On the extension of isometries between the unit spheres of a C$^*$-algebra and $B(H)$

Abstract

Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley’s problem in the setting of $B(H)$ spaces.

1. Introduction

Let $X$ and $Y$ be normed spaces, whose unit spheres are denoted by $S(X)$ and $S(Y)$, respectively. Suppose $T: X\to Y$ is a surjective real linear isometry. The restriction $T|_{S(X)} : S(X) \to S(Y)$ defines a surjective isometry. The so-called Tingley’s problem, named after the contribution of D. Tingley 36, asks if every surjective isometry $f:S(X)\to S(Y)$ arises in this way, or equivalently, if every surjective isometry $f : S(X) \to S(Y )$ admits an extension to a surjective real linear isometry $T: X\to Y$. Tingley’s achievements show that, for finite dimensional normed spaces $X$ and $Y$, every surjective isometry $f: S(X)\to S(Y)$ satisfies $f(-x) = -f(x)$ for every $x\in S(X)$ (see 36, THEOREM on page 377).

A solution to Tingley’s problem has been pursued by many researchers during the last thirty years. Positive answers to Tingley’s problem have been established for $\ell ^p (\Gamma )$ spaces with $1\leq p\leq \infty$ (see 7810 and 11), $L^{p}(\Omega , \Sigma , \mu )$ spaces, where $(\Omega , \Sigma , \mu )$ is a $\sigma$-finite measure space and $1\leq p\leq \infty$ (compare 2930 and 31), and $C_0(L)$ spaces (see 37). Tingley’s problem also admits a positive solution in the case of finite dimensional polyhedral Banach spaces (see 21). The reader is referred to the surveys 12 and 38 for additional details.

In the non-commutative setting, Tingley’s problem has been solved for surjective isometries between the unit spheres of two finite dimensional C$^*$-algebras (see 34) and for surjective isometries between the unit spheres of two finite von Neumann algebras 35. A more recent contribution solves Tingley’s problem for surjective isometries between the unit spheres of spaces, $K(H),$ of compact linear operators on a complex Hilbert space $H$, or more generally, for surjective isometries between the unit spheres of two compact C$^*$-algebras 25, Theorem 3.14. The novelties in 25 are based on the application of techniques of JB$^*$-triples, and in the above-referenced paper Tingley’s problem is also solved for surjective isometries between the unit spheres of two weakly compact JB$^*$-triples of rank greater than or equal to 5. In 17 we establish a complete solution to Tingley’s problem for arbitrary weakly compact JB$^*$-triples. A solution to Tingley’s problem for spaces of trace class operators is presented in 18.

Tingley’s problem for surjective isometries between the unit spheres of two $B(H)$ spaces seems to be the last frontier in the studies on Tingley’s problem. This paper is devoted to providing a complete solution in this case.

The results in 172534 are based, among other techniques, on those theorems describing the (maximal) norm closed proper faces of the closed unit ball of a C$^*$-algebra (see 2) or of a JB$^*$-triple (see 13). Throughout the paper, the closed unit ball of a normed space $X$ will be denoted by $\mathcal{B}_X$. It is shown in 172534 that, for a compact C$^*$-algebra $A$ (respectively, a weakly compact JB$^*$-triple $E$), the norm closed faces of $\mathcal{B}_{A}$ are determined by finite rank partial isometries in $A$ (respectively, by finite rank tripotents in $E$). However, for a general C$^*$-algebra $A$ the maximal proper faces of $\mathcal{B}_{A}$ are determined by minimal partial isometries in $A^{**}$ (see Section 2 for more details). This is a serious obstacle which makes invalid the arguments in 1725 in the case of $B(H)$.

To avoid the difficulties mentioned in the previous paragraph, our first geometric result shows that a surjective isometry $f$ from the unit sphere of a C$^*$-algebra $A$ onto the unit sphere of $B(H)$ maps minimal partial isometries in $A$ into minimal partial isometries in $B(H)$ (see Theorem 2.5). Apart from the just commented geometric tools, our arguments are based on techniques of functional analysis and linear algebra. In our main result we prove that, given two complex Hilbert spaces $H$ and $K$, every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K) \to B(H)$ (see Theorem 3.2). In the final result we show that the same conclusion remains true when $B(H)$ spaces are replaced by $\ell _{\infty }$-sums of $B(H)$ spaces (see Theorem 3.4). The next natural question beyond these conclusions is whether Tingley’s problem admits or not a positive answer for Cartan factors and atomic JBW$^*$-triples.

It should be remarked here that the solution to Tingley’s problem for surjective isometries between the unit spheres of $K(H)$-spaces in 1725 and the solution presented in this note for surjective isometries between the unit spheres of $B(H)$-spaces are completely independent results.

2. Surjective isometries between the unit spheres of two C$^*$-algebras

In this section we carry out a study of the geometric properties of those surjective isometries between the unit spheres of two C$^*$-algebras with special interest on C$^*$-algebras of the form $B(H)$. We begin by gathering some technical results and concepts needed for later purposes.

Proposition 2.1 (3, Lemma 5.1 and 32, Lemma 3.5).

Let $X$, $Y$ be Banach spaces, and let $T : S(X) \to S(Y )$ be a surjective isometry. Then $C$ is a maximal convex subset of $S(X)$ if and only if $T(C)$ is that of $S(Y)$. Then $C$ is a maximal proper (norm-closed) face of $\mathcal{B}_X$ if and only if $f(C)$ is a maximal proper (norm-closed) face of $\mathcal{B}_Y$.■

An interesting generalization of the Mazur-Ulam theorem was established by P. Mankiewicz in 22, who proved that, given two convex bodies $V\subset X$ and $W\subset Y$, every surjective isometry $g$ from $V$ onto $W$ can be uniquely extended to an affine isometry from $X$ onto $Y$. Consequently, every surjective isometry between the closed unit balls of two Banach spaces $X$ and $Y$ extends uniquely to a real linear isometric isomorphism from $X$ into $Y$.

Let $a$ and $b$ be two elements in a C$^*$-algebra $A$. We recall that $a$ and $b$ are orthogonal ($a\perp b$ for short) if $ab^* = b^* a =0$. Symmetric elements in $A$ are orthogonal if and only if their product is zero.

For each element $a$ in a C$^*$-algebra $A$, the symbol $|a|$ will denote the element $(a^* a)^{\frac{1}{2}}\in A$. Throughout this note, for each $x\in A$, $\sigma (x)$ will denote the spectrum of the element $x$. We observe that $\sigma (|a|)\cup \{0\} = \sigma (|a^*|)\cup \{0\}$, for every $a\in A$. Let $a = v |a|$ be the polar decomposition of $a$ in $A^{**}$, where $v$ is a partial isometry in $A^{**}$, which, in general, does not belong to $A$ (compare 27). It is further known that $v^*v$ is the range projection of $|a|$ ($r(|a|)$ for short), and for each $h\in C(\sigma (|a|)),$ with $h(0)=0$ the element $v h(|a|)\in A$ (see 1, Lemma 2.1).

Proposition 2.1 points out the importance of an appropriate description of the maximal proper faces of the closed unit ball $\mathcal{B}_A$ of a C$^*$-algebra $A$. A complete study was established by C. A. Akemann and G. K. Pedersen in 2. When $A$ is a von Neumann algebra, weak$^*$-closed faces in $\mathcal{B}_A$ were originally determined by C. M. Edwards and G. T. Rüttimann in 14, who proved that general weak$^*$-closed faces in $\mathcal{B}_A$ have the form

$$\begin{equation*} F_v=v + (1 - vv^*) \mathcal{B}_{A} (1 - v^*v) = \{x\in \mathcal{B}_A:\ xv^* = vv^*\}, \end{equation*}$$

for some partial isometry $v$ in $A$. Actually, the mapping $v\mapsto F_v$ is an anti-order isomorphism from the complete lattice of partial isometries in $A$ onto the complete lattice of weak$^*$-closed faces of $\mathcal{B}_A$, where the partial order in the set of partial isometries of $A$ is given by $v\leq u$ if and only if $u = v + (1-vv^*) u (1-v^*v)$ (see 14, Theorem 4.6).

However, partial isometries in a general C$^*$-algebra $A$ are not enough to determine all the norm-closed faces in $\mathcal{B}_A$, even more after recalling the existence of C$^*$-algebras containing no partial isometries. In the general case, certain partial isometries in the second dual, $A^{**}$, are required to determine the facial structure of $\mathcal{B}_A$. We recall that a projection $p$ in $A^{**}$ is called open if $A\cap (p A^{**} p)$ is weak$^*$-dense in $p A^{**} p$ (see 24, §3.11 and 28, §III.6). A projection $p \in A^{**}$ is said to be closed if $1-p$ is open. A closed projection $p\in A^{**}$ is compact if $p\leq x$ for some positive norm-one element $x \in A$. A partial isometry $v\in A^{**}$ belongs locally to $A$ if $v^*v$ is a compact projection and there exists a norm-one element $x$ in $A$ satisfying $v = x v^*v$ (compare 2, Remark 4.7).

It is shown in 2, Lemma 4.8 and Remark 4.11 that “the partial isometries that belong locally to $A$ are obtained by taking an element $x$ in $A$ with norm 1 and polar decomposition $x = u |x|$ (in $A^{**}$), and then letting $v = ue$ for some compact projection $e$ contained in the spectral projection $\chi _{_{\{1\}}}(|x|)$ of $|x|$ corresponding to the eigenvalue 1”.

