# Unitarily invariant Norms on Operators

Jor-Ting Chan  and  Chi-Kwong Li  Email<EMAIL_ADDRESS>Department of
Mathematics, The College of William & Mary, Williamsburg, VA 13185, USA.
Email<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be a symmetric norm on ${\mathbb{R}}^{n}$ and let
${{\mathcal{B}}({\mathcal{H}})}$ be the set of all bounded linear operators on
a Hilbert space ${\mathcal{H}}$ of dimension at least $n$. Define a norm on
${{\mathcal{B}}({\mathcal{H}})}$ by $\|A\|_{f}=f(s_{1}(A),\dots,s_{n}(A))$,
where $s_{k}(A)=\inf\\{\|A-X\|:X\in{{\mathcal{B}}({\mathcal{H}})}\hbox{ has
rank less than }k\\}$ is the $k$th singular value of $A$. Basic properties of
the norm $\|\cdot\|_{f}$ are obtained including some norm inequalities and
characterization of the equality case. Geometric properties of the unit ball
of the norm are obtained; the results are used to determine the structure of
maps $L$ satisfying $\|L(A)-L(B)\|_{f}=\|A-B\|_{f}$ for any
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$.

###### Key words and phrases:

Symmetric norm, unitarily invariant norm.

###### 2000 Mathematics Subject Classification:

15A04, 15A60, 47B48

##  1\. Introduction

Let $\mathcal{H}$ be a complex Hilbert space and
${{\mathcal{B}}({\mathcal{H}})}$ the algebra of all bounded linear operators
on $\mathcal{H}$. If ${\mathcal{H}}$ has finite dimension $n$, we can identify
${\mathcal{H}}$ and ${{\mathcal{B}}({\mathcal{H}})}$ respectively as
${\mathbb{C}}^{n}$ and $M_{n}$, the set of all $n\times n$ complex matrices.
Let $\|\cdot\|_{\rm sp}$ be the operator norm on
${{\mathcal{B}}({\mathcal{H}})}$. For any
$A\in{{\mathcal{B}}({\mathcal{H}})}$, let

$s_{k}(A)=\inf\\{\|A-X\|_{\rm sp}:X\in{{\mathcal{B}}({\mathcal{H}})}\hbox{ has
rank less than }k\\}$

be the $k$th singular value of $A$. They are also called the $s$-numbers or
approximation numbers of $A$. See [19] for the background. If $A$ is compact,
then $s_{k}(A)$ is the $k$th largest eigenvalue of $(A^{*}A)^{1/2}$. We can
use the singular values of matrices to define those for a general operator
$A\in{{\mathcal{B}}({\mathcal{H}})}$ as

$s_{k}(A)=\sup\\{s_{k}(X^{*}AY):X,Y:\mathbb{C}^{n}\to{\mathcal{H}},X^{*}X=Y^{*}Y=I_{k}\\}.$

A norm $f$ on ${\mathbb{R}}^{n}$ is called symmetric if $f(Px)=f(x)$ for any
$x\in{\mathbb{R}}^{n}$ and any permutation or diagonal orthogonal matrix $P$.
Suppose $f$ is a symmetric norm on ${\mathbb{R}}^{n}$ and
$n\leq\dim{\mathcal{H}}$. One can define a norm on
${{\mathcal{B}}({\mathcal{H}})}$ by

$\|A\|_{f}=f(s_{1}(A),\dots,s_{n}(A))\quad\mbox{for all $A$}.$

For example, if $f(x)=\sum_{j=1}^{n}(|c_{j}x_{j}|^{p})^{1/p}$ for
$x=(x_{1},\dots,x_{n})^{t}\in{\mathbb{R}}^{n}$, where $(c_{1},\dots,c_{n})$ is
a nonzero vector, then $\|\cdot\|_{f}$ is the $(c,p)$-norm studied in [3].

It is clear that $\|\cdot\|_{f}$ is unitarily invariant, i.e.,

$\|UAV\|=\|A\|\quad\hbox{ for all }A\hbox{ and unitary }U,V\hbox{ in
}{{\mathcal{B}}({\mathcal{H}})}.$

Unitarily invariant norms on $M_{n}$ are well studied. There has also been
considerable interest in unitarily invariant norms on
${{\mathcal{B}}({\mathcal{H}})}$, but most of the study were restricted to
symmetrically norm ideals in ${{\mathcal{B}}({\mathcal{H}})}$ to ensure that
the norm is bounded; for example, see [1, 6, 7, 21, 22].

