_{2}(c_{J^{\prime}k})w^{k}\rangle^{-1}\\\
\longrightarrow\langle(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}}-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))w,w\rangle.$

Thus,

$\lim_{k\longrightarrow\infty}h_{k}(x)=\langle(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}}-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))w,w\rangle.$

Since $w\in V_{2}(c_{J^{\prime}})$, we find that

$\lim_{k\longrightarrow\infty}h_{k}(x)\leq\Lambda_{V_{2}(c_{J^{\prime}})}(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j})).$

To show that this is an equality, let $u\in V_{2}(c_{J^{\prime}})$ be such
that

$\Lambda_{V_{2}(c_{J^{\prime}})}(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))\\\
=\langle(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))u,u\rangle.$

As $P_{2}(c_{J^{\prime}k})u\in V_{2}(c_{Jk})$, we have that

$\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})-\sum_{j\in
J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))w^{k},w^{k}\rangle\\\
\geq\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})\\\ -\sum_{j\in
J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))P_{2}(c_{J^{\prime}k})u,P_{2}(c_{J^{\prime}k})u\rangle\langle
P_{2}(c_{J^{\prime}k})u,P_{2}(c_{J^{\prime}k})u\rangle^{-1}.$

Note that, as $\langle
P_{2}(c_{J^{\prime}k})u,P_{2}(c_{J^{\prime}k})u\rangle\longrightarrow\langle
u,u\rangle=1$, the right-hand side is defined for all $k$ large.

So,

$\langle(-\frac{1}{2}(c_{Jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{Jk})x+P_{2}(c_{Jk})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{Jk})-\sum_{j\in
J}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))P_{2}(c_{J^{\prime}k})u,P_{2}(c_{J^{\prime}k})u\rangle\\\
=\langle(-\frac{1}{2}(c_{J^{\prime}k}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}k})x+P_{2}(c_{J^{\prime}k})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}k})-\sum_{j\in
J^{\prime}}\beta_{jk}(c_{jk}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jk}))P_{2}(c_{J^{\prime}k})u,P_{2}(c_{J^{\prime}k})u\rangle\\\
\longrightarrow\langle(-\frac{1}{2}(c_{J^{\prime}}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J^{\prime}})x+P_{2}(c_{J^{\prime}})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J^{\prime}})-\sum_{j\in
J^{\prime}}\beta_{j}(c_{j}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{j}))u,u\rangle,$

which proves (6.8). ∎

######  Lemma 6.13.

The map $\varphi\colon V\cup V(\infty)\longrightarrow{D}^{\circ}$ is
continuous on $V(\infty)$.

###### Proof.

Let $h_{n}\longrightarrow h$ in $V(\infty)$. We show that each subsequence
$(\varphi(h_{k}))$ of $(\varphi(h_{n}))$ has a convergent subsequence with
limit $\varphi(h)$. As $h_{n}$ is horofunction, we can express it as

$h_{n}(x)=\Lambda_{V_{2}(c_{J_{n}n})}(-\frac{1}{2}(c_{J_{n}n}{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule
height=0.4pt}}\,}\,P_{2}(c_{J_{n}n})x+P_{2}(c_{J_{n}n})x{\,\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt}\hrule height=0.4pt}}\,}\,c_{J_{n}n})-\sum_{j\in
J_{n}}\beta_{jn}(c_{jn}{\,\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt}\hrule
height=0.4pt}}\,}\,c_{jn}))\qquad(x\in V).$

After taking a successive subsequences we may assume that $J_{k}=J$ for all
$k$, $c_{jk}\longrightarrow c_{j}$ and
$\beta_{jk}\longrightarrow\beta_{j}\in[0,\infty]$ for all $j\in J$. Note that
$\min_{j\in J}\beta_{j}=0$, as $\min_{j\in J}\beta_{jk}=0$ for all $k$.

Now let $J^{\prime}=\\{j\in J\colon\beta_{j}<\infty\\}$. By Lemma 6.12, we