It should be noted that a partial isometry $v$ in $A^{**}$ belongs locally to $A$ if and only if it is compact in the sense introduced by C. M. Edwards and G. T. Rüttimann in 15, Theorem 5.1.

The facial structure of the unit ball of a C$^*$-algebra is completely described by the following result due to C. A. Akemann and G. K. Pedersen.

Theorem 2.2 (2, Theorem 4.10).

Let $A$ be a C$^*$-algebra. The norm-closed faces of the unit ball of $A$ have the form

$$\begin{equation*} F_v=\left(v + (1 - vv^*) \mathcal{B}_{A^{**}} (1 - v^*v)\right)\cap \mathcal{B}_{A} = \{x\in \mathcal{B}_A:\ xv^* = vv^*\}, \end{equation*}$$

for some partial isometry $v$ in $A^{**}$ belonging locally to $A$. Actually, the mapping $v\mapsto F_v$ is an anti-order isomorphism from the complete lattice of partial isometries in $A^{**}$ belonging locally to $A$ onto the complete lattice of norm-closed faces of $\mathcal{B}_A$.■

A non-zero partial isometry $e$ in a C$^*$-algebra $A$ is called minimal if $ee^*$ (equivalently, $e^* e$) is a minimal projection in $A$, that is, $ee^* A e e^* =\mathbb{C} ee^*.$ By Kadison’s transitivity theorem minimal partial isometries in $A^{**}$ belong locally to $A$, and hence every maximal proper face of the unit ball of a C$^*$-algebra $A$ is of the form

$$\begin{equation*} \left(v + (1 - vv^*) \mathcal{B}_{A^{**}} (1 - v^*v)\right)\cap \mathcal{B}_{A} \end{equation*}$$

for a unique minimal partial isometry $v$ in $A^{**}$ (compare 2, Remark 5.4 and Corollary 5.5).

Our main goal in this section is to show that a surjective isometry $f : S(A)\to S(B)$ between the unit spheres of two C$^*$-algebras maps minimal partial isometries into minimal partial isometries. In a first step we shall show that, for each minimal partial isometry $e$ in $A$, 1 is isolated in the spectrum of $|f(e)|$.

Theorem 2.3.

Let $A$ and $B$ be C$^*$-algebras, and suppose that $f: S(A) \to S(B)$ is a surjective isometry. Let $e$ be a minimal partial isometry in $A$. Then $1$ is isolated in the spectrum of $|f(e)|$.

Proof.

Since $e$ also is a minimal partial isometry in $A^{**}$ and belongs (locally) to $A$, the set $F_{e} =e + (1 - ee^*) \mathcal{B}_A (1 - e^*e)$ is a maximal proper face of $\mathcal{B}_{A}$. Applying Proposition 2.1 and Theorem 2.2 we deduce the existence of a minimal partial isometry $w$ in $B^{**}$ such that

$$\begin{equation} f(F_e) =\! F_{w} =\!\left(w + (1 - ww^*) \mathcal{B}_{B^{**}} (1 - w^*w)\right)\cap \mathcal{B}_B \!=\{x\in \mathcal{B}_A: xw^* = ww^*\}. {\tag{1}} \end{equation}$$

Since $f(e)\in f(F_e) = F_{w}$ we have $f(e) = w + (1 - ww^*) f(e) (1 - w^*w)$.

Arguing by contradiction, we assume that $1$ is not isolated in $\sigma ( |f(e)|)$. Let $f(e) = r |f(e)|$ denote the polar decomposition of $f(e)$.

By assumptions we can find $t_0\in \sigma (|f(e)|)$ satisfying $\frac{3}{\sqrt {10}}<{t_0}<1$. Let us consider the functions ${h_1}$ and $h_2$ in the unit sphere of $C_0(\sigma (|f(e)|))$ given by

$$\begin{align*} h_1(t)&:= \begin{cases} \frac{t}{t_0} & \text{if $0\leq t\leq t_0$,} \\ \text{affine} & \text{if $t_0\leq t\leq \frac{1}{2} (t_0+1)$,} \\ 0 & \text{if $\frac{1}{2} (t_0+1)\leq t\leq 1$;} \\ \end{cases}\\ h_2(t)&:= \begin{cases} 0 & \text{if $0\leq t\leq \frac{1}{2} (t_0+1)$,} \\ \text{affine} & \text{if $\frac{1}{2} (t_0+1)\leq t\leq 1$,} \\ 1 & \text{if $ t=1$.} \\ \end{cases} \end{align*}$$

We set $\widehat{x} = r h_1 (|f(e)|)$ and $\widehat{y} = r h_2 (|f(e)|)$. Obviously, $h_1 (|f(e)|)$ and $h_2 (|f(e)|)$ are positive elements in $S(B)$ satisfying $h_1 (|f(e)|) h_2 (|f(e)|)=0.$ Since

$$\begin{equation*} \widehat{x} \widehat{y}^* = r h_1 (|f(e)|) h_2 (|f(e)|) r^* =0, \end{equation*}$$

and

$$\begin{equation*} \widehat{y}^* \widehat{x} = h_2 (|f(e)|) r^* r h_1 (|f(e)|) =h_2 (|f(e)|) h_1 (|f(e)|) =0, \end{equation*}$$

it follows that $\hat{x}\perp \hat{y}.$

Let $x=f^{-1}(-\hat{x})\in S(A)$ and $y=f^{-1}(\hat{y})\in S(A)$. Since $f$ is an isometry we deduce that

$$\begin{equation*} 1=\|\hat{x}+\hat{y}\|=\|\hat{y}- (-\hat{x})\|= \|f(y)-f(x)\|= \|y-x\|, \end{equation*}$$

and

$$\begin{equation*} 1+t_0=\|f(e)+\hat{x}\|=\|f(e)-f(x)\|=\|e-x\|. \end{equation*}$$

We recall that, from 1, $f(e) = w + k$ where $k=(1 - ww^*) f(e) (1 - w^*w)$ satisfies $k^* w = 0 = w k^*$, which proves that $k\perp w$. Let $r_0$ denote the (unique) partial isometry appearing in the polar decomposition of $k$. Since $r$ is the partial isometry in the polar decomposition of $f(e)$, $w\perp k$, and $f(e) = w + k$, it follows that $r = w + r_0$ with $r_0\perp w$. We also know that $|f(e)| = w^*w + |k|$, and hence a simple application of the continuous functional calculus (having in mind that $h_2 (1) =1$) shows that $h_2 (|f(e)|) = w^* w + h_2 (|k|)$, with $w^* w \perp h_2 (|k|)$. We therefore have

$$\begin{equation} \hat{y} w^* = r h_2 (|f(e)|) w^* = r w^* w w^*+ r h_2 (|k|) w^* = r w^* =(w+r_0) w^* = w w^*, {\tag{2}} \end{equation}$$

which implies that $\hat{y} \in F_w,$ and consequently $y =f^{-1}(\hat{y})\in F_e$ (see 1).

We claim that

$$\begin{equation} \| ee^* (e-x) e^*e\|>1. {\tag{3}} \end{equation}$$

The element $e-x$ has norm $1+t_0>1$. Suppose that $\{H_i\}_{I}$ is a family of complex Hilbert spaces and $\pi : A\to \bigoplus _i^{\ell _{\infty }} B(H_i)$ is an isometric $^*$-homomorphism with weak$^*$-dense range (we can consider, for example, the atomic representation of $A$ 24, 4.3.7, where the family $I$ is precisely the set of all pure states of $A$ and $\pi$ is the direct sum of all the irreducible representations associated with the pure states 24, Theorem 3.13.2). For each $j\in I$, let $P_j$ denote the projection of $\bigoplus _i^{\ell _{\infty }} B(H_i)$ onto $B(H_j)$ and let $\pi _j = P_j \circ \pi$. Clearly, $\pi _j$ is a $^*$-homomorphism with weak$^*$-dense range. Since $e$ is a minimal partial isometry, there exists a unique $i_0\in I$ such that $\pi _{i_0} (e)$ is a non-zero (minimal) partial isometry and $\pi _j (e) =0$, for every $j\neq i_0$. We also know that $\|x\|=1$, and thus $\| \pi _{i_0} (e-x)\|= 1+{t_0}.$

Let $\pi _{i_0} (e-x)= u |\pi _{i_0} (e-x)|$ be the polar decomposition of $\pi _{i_0} (e-x)$ in $B(H_{i_0})$. Take $0<\varepsilon <t_0 -\frac{3}{\sqrt {10}}$. Since $\| |\pi _{i_0} (e-x)| \|=1+t_0$, we can find a minimal projection $q= \xi \otimes \xi \in B(H_{i_0})$ with $\|\xi \|=1$ in $H_{i_0}$ satisfying $q\leq u^* u$ and

$$\begin{equation} 1+{t_0} -\varepsilon <\langle |\pi _{i_0} (e-x)| (\xi ) / \xi \rangle , {\tag{4}} \end{equation}$$

and

$$\begin{align} 1+{t_0} -\varepsilon & < \langle |\pi _{i_0} (e-x)| (\xi ) / \xi \rangle \\ & \leq \langle |\pi _{i_0} (e-x)| (\xi ) / |\pi _{i_0} (e-x)| (\xi )\rangle ^{\frac{1}{2}} \ \|\xi \|\\ & = \langle |\pi _{i_0} (e-x)|^2 (\xi ) / \xi \rangle ^{\frac{1}{2}} \\ & = \langle \pi _{i_0} (e-x)^* \pi _{i_0} (e-x) (\xi ) / \xi \rangle ^{\frac{1}{2}}. {\tag{5}} \end{align}$$

The element $v = u q$ is a minimal partial isometry in $B(H_{i_0})$.