In this paper, we study norm properties and distance preserving maps related
to $\|\cdot\|_{f}$ defined above. In Section 2, we study basic properties for
$\|\cdot\|_{f}$, and obtain results analogous to those on matrices and compact
operators; e.g., see [5, 10, 14, 17]. In Section 3, we obtain some geometrical
properties of the unit norm ball of $\|\cdot\|_{f}$ and use the result to
determine the structure of maps
$L:{{\mathcal{B}}({\mathcal{H}})}\rightarrow{{\mathcal{B}}({\mathcal{H}})}$
such that $\|A-B\|_{f}=\|L(A)-L(B)\|_{f}$ for all
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$. These extend the results on isometries
and distance preserving maps of unitarily invariant norms in the literature;
e.g., see [1, 2, 3, 8, 11, 13, 15, 22].

##  2\. Basic norm properties and inequalities

The construction of $\|\cdot\|_{f}$ on ${{\mathcal{B}}({\mathcal{H}})}$ also
arises naturally from other considerations. For any norm $\|\cdot\|$ on
$M_{n}$ with $n\leq\dim{\mathcal{H}}$, one can define a norm on
${{\mathcal{B}}({\mathcal{H}})}$ by

$\|A\|_{\rm
ui1}=\sup\\{\|X^{*}AY\|:X,Y:\mathbb{C}^{n}\to{\mathcal{H}},X^{*}X=Y^{*}Y=I_{n}\\}.$

Then $\|\cdot\|_{\rm ui1}$ is unitarily invariant. Clearly, if $n=1$ and
$\|[a]\|=|a|$, then $\|\cdot\|_{\rm ui1}$ is the operator norm. In fact, one
may assume that the norm $\|\cdot\|$ on $M_{n}$ is unitarily invariant. Else,
we may modify the norm as

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|A\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=\max\\{\|UAV\|:U,V\in
M_{n}\hbox{ are unitary}\\}.$

Then ${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\ \cdot\
\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$ is unitarily invariant;
$\|\cdot\|$ and ${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\ \cdot\
\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$ induce the same
$\|\cdot\|_{\rm ui1}$ on ${{\mathcal{B}}({\mathcal{H}})}$. By a result of von
Neumann in [18] (see also [15, 17]),

(U1) there is a symmetric norm $f$ on ${\mathbb{R}}^{n}$ such that $\|A\|_{\rm
ui1}=f(s_{1}(A),\dots,s_{n}(A))$ for all $A\in M_{n}$.

As a result, $\|A\|_{\rm ui1}=\|A\|_{f}$ for all
$A\in{{\mathcal{B}}({\mathcal{H}})}$. So, instead of using a symmetric norm
$f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ to define $\|\cdot\|_{f}$ on
${{\mathcal{B}}({\mathcal{H}})}$, one can start with an arbitrary norm
$\|\cdot\|$ on $M_{n}$ and the corresponding $\|\cdot\|_{\rm ui1}$ defined
above will coincide with $\|\cdot\|_{f}$ for some symmetric norm $f$.

Also, let $\|\cdot\|$ be any norm on ${{\mathcal{B}}({\mathcal{H}})}$. If
$n\leq\dim{\mathcal{H}}$, one can define

$\|A\|_{\rm ui2}=\sup\\{\|X^{*}AY\|:X,Y\hbox{ are rank $n$ partial
isometries}\\}.$

Then $\|\cdot\|_{\rm ui2}$ is a unitarily invariant norm on
${{\mathcal{B}}({\mathcal{H}})}$ such that

$\|A\|\leq\|A\|_{\rm ui2}\qquad\hbox{ for all
}A\in{{\mathcal{B}}({\mathcal{H}})}\hbox{ with }{\rm rank}\,(A)\leq n.$

Take two orthonormal families $\\{x_{1},\dots,x_{n}\\}$ and
$\\{y_{1},\dots,y_{n}\\}$ in ${\mathcal{H}}$ and define
$f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ by

$f(a_{1},\dots,a_{n})=\left\|\mbox{$\sum_{j=1}^{n}a_{j}x_{j}y_{j}^{*}$}\right\|_{\rm
ui2}\quad\mbox{for all $(a_{1},\ldots,a_{n})\in{\mathbb{R}}^{n}$}.$

Then one can check that $f$ is a symmetric norm on ${\mathbb{R}}^{n}$ and

$\|A\|_{\rm ui2}=\|A\|_{f}\qquad\hbox{ for all
}A\in{{\mathcal{B}}({\mathcal{H}})}.$

So, $\|\cdot\|_{f}$ can be generated from any norm on
${{\mathcal{B}}({\mathcal{H}})}$.