We observe that $\pi (e) = \pi _{i_0} (e)$ and $v$ are not orthogonal. Otherwise, $\pi _{i_0} (e)^* \pi _{i_0} (e)$ $\perp v^* v=q$, and hence $\pi _{i_0} (e) q = 0 = q \pi _{i_0} (e)^*,$ which, by 5, implies that

$$\begin{align*} \left(1+{t_0} -\varepsilon \right)^2 & < \langle \pi _{i_0} (e-x)^* \pi _{i_0} (e-x) (\xi ) / \xi \rangle \\ & = \langle q \pi _{i_0} (e-x)^* \pi _{i_0} (e-x) q (\xi ) / \xi \rangle \\ & = \langle q \pi _{i_0} (x)^* \pi _{i_0} (x) q (\xi ) / \xi \rangle \\ & \leq \| \pi _{i_0} (x)^* \pi _{i_0} (x) \| \\ &= \|\pi _{i_0} (x)\|^2 \\ & \leq \|x\|^2 \\ &= 1, \end{align*}$$

which is impossible.

Therefore, $\pi _{i_0} (e)$ and $v$ are two minimal partial isometries in $B(H_{i_0})$ which are not orthogonal. They must be of the form $\pi _{i_0} (e) = \eta _1 \otimes \xi _1$ and $v= \widetilde{\eta }_1 \otimes \widetilde{\xi }_1$ for suitable $\xi _1,\eta _1, \widetilde{\xi }_1,\widetilde{\eta }_1\in S(H_{i_0})$ with $|\langle \xi _1 / \widetilde{\xi }_1 \rangle |+ |\langle \widetilde{\eta }_1 / \eta _1 \rangle |\neq 0$. Let us consider two orthonormal systems $\{\eta _1, \eta _2\}$ and $\{\xi _1, \xi _2\}$ such that

$$\begin{equation*} v=\alpha \pi _{i_0} (e)+ \beta v_{12}+ \delta v_{22}+\gamma v_{21}, \end{equation*}$$

where $\pi _{i_0} (e):= v_{11}= \eta _1 \otimes \xi _1$, $v_{12} = {\eta }_2 \otimes {\xi }_1$, $v_{21} = {\eta }_1 \otimes {\xi }_2$, $v_{22} = {\eta }_2 \otimes {\xi }_2$, $\alpha = \langle \xi _1 / \widetilde{\xi }_1 \rangle \langle \widetilde{\eta }_1 / \eta _1 \rangle ,$ $\beta = \langle \xi _1 / \widetilde{\xi }_1 \rangle \langle \widetilde{\eta }_1 / \eta _2 \rangle ,$ $\gamma = \langle \xi _2 / \widetilde{\xi }_1 \rangle \langle \widetilde{\eta }_1 / \eta _1 \rangle ,$ $\delta = \langle \xi _2 / \widetilde{\xi }_1 \rangle \langle \widetilde{\eta }_1 / \eta _2 \rangle \in \mathbb{C}.$ It is easy to check that $|\alpha |^2+ |\beta |^2+|\gamma |^2+ |\delta |^2$ $=|\langle \xi _1 / \widetilde{\xi }_1 \rangle |^2 \|\widetilde{\eta }_1\|^2+ |\langle \xi _2 / \widetilde{\xi }_1 \rangle |^2 \|\widetilde{\eta }_1 \|^2 = \| \widetilde{\xi }_1\|^2=1$, and $\alpha \delta = \beta \gamma$.

For each $(i,j)\in \{1,2\}^2$, let $\varphi _{ij}\in S(B(H_{i_0})_*)$ be the unique extreme point of the unit ball $\mathcal{B}_{B(H_{i_0})_*}$ defined by $\varphi _{ij} (z) := \langle z(\xi _i) / \eta _j\rangle$ ($z\in B(H_{i_0})$). We shall also consider $\varphi _v \in S(B(H_{i_0})_*)$, defined by $\varphi _{v} (z) := \langle z(\widetilde{\xi }_1) / \widetilde{\eta }_1\rangle$ ($z\in B(H_{i_0})$). Each $\varphi _{ij}$ is supported by $v_{ij}\in S(B(H_{i_0}))$, while $\varphi _v$ is supported by $v$.

Clearly, the identity

$$\begin{equation*} \varphi _{v}(\pi _{i_0} (e))= \langle \pi _{i_0} (e)(\widetilde{\xi }_1) / \widetilde{\eta }_1\rangle = \langle \widetilde{\xi }_1 / \xi _1 \rangle \langle \eta _1 / \widetilde{\eta }_1 \rangle =\bar{\alpha } \end{equation*}$$

holds, and similarly we have

$$\begin{equation*} \varphi _{v}\pi _{i_0}(x)=\bar{\alpha } z_{11}+ \bar{\beta } z_{12}+ \bar{\delta } z_{22}+\bar{\gamma } z_{21}, \end{equation*}$$

where $z_{ij}=\varphi _{ij} \pi _{i_0}(x)$ for all $(i,j)\in \{1,2\}^2$. We also know that $\widetilde{\xi }_1\otimes \widetilde{\xi }_1= v^* v =q =\xi \otimes \xi \leq u^* u,$ and thus $\widetilde{\xi }_1 = \mu _0 \xi$, for a suitable $\mu _0\in \mathbb{C}$ with $|\mu _0|=1$. We deduce from 4 that

$$\begin{align*} 1+{t_0} -\varepsilon &<\langle |\pi _{i_0} (e-x)| (\xi ) / \xi \rangle \\ & = \langle u^*u |\pi _{i_0} (e-x)| (\xi ) / \xi \rangle \\ & = \langle u |\pi _{i_0} (e-x)| (\xi ) / u q (\xi ) \rangle \\ &= \langle \pi _{i_0} (e-x) (\xi ) / v (\xi ) \rangle \\ &= \langle \pi _{i_0} (e-x) (\widetilde{\xi }_1) / v (\widetilde{\xi }_1) \rangle \\ &= \langle \pi _{i_0} (e-x) (\widetilde{\xi }_1) / \widetilde{\eta }_1 \rangle \\ &=\varphi _v (\pi _{i_0} (e-x))\\ &=\varphi _v (\pi _{i_0} (e)) + \varphi _v (\pi _{i_0} (x))\\ &\leq |\alpha | + |\varphi _v (\pi _{i_0} (x)) |\\ &\leq |\alpha | +1, \end{align*}$$

which proves

$$\begin{equation*} {t_0} -\varepsilon < |\alpha |\leq 1. \end{equation*}$$

Now, the equality $|\alpha |^2+ |\beta |^2+|\gamma |^2+ |\delta |^2=1$ implies that

$$\begin{equation*} |\beta |^2,|\gamma |^2, |\delta |^2\leq 1-({t_0} -\varepsilon )^2<\frac{1}{10}, \end{equation*}$$

and since $|z_{ij}|\leq 1$, we have

$$\begin{align*} 1+{t_0} -\varepsilon & < \varphi _v \pi _{i_0} (e-x)\\ & = |\varphi _v \pi _{i_0} (e-x)|\\ & = |\bar{\alpha }-( \bar{\alpha } z_{11}+ \bar{\beta } z_{12}+ \bar{\delta } z_{22}+\bar{\gamma } z_{21}) | \\ & \leq |\alpha | |1-z_{11}|+ |z_{12}||\beta | + |z_{21}||\gamma |+ |z_{22}||\delta | \\ & \leq |1-z_{11}|+ |\beta | + |\gamma |+ |\delta |\\ & \leq |1-z_{11}|+\frac{3}{\sqrt {10}}. \end{align*}$$

Let us observe that $\pi _{i_0} (e) \pi _{i_0} (e)^* = v_{11}v_{11}^*= \eta _1 \otimes \eta _1$ and $\pi _{i_0} (e)^* \pi _{i_0} (e)= v_{11}^*v_{11}= \xi _1 \otimes \xi _1$. Therefore

$$\begin{align*} 1 & <1+{t_0} -\varepsilon -\frac{3}{\sqrt {10}}\\ &\leq |1-z_{11}|\\ & = |\varphi _{11} \pi _{i_0} (e-x) |\\ & = \Big |\varphi _{11} \Big ((\pi _{i_0} (e) \pi _{i_0} (e)^*) \pi _{i_0} (e-x) (\pi _{i_0} (e)^* \pi _{i_0} (e)) \Big )\Big |\\ & \leq \Big \| (\pi _{i_0} (e) \pi _{i_0} (e)^*) \pi _{i_0} (e-x) (\pi _{i_0} (e)^* \pi _{i_0} (e)) \Big \| \\ & = \| \pi _{i_0} ( (e e^*) (e-x) (e^* e)) \|\leq \| (e e^*) (e-x) (e^* e)\|, \end{align*}$$

which proves the claim in 3.

Finally, since $y\in F_e = e + (1 - ee^*) \mathcal{B}_A (1 - e^*e)$ we can write

$$\begin{equation*} y = e + (1 - ee^*) y (1 - e^*e), \end{equation*}$$

and we deduce from 3 that

$$\begin{align*} 1 & = \|y-x\| \\ & \geq \| ee^* (y-x) e^* e \|\\ & = \| ee^* (e + (1 - ee^*) y (1 - e^*e)-x) e^* e \| \\ &= \| ee^* (e - x) e^* e \| >1, \end{align*}$$

leading to the desired contradiction.

■

In the problem of dealing with maximal faces of the unit ball of a C$^*$-algebra $A$ we need to handle minimal partial isometries in $A^{**}$ (compare Theorem 2.2). We now present a technical result, which will be used later to facilitate the arguments depending on the facial structure of $\mathcal{B}_A$.

Lemma 2.4.

Let $A$ be a C$^*$-algebra. The following statements hold:

(a)

Every minimal projection $p$ in $A^{**}\backslash A$ is orthogonal to all minimal projections in $A$.

(b)

Every minimal partial isometry $u$ in $A^{**}\backslash A$ is orthogonal to all minimal partial isometries in $A$.

Proof.

(a)

Suppose $p$ is a minimal projection in $A^{**}\backslash A$. Let $q$ denote a minimal projection in $A$. Arguing by contradiction we assume that $p q\neq 0$.

As in the proof of Theorem 2.3 let $\pi : A\to \bigoplus _i^{\ell _{\infty }} B(H_i)$ be an isometric $^*$-homomorphism with weak$^*$-dense range, where $\{H_i\}_{I}$ is a family of complex Hilbert spaces (consider, for example, the atomic representation of $A$ 24, 4.3.7). By the weak$^*$-density of $A$ in $\bigoplus _i^{\ell _{\infty }} B(H_i)$ and the separate weak$^*$-continuity of the product of every von Neumann algebra, $\pi (q)$ is a minimal projection in $\bigoplus _i^{\ell _{\infty }} B(H_i)$. Clearly, the images of the mappings $L_{\pi (q)}: x \mapsto \pi (q) x$ and $R_{\pi (q)}: x\mapsto x \pi (q)$ ($\forall x\in \bigoplus _i^{\ell _{\infty }} B(H_i)$) are contained in suitable Hilbert spaces. It follows that the left and right multiplication operators $L_q$ and $R_q$ by $q$ on $A$ factors through a Hilbert space, and thus they are weakly compact (compare 6). Consequently, the spaces $(1-q) A q$, $q A (1-q)$, and $q A q = \mathbb{C} q$ are all reflexive. Applying the Krein-Šmulian theorem we deduce that $(1-q) A q$, $q A (1-q)$, and $q A q = \mathbb{C} q$ are weak$^*$-closed in $A^{**}$, showing that$$\begin{gather*} (1-q) A^{**} q = (1-q) A q,\\ q A^{**} (1-q) = q A (1-q) \subseteq A, \end{gather*}$$

and$$\begin{equation*} q A^{**} q = q A q = \mathbb{C} q. \end{equation*}$$

We now recall a useful matricial representation theorem. Let $C$ denote the C$^*$-subalgebra of $A^{**}$ generated by $p$ and $q$. Since $p$ and $q$ are minimal projections in $A^{**}$, Theorem 1.3 in 26 (see also 23, §3) assures the existence of $t\in [0,1]$ and a $^*$-isomorphism $\Phi : C \to M_2 ({\mathbb{C}})$ such that $\Phi (q) = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $\Phi (p) = \begin{pmatrix} t & \sqrt {t(1-t)} \\ \sqrt {t(1-t)} & 1-t \end{pmatrix}.$ Since $pq \neq 0$ we know that $t\neq 0$. Clearly, $\Phi ^{-1} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \in q C (1-q) \oplus (1-q) C q \subset A,$ and $\Phi ^{-1} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = q \in A$. Then$$\begin{equation*} \Phi ^{-1} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \Phi ^{-1} \left( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right)\in \Phi ^{-1} (\Phi (A\cap C)) =A\cap C. \end{equation*}$$

By linearity $p = \Phi ^{-1} \begin{pmatrix} t & \sqrt {t(1-t)} \\ \sqrt {t(1-t)} & 1-t \end{pmatrix} \in A$ which is impossible.

(b)

Suppose now that $u$ is a minimal partial isometry in $A^{**}\backslash A$ and $v$ is a minimal partial isometry in $A$. We shall first show that $uu^*, u^*u\in A^{**}\backslash A$. Indeed, since every minimal partial isometry in $A^{**}$ belongs locally to $A$ (compare Kadison’s transitivity theorem and 2, Remark 5.4 and Corollary 5.5), there exists a norm $1$ element $x\in A$ satisfying $x= u + (1-uu^*) x (1-u^*u)$. If $uu^*$ (respectively, $u^* u$) lies in $A$, then $u = uu^* x \in A$ (respectively, $u = x u^* u \in A$) which is impossible.

We have therefore shown that $uu^*, u^*u\in A^{**}\backslash A$ are minimal projections, while $vv^*, v^*v$ are minimal projections in $A$. It follows from (a) that $uu^*, u^*u \perp vv^*, v^*v$. Finally, the identities $u^* v = u^* u u^* v v^* v =0$ and $v u^* = v v^* v u^* u u^* =0$ prove that $u\perp v$.

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We are now in position to show that a surjective isometry between the unit spheres of two C$^*$-algebras maps minimal partial isometries to minimal partial isometries.

Theorem 2.5.

Let $A$ be a C$^*$-algebra, and let $H$ be a complex Hilbert space. Suppose that $f: S(A) \to S(B(H))$ is a surjective isometry. Let $e$ be a minimal partial isometry in $A$. Then $f(e)$ is a minimal partial isometry in $B(H)$. Moreover, there exists a surjective real linear isometry

$$\begin{equation*} T_{e} : (1 - ee^*) A (1 - e^*e)\to {(1 - f(e)f(e)^*) B(H) (1 - f(e)^*f(e))} \end{equation*}$$

such that

$$\begin{equation*} f(e + x) = f(e) + T_e (x) \text{ for all $x$ in } \mathcal{B}_{(1 - ee^*) A (1 - e^*e)}. \end{equation*}$$

In particular the restriction of $f$ to the face $F_{e} =e + (1 - ee^*) \mathcal{B}_A (1 - e^*e)$ is a real affine function.

Proof.

Arguing as in the beginning of the proof of Theorem 2.3, the set

$$\begin{equation*} F_{e} =e + (1 - ee^*) \mathcal{B}_A (1 - e^*e) \end{equation*}$$

is a maximal proper face of $\mathcal{B}_{A}$, and thus, by Proposition 2.1 and Theorem 2.2, there exists a minimal partial isometry $w$ in $B(H)^{**}$ such that

$$\begin{equation} f(F_e) = F_{w} =\left(w + (1 - ww^*) \mathcal{B}_{B(H)^{**}} (1 - w^*w)\right)\cap \mathcal{B}_{B(H)}.{\tag{6}} \end{equation}$$

We claim that $w\in B(H)$. Suppose, on the contrary that $w\in B(H)^{**}\backslash B$.

Theorem 2.3 implies that 1 is an isolated point in $\sigma (|f(e)|)$, and hence the function $\chi _{_{\{1\}}}$ belongs to $C_0(\sigma (|f(e)|))$. Let $f(e) = r |f(e)|$ denote the polar decomposition of $f(e)$. An application of the continuous functional calculus proves that $\hat{v}= r \ \chi _{_{\{1\}}} (|f(e)|)$ is a partial isometry in $B(H).$ Furthermore, since $f(e)\in f(F_e) = F_w$, we deduce that $\hat{v} \in F_w$ and

$$\begin{equation} \hat{v} = {w} + (1-{w} {w}^*) \hat{v} (1-{w}^* {w}) {\tag{7}} \end{equation}$$

(compare the arguments in the proof of 2).