There are other interesting connections between unitarily invariant norms and
symmetric norms in the finite dimensional case, see [13] and [14]. In
particular, if $\|\cdot\|$ is a unitarily invariant norm on $M_{n}$, then (U1)
holds, and

  * (U2)

there is a compact set ${\mathcal{S}}\subseteq{\mathbb{R}}^{n}$ consisting of
vectors $c=(c_{1},\dots,c_{n})$ with $c_{1}\geq\cdots\geq c_{n}\geq 0$ such
that

$\|A\|=\max\\{\|A\|_{c}:c\in{\mathcal{S}}\\},$

where $\|A\|_{c}=\sum_{j=1}^{n}c_{j}s_{j}(A)$ is the $c$-norm of $A$.

From the above discussion we have the following proposition, which will be
useful in our study.

######  Proposition 2.1.

Let $f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ be a symmetric norm. Suppose
${\mathcal{H}}$ is a Hilbert space with $\dim{\mathcal{H}}\geq n$, and
$\|\cdot\|$ is a norm on ${{\mathcal{B}}({\mathcal{H}})}$ defined by

(2.1) $\|A\|=f(s_{1}(A),\dots,s_{n}(A))\quad\hbox{ for all
}A\in{{\mathcal{B}}({\mathcal{H}})}.$

Then there is a compact set ${\mathcal{S}}\subseteq{\mathbb{R}}^{n}$
consisting of vectors $c=(c_{1},\dots,c_{n})$ with $c_{1}\geq\cdots\geq
c_{n}\geq 0$ such that

(2.2) $\|A\|=\max\\{\|A\|_{c}:c\in{\mathcal{S}}\\}.$

By the above result, we can use the properties of symmetric norms and
$c$-norms to study unitarily invariant norm $\|\cdot\|$ defined on
${{\mathcal{B}}({\mathcal{H}})}$ as in (2.1). For example, it is easy to
extend [10, Theorem 1] concerning the comparison of a unitarily invariant norm
and the numerical radius defined by

$r(A)=\sup\\{|{\langle}Ax,x{\rangle}|:x\in{\mathcal{H}},{\langle}x,x{\rangle}=1\\}.$

To present our result and proof, we need the following definition. For two
nonnegative vectors $x,y\in{\mathbb{R}}^{n}$ we say that $x$ is weakly
majorized by $y$, denoted by $x\prec_{w}y$, if the sum of the $k$ largest
entries of $x$ is not larger than that of $y$ for every $1\leq k\leq n$. See
[16].

######  Proposition 2.2.

Suppose $\|\cdot\|$ is a norm on ${{\mathcal{B}}({\mathcal{H}})}$ defined as
in (2.1). Let

$\alpha=f(e_{1})\quad\hbox{ and
}\quad\beta=\begin{cases}2f(e_{1}+\cdots+e_{k})&\hbox{ if }n=2k,\cr
2f(e_{1}+\cdots+e_{k}+e_{k+1}/2)&\hbox{ if }n=2k+1.\cr\end{cases}$

Then for any $A\in{{\mathcal{B}}({\mathcal{H}})}$,

$\alpha r(A)\leq\|A\|\leq\beta r(A).$

The constants $\alpha$ and $\beta$ are the best possible in view of the
following:

  * •

$\alpha r(A)=\alpha s_{1}(A)=\|A\|$ if $A=xx^{*}$ for a unit vector $x$;

  * •

$\|A\|=\beta r(A)$ if $A=\begin{cases}\sum_{j=1}^{k}2x_{j}y_{j}^{*}&\hbox{ if
}n=2k,\cr\sum_{j=1}^{k}2x_{j}y_{j}^{*}+x_{k+1}x_{k+1}^{*}&\hbox{ if
}n=2k+1,\cr\end{cases}$
for an orthonormal set
${\mathcal{F}}=\\{x_{1},\dots,x_{k},y_{1},\dots,y_{k}\\}\subseteq{\mathcal{H}}$
and $x_{k+1}\in{\mathcal{F}}^{\perp}$ if $n=2k+1$.