In $B(H)$ we can always find a minimal partial isometry $\hat{w}\in B(H)$ satisfying

$$\begin{equation} \hat{v} = \hat{w} + (1-\hat{w} \hat{w}^*) \hat{v} (1-\hat{w}^* \hat{w}). {\tag{8}} \end{equation}$$

Since, by assumptions $w\in B(H)^{**}\backslash B(H)$, Lemma 2.4 implies that $w\perp \hat{w}$,

$$\begin{equation*} \hat{w} = (1-{w} {w}^*) \hat{w} (1-{w}^* {w}) \end{equation*}$$

and hence, by 7 we get

$$\begin{equation*} \hat{v} - \hat{w} = {w} + (1-{w} {w}^*) (\hat{v}- \hat{w}) (1-{w}^* {w}). \end{equation*}$$

By hypothesis,

$$\begin{equation*} 2=\|f(e)+\hat{w}\|=\|f(e)-(-\hat{w})\| =\|e-f^{-1}(-\hat{w})\|, \end{equation*}$$

where $e$ is a minimal partial isometry in $A$. Proposition 2.2 in 17 proves that

$$\begin{equation*} \hat{e}=f^{-1}(-\hat{w})=-e+(1-ee^*) \hat{e} (1-e^*e). \end{equation*}$$

By construction $f(e) = \hat{v} + (1-\hat{v} \hat{v}^*) f(e) (1-\hat{v}^* \hat{v})$, and by 8,

$$\begin{equation*} f(e) = \hat{w} + (1-\hat{w} \hat{w}^*) \hat{v} (1-\hat{w}^* \hat{w}) + (1-\hat{v} \hat{v}^*) f(e) (1-\hat{v}^* \hat{v}), \end{equation*}$$

and consequently $\|f(e)-\hat{w}\|\leq 1$. Having in mind that

$$\begin{equation*} f(e)-\hat{w} = f(e) - (1-{w} {w}^*) \hat{w} (1-{w}^* {w})\in F_w + (1-{w} {w}^*) \mathcal{B}_{B(H)} (1-{w}^* {w}) \end{equation*}$$

we get

$$\begin{equation*} f(e)-\hat{w} = w + (1-w w^*) (f(e)-\hat{w}) (1-w^* w) \in F_{w}. \end{equation*}$$

We deduce from 6 that $z=f^{-1}(f(e)-\hat{w})\in F_{e}$, and thus $z=e+(1-ee^*)z (1-e^*e),$ which leads to

$$\begin{align*} 1 & =\|f(e)\|\\ & = \|f(e)-\hat{w}+\hat{w}\|\\ & = \|(f(e)-\hat{w})-(-\hat{w})\|\\ & = \|f(z) -f(\hat{e})\|\\ & = \|z-\hat{e}\|\\ & = \|e+(1-ee^*)z (1-e^*e) -(-e+(1-ee^*) \hat{e} (1-e^*e))\|\\ & = \|2e+(1-ee^*) (z-\hat{e}) (1-e^*e) \|\\ & = 2, \end{align*}$$

and hence to a contradiction. Therefore, $w\in B(H)$, and

$$\begin{equation*} F_{w} = w + (1 - ww^*) \mathcal{B}_{B(H)} (1 - w^*w). \end{equation*}$$

We can argue now as in the proof of 25, Proposition 3.1 to conclude. We insert a short argument here for completeness reasons. We have established that

$$\begin{equation*} f\left( e + \mathcal{B}_{(1 - ee^*) A (1 - e^*e)} \right)=f(F_e) =F_w = w + \mathcal{B}_{(1 - ww^*) B(H) (1 - w^*w)}. \end{equation*}$$

Let $\mathcal{T}_{x_0}$ denote the translation with respect to $x_0$, that is, $\mathcal{T}_{x_0} (x) = x+x_0$. The mapping $f_{e} = \mathcal{T}_{w}^{-1}|_{f(F_e)} \circ f|_{F_e} \circ \mathcal{T}_{e}|_{\mathcal{B}_{(1 - ee^*) A (1 - e^*e)}}$ is a surjective isometry from $\mathcal{B}_{(1 - ee^*) A (1 - e^*e)}$ onto $\mathcal{B}_{(1 - ww^*) B(H) (1 - w^*w)}$. Mankiewicz’s theorem (see 22) implies the existence of a surjective real linear isometry $T_{e} : (1 - ee^*) A (1 - e^*e)\to {(1 - ww^*) B(H) (1 - w^*w)}$ such that $f_{e} = T_{e}|_{S((1 - ee^*) A (1 - e^*e))}$ and hence

$$\begin{equation*} f(e + x) = w + T_e (x) \text{ for all $x$ in } \mathcal{B}_{(1 - ee^*) A (1 - e^*e)}. \end{equation*}$$

In particular, $f(e) = w$.

For the final statement we simply write

$$\begin{equation*} f|_{F_e}= \mathcal{T}_{w}|_{\mathcal{B}_{(1 - ww^*) B(H) (1 - w^*w)}} \circ f_{e} \circ \mathcal{T}_{e}^{-1}|_{F_e}= \mathcal{T}_{w}|_{\mathcal{B}_{(1 - ww^*) B(H) (1 - w^*w)}} \circ T_{e} \circ \mathcal{T}_{e}^{-1}|_{F_e} \end{equation*}$$

as a composition of real affine functions.

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The next technical lemma is obtained with basic techniques of linear algebra.

Lemma 2.6.

Let $H$ be a complex Hilbert space, and let $v_1,v_2, v_3$, $e_1$, and $e_2$ be minimal partial isometries in $B(H)$ satisfying $e_1\perp e_2$, $v_1\perp v_2,$ $v_1\perp v_3$,

$$\begin{equation*} -v_1+v_3 = e_1 -e_2 \quad \text{and} \quad v_1+v_2 = e_1 +e_2. \end{equation*}$$

Then $v_1 = e_2$ and $v_2=e_1= v_3$.

Proof.

Since $v_1\perp v_2,v_3$, by multiplying the identities $-v_1+v_3 = e_1 -e_2$ and $v_1+v_2 = e_1 +e_2$ on the left by $v_1^*$ we get

$$\begin{equation*} -v_1^* v_1 = v_1^* e_1 - v_1^* e_2, \quad \text{and} \quad v_1^* v_1 = v_1^* e_1 + v_1^*e_2, \end{equation*}$$

which shows that $v_1^* e_1 =0$. Multiplying by $v_1^*$ on the right we prove $e_1 v_1^* =0$. We have therefore shown that $v_1\perp e_1$.

Applying that $e_1\perp e_2$ and $v_1\perp v_2$ we get $e_1 e_1^* + e_2 e_2^* = v_1 v_1^* + v_2 v_2^*$, where $e_1 e_1^*$ and $v_1 v_1^*$ are orthogonal rank $1$ projections. Thus, $e_1 e_1^* = e_1 e_1^* v_2 v_2^*$, and by minimality $v_2 v_2^* = e_1 e_1^*$. We can similarly prove $v_2^* v_2 = e_1^* e_1.$ Finally,

$$\begin{equation*} v_2 = v_2 v_2^* v_2 = v_2 v_2^* (v_1+v_2) = v_2 v_2^* (e_1+e_2) = e_1 e_1^* (e_1+e_2) = e_1, \end{equation*}$$

and the rest is clear.

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Next, we shall establish several consequences of the above theorem.

Theorem 2.7.

Let $f: S(B(K)) \to S(B(H))$ be a surjective isometry where $H$ and $K$ are complex Hilbert spaces with dimension greater than or equal to $3$. Then the following statements hold:

(a)

For each minimal partial isometry $v$ in $B(K)$, the mapping$$\begin{equation*} T_v : (1 - vv^*) B(K) (1 - vv^*)\to {(1 - f(v)f(v)^*) B(H) (1 - f(v)^*f(v))} \end{equation*}$$

given by Theorem 2.5 is complex linear or conjugate linear.

(b)

For each minimal partial isometry $v$ in $B(K)$ we have $f(-v) = -f(v)$ and $T_{v} = T_{-v}$. Furthermore, $T_v$ is weak$^*$-continuous and $f(e) = T_v(e)$ for every minimal partial isometry $e\in (1 - vv^*) B(K) (1 - v^* v)$.

(c)

For each minimal partial isometry $v$ in $B(K)$ the equality $f(w) = T_v (w)$ holds for every partial isometry $w\in (1 - vv^*) B(K) (1 - v^*v)\backslash \{0\}$.

(d)

Let $w_1,\ldots ,w_n$ be mutually orthogonal non-zero partial isometries in $B(K)$, and let $\lambda _1,\ldots ,\lambda _n$ be positive real numbers with $\lambda _1=1,$ and $\lambda _j\leq 1$ for all $j$. Then$$\begin{equation*} f\left(\sum _{j=1}^n \lambda _j w_j\right) = \sum _{j=1}^n \lambda _j f\left(w_j\right). \end{equation*}$$

(e)

For each minimal partial isometry $v$ in $B(K)$ we have $f(x) = T_v (x)$ for every $x\in S(\mathcal{B}_{(1 - vv^*) B(K) (1 - v^* v)})$.

(f)

For each partial isometry $w$ in $B(K)$ the element $f(w)$ is a partial isometry.

(g)

Suppose $v_1,v_2$ are mutually orthogonal minimal partial isometries in $B(K)$; then $T_{v_1} (x) = T_{v_2} (x)$ for every $x\in (1 - v_1v_1^*) B(K) (1 - v_1v_1^*) \cap (1 - v_2v_2^*) B(K) (1 - v_2v_2^*)$.

(h)

Suppose $v_1,v_2$ are mutually orthogonal minimal partial isometries in $B(K)$; then exactly one of the following statements holds:

(1)

The mappings $T_{v_1}$ and $T_{v_2}$ are complex linear.

(2)

The mappings $T_{v_1}$ and $T_{v_2}$ are conjugate linear.

Proof.

(a)

Let $v$ be a minimal partial isometry in $B(K)$. Suppose that$$\begin{equation*} T_v : (1 - vv^*) B(K) (1 - v^* v)\to {(1 - f(v)f(v)^*) B(H) (1 - f(v)^*f(v))} \end{equation*}$$

is the surjective real linear isometry given by Theorem 2.5. Having in mind that $(1 - vv^*) B(K) (1 - v^*v)\cong B((1 - v^*v)(K), (1 - vv^*)(K))$ and $(1 - f(v)f(v)^*) B(H) (1 - f(v)^*f(v))\cong B((1 - f(v)^*f(v))(H), (1 - f(v)f(v)^*)(H))$ are Cartan factors of type 1 and rank $\geq 2$, Proposition 2.6 in 5 assures that $T_v$ is complex linear or conjugate linear.

(b)

We keep the notation in (a). By Theorem 2.5 $f(v)$ and $f(-v)$ are minimal partial isometries. By assumptions $\|f(v)-f(-v)\| = \|v +v\| =2$, and hence by 25, Lemmas 3.4 and 3.5 or 17, Proposition 2.2 we have $f(-v) = -f(v)$. Let $T_v, T_{-v} : (1 - vv^*) B(K) (1 - v^* v)\to {(1 - f(v)f(v)^*) B(H) (1 - f(v)^*f(v))}$ be the surjective real linear isometries given by Theorem 2.5. Lemma 2.5 in 5 proves that $T_v$ and $T_{-v}$ both are weak$^*$-continuous, while 5, Proposition 2.6 implies that $T_v$ and $T_{-v}$ preserve products of the form $(a,b,c)\mapsto ab^* c + c b^* a$ ($a,b,c\in (1 - vv^*) B(K) (1 - v^* v)$).

By the hypothesis on $H$, we can find another minimal partial isometry $e\in (1 - vv^*) B(K) (1 - v^* v)$. Since $e+v,e-v\in F_e$, applying Theorem 2.5 we deduce that$$\begin{equation*} f(v) + T_v(e) = f(e+v) = f(e) + T_e (v) \end{equation*}$$

and$$\begin{equation*} -f(v) + T_{-v} (e) = f(-v) + T_{-v} (e) = f(e-v) = f(e) - T_e (v), \end{equation*}$$

where $T_v(e),$ $T_{-v}(e)$ and $T_e (v)$ are minimal partial isometries with $f(v) \perp T_v(e)$, $-f(v)=f(-v) \perp T_{-v} (e)$, and $f(e) \perp T_e (v)$. It follows from Lemma 2.6 above that $T_e(v) = f(v),$ $T_v (e) = f(e)=T_{-v} (e)$, and $f(-v) = -T_{e} (v) =-f(v).$

We have also shown that $T_v (e) =T_{-v} (e)$ for every minimal partial isometry $e\in (1 - vv^*) B(K) (1 - v^* v)$. That is, $T_v$ and $T_{-v}$ are surjective complex linear or conjugate linear surjective isometries between Cartan factors of type 1 and rank $\geq 2$. Since $T_v$ and $T_{-v}$ coincide on minimal partial isometries, we deduce by linearity that $T_v$ and $T_{-v}$ both are complex linear or conjugate linear and coincide on finite linear combinations of mutually orthogonal minimal partial isometries. Finally, since $(1 - vv^*) B(K) (1 - v^*v)\cong B((1 - v^*v)(K), (1 - vv^*)(K))$ coincides with the weak$^*$-closed span of its minimal tripotents, we conclude that $T_v = T_{-v}$.

(c)

Let $w$ be a non-zero partial isometry in $(1 - vv^*) B(K) (1 - v^*v)$. Take a minimal partial isometry $w_0$ such that $w= w_0 +(1 - w_0 w_0^*) w (1 - w_0^* w_0)$. We set $w_0^{\perp } =(1 - w_0 w_0^*) w (1 - w_0^* w_0)$. Applying Theorem 2.5 and (b) we get$$\begin{align*} f(v) + T_v(w) & = f(v + w)\\ & = f(v + w_0 +w_0^{\perp }) \\ & = f(w_0) + T_{w_0}(v) + T_{w_0} (w_0^{\perp })\\ & = f(w_0) + f(v) + T_{w_0} (w_0^{\perp })\\ & = f(v) + f(w_0 + w_0^{\perp }) \\ & = f(v) + f(w), \end{align*}$$

which proves (c).

(d)

Let $w_1,\ldots ,w_n$ be mutually orthogonal non-zero partial isometries in $B(K)$, and let $\lambda _1,\ldots ,\lambda _n$ be positive real numbers with $\lambda _1=1$. Pick again a minimal partial isometry $w_0$ such that $w_1= w_0 +(1 - w_0 w_0^*) w_1 (1 - w_0^* w_0)$. Theorem 2.5 proves that$$\begin{align*} f\left(\sum _{j=1}^n \lambda _j w_j\right) & = f\left(w_0 + (1 - w_0 w_0^*) w_1 (1 - w_0^* w_0)+ \sum _{j=2}^n \lambda _j w_j\right) \\ & = f(w_0) + T_{w_0}\Big ( (1 - w_0 w_0^*) w_1 (1 - w_0^* w_0)+ \sum _{j=2}^n \lambda _j w_j \Big ) \\ & = f(w_0) + T_{w_0} ((1 - w_0 w_0^*) w_1 (1 - w_0^* w_0)) + \sum _{j=2}^n \lambda _j T_{w_0}( w_j ) \\ & = \text{(by \textrm{(c)})} = \sum _{j=1}^n \lambda _j f\left(w_j\right). \end{align*}$$

(e)

Since elements in $(1 - vv^*) B(K) (1 - v^*v)$ can be approximated in norm by finite real linear combinations of mutually orthogonal partial isometries in $(1 - vv^*) B(K) (1 - v^*v)$ (see 20, Lemma 3.11), and $f$ and $T_v$ are isometries, we derive from (c) and (d) that $f(x) = T_v (x)$ for every $x\in S(\mathcal{B}_{(1 - vv^*) B(K) (1 - v^* v)})$.

(f)

Let $w$ be a partial isometry in $B(K)$. As before, let $w_0$ be a minimal partial isometry in $B(K)$ satisfying $w= w_0 + w_0^\perp$, with $w_0^{\perp } =(1 - w_0 w_0^*) w (1 - w_0^* w_0)$. Having in mind that $T_{w_0}$ is a surjective real linear isometry between spaces isometrically isomorphic to $B((1-w_0 w_0^*)(K))$ and $B((1-f(w_0) f(w_0)^*)(H))$, Theorem 5.1 in 4 assures that $T_{w_0}$ preserves triple products of the form $\{a,b,c\} =\frac{1}{2} (ab^*c + c b^* a),$ and thus $T_{w_0} ( w_0^{\perp })$ is a partial isometry. By Theorem 2.5, we get$$\begin{equation*} f(w) = f(w_0 + w_0^{\perp }) = f(w_0) + T_{w_0} ( w_0^{\perp }) \end{equation*}$$

is the sum of two orthogonal partial isometries in $B(H)$, and then $f(w)$ is a partial isometry.

(g)

Suppose $v_1,v_2$ are mutually orthogonal partial isometries in $B(K)$. Let us pick a non-zero $x\in (1 - v_1v_1^*) B(K) (1 - v_1^*v_1) \cap (1 - v_2v_2^*) B(K) (1 - v_2^*v_2)$. The equality $T_{v_1} (\frac{x}{\|x\|})= f (\frac{x}{\|x\|})= T_{v_2} (\frac{x}{\|x\|})$ holds by (e), and by linearity $T_{v_1} (x) = T_{v_2} (x)$.

Finally statement (h) follows straightforwardly from (a) and (g) because the dimensions of $H$ and $K$ are greater than or equal to 3.

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3. Synthesis of a surjective real linear isometry

In order to produce a real linear extension of our surjective isometry between $B(H)$ spaces, the next identity principle, which generalizes 25, Proposition 3.9, will play a central role.

Proposition 3.1.

Let $H$ and $K$ be complex Hilbert spaces. Suppose that $f: S(B(K)) \to S(B(H))$ is a surjective isometry, and $T:B(K)\to B(H)$ is a weak$^*$-continuous real linear operator such that $f(v)=T(v)$, for every minimal partial isometry $v$ in $B(K)$. Then $T$ and $f$ coincide on $S(B(K))$.

Proof.

Take a minimal partial isometry $e$ in $B(K)$. By Theorem 2.7(b) and the hypothesis $T_e (v) = f(v) = T(v)$, for every minimal partial isometry $v$ in $(1 - ee^*) B(K) (1 - e^* e)$. Finite real linear combinations of mutually orthogonal minimal partial isometries in $(1 - ee^*) B(K) (1 - e^* e)$ are weak$^*$-dense in $B(K)$, we therefore deduce from the weak$^*$-continuity of $T_e$ and $T$ that $T_e = T$ on $(1 - ee^*) B(K) (1 - e^* e)$.

Pick a non-zero partial isometry $w$ in $B(K)$, and a minimal partial isometry $w_0$ such that $w=w_0 +(1 - w_0 w_0^*) w (1 - w_0^* w_0).$ By Theorem 2.5, the hypothesis and what we have proved in the first paragraph we obtain

$$\begin{align*} f(w) & = f(w_0) + T_{w_0} ((1 - w_0 w_0^*) w (1 - w_0^* w_0)) \\ &= T(w_0) + T((1 - w_0 w_0^*) w (1 - w_0^* w_0))\\ & = T(w). \end{align*}$$

We have thus established that $T(w) = f(w)$ for every partial isometry $w$ in $B(K)$. Repeating the arguments in the proof of Theorem 2.7(e) we conclude that $T$ and $f$ coincide on $S(B(K))$.

■

We have developed enough tools to prove our main result.

Theorem 3.2.

Let $H$ and $K$ be complex Hilbert spaces. Suppose that $f: S(B(K))$ $\to S(B(H))$ is a surjective isometry. Then there exists a surjective complex linear or conjugate linear surjective isometry $T: B(K) \to B(H)$ satisfying $f(x) = T(x)$, for every $x\in S(B(K))$.

Proof.

By Riesz’s lemma $H$ is finite dimensional if and only if $K$ is. When $H$ and $K$ are finite dimensional, the desired conclusion follows from 33 or 34.

We assume now that $H$ and $K$ are infinite dimensional. We shall apply the technique in 25, Theorem 3.13 to define our real linear isometry. Let $p_1,p_2$, and $p_3$ be three minimal projections in $B(K)$. Given $j\in \{1,2,3\}$, let $T_{p_j} : (1-p_j)B(K)(1-p_j) \to (1-f(p_j)f(p_j)^*)B(H)(1-f(p_j)^*f(p_j))$ denote the surjective real linear isometry given by Theorem 2.5.

By Theorem 2.7(b) and (h), the operators $T_{p_1}$, $T_{p_2}$, and $T_{p_3}$ are weak$^*$-continuous, and they are all complex linear or conjugate linear. We assume that we are in the first case (the second case produces a conjugate linear map).

We can mimic the construction done in 25, Theorem 3.13 with the appropriate adaptations via the stronger properties developed in Section 2. Clearly, $B(K)$ admits the following decomposition:

$$\begin{equation*} B(K) = \mathbb{C} p_1 \oplus p_1 B(K) p_2\oplus p_2 B(K) p_1 \oplus p_1 B(K) (1-p_1-p_2) \end{equation*}$$

$$\begin{equation*} \oplus (1-p_1-p_2) B(K) p_1 \oplus (1-p_1) B(K) (1-p_1). \end{equation*}$$ We define a mapping $T : B(K) \to B(H)$ given by

$$\begin{equation*} T(x) = T_{p_3} (p_1 x p_1) + T_{p_3} (p_1 x p_2 + p_2 x p_1) + T_{p_2}(p_1 x (1-p_1-p_2) +(1-p_1-p_2) x p_1) \end{equation*}$$

$$\begin{equation*} + T_{p_1}((1-p_1)x (1-p_1)). \end{equation*}$$ The mapping $T$ is well defined, complex linear, and weak$^*$-continuous thanks to the uniqueness of the above decomposition and the linearity and weak$^*$-continuity of the mappings $T_{p_1}$, $T_{p_2}$, and $T_{p_3}$ (compare Theorem 2.7(b)).

We shall conclude the proof by applying Proposition 3.1. For this purpose we shall show that

$$\begin{equation} T(e) = f(e) \text{ for every minimal partial isometry $v$ in $B(K)$.} {\tag{9}} \end{equation}$$

Let $e$ be a minimal partial isometry in $B(K)$. Since $\dim (K)=\infty$ there exists a minimal projection $p_4$ satisfying $p_4 \perp p_1, p_2, p_3, e$. The relations of orthogonality imply that

$$\begin{equation*} p_1 e p_1, p_1 e p_2 + p_2 e p_1, p_1 e (1-p_1-p_2) + (1-p_1-p_2) e p_1, (1-p_1)e (1-p_1)\perp p_4. \end{equation*}$$

Equivalently, the elements $p_1 e p_1,$ $p_1 e p_2 + p_2 e p_1,$ $p_1 e (1-p_1-p_2) + (1-p_1-p_2) e p_1,$ $(1-p_1)e (1-p_1)$ all belong to $(1-p_4) B(K) (1-p_4).$ By definition, Theorem 2.7(g), and the previous observation we get

$$\begin{align*} T(e) & = T_{p_3} (p_1 e p_1) + T_{p_3} (p_1 e p_2 + p_2 e p_1)\\ &\qquad + T_{p_2}(p_1 e (1-p_1-p_2) +(1-p_1-p_2) e p_1)\\ & \qquad + T_{p_1}((1-p_1)e (1-p_1))\\ & = T_{p_4} (p_1 e p_1) + T_{p_4} (p_1 e p_2 + p_2 e p_1) + T_{p_4}(p_1 e (1-p_1-p_2)\\ &\qquad +(1-p_1-p_2) e p_1)+ T_{p_4}((1-p_1)e (1-p_1))\\ &= T_{p_4} (e)\\ &= \text{(Theorem \xhref[statement]{#ltxid16}{2.7}\textrm{(e)})}= f(e) , \end{align*}$$

which proves 9 and finishes the arguments.

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We are now in position to extend Theorem 3.2 for $\ell _{\infty }$-sums of $B(H)$ spaces. In the proof presented here we revise the arguments in the proof of 25, Theorem 3.12 and we insert the appropriate modifications.

Since Proposition 2.1, Theorems 2.2 and 2.3, and Lemma 2.4 are valid for general Banach spaces and general C$^*$-algebras, respectively, the proof of Theorem 2.5 can be literally applied to prove the following result.

Theorem 3.3.

Let $(H_i)_{i\in I}$ be a family of complex Hilbert spaces, let $A$ be a C$^*$-algebra, and suppose that $f: S\left(A \right) \to S\left(\bigoplus _i^{\ell _{\infty }} B(H_i) \right)$ is a surjective isometry. Let $e$ be a minimal partial isometry in $A$. Then $f(e)$ is a minimal partial isometry in $B= \bigoplus _i^{\ell _{\infty }} B(H_i)$, and there exists a surjective real linear isometry

$$\begin{equation*} T_{e} : (1 - ee^*) A (1 - e^*e)\to {(1 - f(e)f(e)^*) B (1 - f(e)^*f(e))} \end{equation*}$$

such that

$$\begin{equation} f(e + x) = f(e) + T_e (x) \text{ for all $x$ in } \mathcal{B}_{(1 - ee^*) A (1 - e^*e)}. {\tag{10}} \end{equation}$$

In particular, the restriction of $f$ to the face $F_{e} =e + (1 - ee^*) \mathcal{B}_A (1 - e^*e)$ is a real affine function.■

It should be noted here that statements (a) and (h) in Theorem 2.7 need not be true when $B(K)$ and $B(H)$ are replaced with von Neumann algebras of the form $\bigoplus _j^{\ell _{\infty }} B(K_j)$ and $\bigoplus _i^{\ell _{\infty }} B(H_i)$, respectively. However, this obstacle will be avoided in the proof of our last result with an appropriate version of Theorem 2.7.

Theorem 3.4.

Let $(H_i)_{i\in I}$ and $(K_j)_{j\in J}$ be two families of complex Hilbert spaces. Suppose $f: S\left(\bigoplus _j^{\ell _{\infty }} B(K_j) \right) \to S\left(\bigoplus _i^{\ell _{\infty }} B(H_i) \right)$ is a surjective isometry. Then there exists a surjective real linear isometry $T: S\left(\bigoplus _j^{\ell _{\infty }} B(K_j) \right) \!\to \! S\left(\bigoplus _i^{\ell _{\infty }} B(H_i) \right)$ satisfying $T|_{S(E)} = f$.

Proof.

To simplify the notation, set $A= \bigoplus _j^{\ell _{\infty }} B(K_j)$ and $B=\bigoplus _i^{\ell _{\infty }} B(H_i)$. If $\sharp J \geq 2$, we can pick two different subindexes $j_1$ and $j_2$ in $J$. Let $p_{1}\in B(K_{j_1})$ and $p_{2}\in B(K_{j_2})$ be minimal projections, and for $k=1,2$ let

$$\begin{equation*} T_{p_k} : (1-p_{j_k})A (1-p_{j_k}) \to (1-f(p_{j_k})f(p_{j_k})^*)B (1-f(p_{j_k})^*f(p_{j_k})) \end{equation*}$$

be the surjective real linear isometry given by Theorem 3.3.

Let us observe that we can write $A = A_1 \oplus A_2$, where $A_2=\bigoplus _{j\neq j_1}^{\ell _{\infty }} B(K_j)$ and $A_1=B(K_{j_1})$. The symbol $\pi _i$ will stand for the projection of $A$ onto $A_i$. The mapping $T: A\to B$, $T(x) := T_{p_2} (\pi _1 (x)) + T_{p_1} (\pi _2 (x))$ is well defined, real linear and continuous. Lemma 2.13 in 16 proves that $T_{p_1}$ and $T_{p_2}$ (and hence $T$) are weak$^*$-continuous.

We shall prove next that for each minimal partial isometry $v\in A$

$$\begin{equation} \begin{gathered} \text{the surjective real linear isometry $T_v$ given by Theorem \xhref[statement]{#ltxid37}{3.3}}\\ \text{preserves triple products of the form $\{a,b,c\} = 2^{-1} (a b^* c+ c b^* a)$.} \end{gathered} {\tag{11}} \end{equation}$$

Take a minimal partial isometry $v$ in $A$. We deduce from the structure of $A$ and the minimality of $v$ the existence of a unique index $j_0$ in $J$ such that $v$ is a minimal partial isometry in $B(K_{j_0})$. Therefore,

$$\begin{equation*} (1-vv^*) A (1-v^*v) = \bigoplus _{j\neq j_0}^{\ell _{\infty }} B(K_j)\oplus ^{\infty } \Big ((1-vv^*) B(K_{j_0}) (1-v^*v) \Big ), \end{equation*}$$

and precisely one of the next statements holds:

(i)

$(1-vv^*) A (1-v^*v) = \bigoplus _{j\neq j_0}^{\ell _{\infty }} B(K_j)$ if $\dim (K_{j_0})=1$;

(ii)

$(1-vv^*) A (1-v^*v) \cong \bigoplus _{j\neq j_0}^{\ell _{\infty }} B(K_j) \oplus ^{\infty } \mathbb{C}$ if $\dim (K_{j_0})=2$;

(iii)

$(1-vv^*) A (1-v^*v) = \bigoplus _{j\neq j_0}^{\ell _{\infty }} B(K_j) \oplus ^{\infty } B(K_{j_0}^{1},K_{j_0}^{2})$ if $\dim (K_{j_0})\geq 3$, where $K_{j_0}^{2} = (1-vv^*) (K_{j_0})$ and $K_{j_0}^{1} = (1-v^*v) (K_{j_0})$ are complex Hilbert spaces of dimension larger than or equal to 2.

A similar decomposition holds for $(1-f(v) f(v)^*) B (1-f(v)^*f(v))$. Arguing as in the proof of 5, Theorem 3.1, we can deduce, via 5, Propositions 1.1 and 2.6, that $T_{v}$ preserves triple products. In particular, for each $k=1,2$ the surjective real linear isometry $T_{p_k}$ preserves triple products. Therefore, $T_v$ preserves (minimal) partial isometries and orthogonality among elements in their respective domains (because, by 19, page 18, $a\perp b$ in $A$ if and only if $\{a,a,b\}=0$).

Let $K(A) = \bigoplus _j^{c_0} K(K_j)$, and $K(B) = \bigoplus _i^{c_0} K(H_i)$, where for a complex Hilbert space $H$, $K(H)$ denotes the C$^*$-algebra of all compact operators on $H$. Clearly, $K(A)$ and $K(B)$ are compact C$^*$-algebras. Let $v$ be an arbitrary minimal partial isometry in $A$. Theorem 3.3 assures that $f(v)$ and $f(-v)$ are minimal partial isometries in $B$. So, $f(v)$ and $f(-v)$ belong to $K(B)$. By hypothesis, we have $\|f(v)-f(-v)\| = \|v +v\| =2$, and thus Lemma 3.5 in 25 implies that

$$\begin{equation*} f(-v) = -f(v) + (1-f(v)f(v)^*) K(B) (1-f(v)^* f(v)), \end{equation*}$$

which combined with the minimality of $f(-v)$ proves that

$$\begin{equation} f(-v) = -f(v) \text{ for every minimal partial isometry } v\in A. {\tag{12}} \end{equation}$$

We claim that

$$\begin{equation} T(v) = f(v) \text{ for every minimal partial isometry } v\in A. {\tag{13}} \end{equation}$$

Indeed, every minimal partial isometry $v$ in $A$ lies in $A_1$ or in $A_2$. If $v\in A_1$ (respectively, in $A_2$) we have $T(v) = T_{p_2} (v)$ (respectively, $T(v) = T_{p_1} (v)$) by definition.

Suppose first that $v\in A_2$. By Theorem 3.3 and 11 we know that $f(v)$ and $T(v)=T_{p_1} (v)$ are minimal partial isometries in $B$. Let $T_v$ denote the surjective real linear isometry given by Theorem 3.3. We know from the just quoted theorem and 12 that

$$\begin{equation*} f(v) + T_v (p_1) = f(p_1 + v) = f(p_1) + T_{p_1}(v)= f(p_1) + T(v) \end{equation*}$$

and

$$\begin{equation*} f(v) - T_v ( p_1) =f(v) + T_v (- p_1) = f(- p_1 + v) \!=\! f(-p_1) + T_{-p_1}(v)=-f(p_1) + T_{-p_1}(v). \end{equation*}$$

Adding the last two identities we get $2 f(v) = T(v) + T_{-p_1}(v)$. Since $f(v), T(v),$ and $T_{-p_1}(v)$ are minimal partial isometries with $\|T(v) + T_{-p_1}(v) \| =\|2 f(v) \|= 2$, a new application of 25, Lemma 3.5 implies that $T(v) = T_{-p_1}(v)$, and thus $f(v) = T(v) = T_{-p_1}(v)= T_{p_1}(v)$, as claimed.

If $v\in A_1$, similar arguments can be applied to show that

$$\begin{equation*} f(v) + T_v (p_2) = f(p_2 + v) = f(p_2) + T_{p_2}(v)= f(p_2) + T(v) \end{equation*}$$

and

$$\begin{equation*} f(v) - T_v ( p_2) =f(v) + T_v (- p_2) = f(- p_2 + v) \!=\! f(-p_2) + T_{-p_2}(v)=-f(p_2) + T_{-p_2}(v), \end{equation*}$$

which implies that $2 f(v) = T(v) + T_{-p_2}(v)$. Since $f(v),$ $T(v)=T_{p_2}(v),$ and $T_{-p_2}(v)$ are minimal partial isometries with $\|T(v) + T_{-p_2}(v) \| =\|2 f(v) \|= 2$, it follows from 25, Lemma 3.5 that $T(v) = T_{-p_2}(v)$, and $f(v) = T(v) = T_{-p_2}(v)= T_{p_2}(v)$. This finishes the proof of the claim in 13.

On the other hand, let $v$ and $w$ be minimal partial isometries in $A$ with $v\perp w$. The elements $T_{ w} (v)$ and $T_{ v} (w)$ are minimal partial isometries (compare 11), and we deduce from Theorem 3.3, 12, and 13 that

$$\begin{equation*} \pm T(w) + T_{\pm w} (v) = \pm f(w) + T_{\pm w} (v) = f(v\pm w) = f(v) \pm T_v(w) = T(v) \pm T_v(w), \end{equation*}$$

which implies that $T_{ w} (v) + T_{- w} (v) = 2T(v)= 2 f(v)$ (that is, $\|T_{ w} (v) + T_{- w} (v)\| = 2$), and then $T_{ w} (v) = T_{- w} (v) = T(v)$ (just apply 25, Lemma 3.5).

We have shown that for each minimal partial isometry $v$ in $A$ we have $T(v) = f(v).$ Furthermore, if $w$ is another minimal partial isometry with $v\perp w$, then $T_{ w} (v) = T(v)$. Let $e$ be an arbitrary partial isometry in $A$, and let us find a minimal partial isometry $v\in A$ such that $e = v + (1-vv^*) e (1-v^*v)$. It is known that $\displaystyle e_0 =(1-vv^*) e (1-v^*v) =w^* - \sum _{\lambda } v_\lambda$, where $\{v_{\lambda }\}_{{\lambda }}$ is a family of mutually orthogonal minimal partial isometries in $A$ with $v\perp v_{\lambda }$ for every ${\lambda }$. By Theorem 3.3 and the weak$^*$-continuity of $T$ and $T_v$ we have

$$\begin{align*} f\left( e \right) & = f(v)+ T_{v} \left(e_0 \right)\\ & = f(v) + w^*-\sum _{\lambda } T_{v} (v_{\lambda }) \\ & = T(v) + w^*-\sum _{\lambda } T (v_{\lambda })\\ & = T(v) + T(e_0)\\ & = T(e). \end{align*}$$

That is, $f(e) = T(e)$ for every partial isometry $e$ in $A$. We have further shown that $T(e) =T_v(e)$ for every pair of partial isometries $e,v$ in $A$ with $v\perp e$ and $v$ minimal.

We can now argue as in the proof of Theorem 2.7(d) to show that

$$\begin{equation*} f\left(\sum _{j=1}^n \lambda _j w_j\right) = \sum _{j=1}^n \lambda _j f\left(w_j\right)= T\left(\sum _{j=1}^n \lambda _j w_j\right), \end{equation*}$$

for every set $\{w_1,\ldots ,w_n\}$ of mutually orthogonal non-zero partial isometries in $A$, and $\lambda _1,\ldots ,\lambda _n$ in $\mathbb{R}^+$ with $\lambda _1=1,$ and $\lambda _j\leq 1$ for all $j$. Indeed, pick a minimal partial isometry $v_1$ such that $w_1 = v_1 + w_1^{\perp }$, where $w_1^{\perp } \perp v_1$, by replacing $w_1$ with $v_1$ we can always assume that $w_1$ is minimal. By Theorem 3.3 and the above properties we have

$$\begin{align*} f\left(\sum _{j=1}^n \lambda _j w_j\right) & = f(w_1) + T_{w_1} \left(\sum _{j=2}^n \lambda _j w_j\right)\\ & = f(w_1) + \sum _{j=2}^n \lambda _j T_{w_1} \left(w_j\right)\\ & = T(w_1) + \sum _{j=2}^n \lambda _j T \left(w_j\right)\\ & = \sum _{j=1}^n \lambda _j f\left(w_j\right)\\ & = \sum _{j=1}^n \lambda _j T\left(w_j\right)\\ & = T\left(\sum _{j=1}^n \lambda _j w_j\right) . \end{align*}$$

Since every element in $S(A)$ can be approximated in norm by finite real linear combinations of mutually orthogonal partial isometries in $A$ (see 20, Lemma 3.11), we derive from the above properties that $f(x) = T (x)$ for every $x\in S(A)$.

If $\sharp I \geq 2$ we can apply the above arguments to $f^{-1}$. Finally, if we assume that $\sharp J = \sharp I= 1$, then the desired statement follows from Theorem 3.2.

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