Proof. Denote by $\\{E_{jk}\\}_{j,k=1}^{n}$ the standard basis for $M_{n}$.
Let $s(Y)$ be the vector of singular values of $Y\in M_{n}$. If
$A\in{{\mathcal{B}}({\mathcal{H}})}$ has $r(A)=1$, then for any
$X,Y:{\mathbb{C}}^{n}\to{\mathcal{H}}$ such that $X^{*}X=Y^{*}Y=I_{n}$, we
have from the proof of Theorem 1 in [10],

$s(E_{11})\prec_{w}s(X^{*}AY)\prec_{w}s(F),$

where $F=2(E_{12}+E_{34}+\cdots+E_{2k-1,2k})$ if $n=2k$ and
$F=2(E_{12}+E_{34}+\cdots+E_{2k-1,2k})+E_{nn}$ if $n=2k+1$. It is known that
if $R,T\in M_{n}$ with $s(R)\prec_{w}s(T)$ then $|R|\leq|T|$ for any unitarily
invariant norm $|\cdot|$; see [14, 17]. We get the inequalities. The equality
cases can be checked directly. ∎

We extend the definitions in ([13, pp. 54-55]) for $\|\cdot\|$ defined as in
(2.1):

(i) $\|\cdot\|$ is a uniform norm if $\|AB\|\leq s_{1}(A)\|B\|$ and
$\|AB\|\leq s_{1}(B)\|A\|$,

(ii) $\|\cdot\|$ is a cross norm if $\|A\|=s_{1}(A)$ for any rank one operator
$A\in{{\mathcal{B}}({\mathcal{H}})}$.

(iii) $\|\cdot\|$ on ${{\mathcal{B}}({\mathcal{H}})}$ is submultiplicative if
$\|AB\|\leq\|A\|\|B\|$ for all $A,B\in{{\mathcal{B}}({\mathcal{H}})}$;

if, in addition, $\|I\|=1$, it is an algebra (or a ring) norm.

We have the following results.

######  Proposition 2.3.

Suppose $\|\cdot\|$ is a norm on ${{\mathcal{B}}({\mathcal{H}})}$ defined as
in (2.1).

  * (a)

Then $\|\cdot\|$ is a uniform norm.

  * (b)

The norm $\|\cdot\|$ is a cross norm if and only if $f(e_{1})=1$.

Proof. (a) As observed above, we may assume that $\|\cdot\|$ is induced by a
unitarily invariant norm ${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\
\cdot\ \right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$ on $M_{n}$. For
any $A,B\in{{\mathcal{B}}({\mathcal{H}})}$ and
$X,Y:{\mathbb{C}}^{n}\to{\mathcal{H}}$ such that $X^{*}X=Y^{*}Y=I_{n}$, let
$P$ be any rank $n$ partial isometry whose image contains that of $BY$. Then
by [5, Lemma 2.2],

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|X^{*}ABY\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|(X^{*}AP)(P^{*}BY)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leq
s_{1}(X^{*}AP){\left|\kern-1.07639pt\left|\kern-1.07639pt\left|P^{*}BY\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leq
s_{1}(A)\|B\|.$

The result follows.

(b) Evidently, $f(e_{1})=1$ if and only if $1=\|xy^{*}\|=f(e_{1})$ for any
unit vectors $x,y$. ∎

######  Proposition 2.4.

Suppose $\|\cdot\|$ is a norm on ${{\mathcal{B}}({\mathcal{H}})}$ defined as
in (2.1). Then $\|\cdot\|$ is submultiplicative if and only if any one of the
following holds.

  * (1)

$\|A\|\geq s_{1}(A)$ for all $A\in{{\mathcal{B}}({\mathcal{H}})}$.

  * (2)

There are unit vectors $x,y\in{\mathcal{H}}$ such that $\|xy^{*}\|\geq 1$.

  * (3)

$f(e_{1})\geq 1$, equivalently, there is $(c_{1},\dots,c_{n})\in{\mathcal{S}}$
with $c_{1}\geq 1$.

Moreover, $\|\cdot\|$ is an algebra norm if and only if it is the operator
norm.

Proof. The equivalence of (1) – (3) can be proved as in the finite dimensional
case, e.g., see [14]. Suppose $\|\cdot\|$ is multiplicative. Then for any unit
vector $x\in{\mathcal{H}}$, $\|xx^{*}\|^{2}\geq\|(xx^{*})^{2}\|=\|xx^{*}\|$ so
that $1\leq\|xx^{*}\|=f(e_{1})$. Thus, (3) holds. Suppose (1) holds. Then by
Proposition 2.3 (a), $\|AB\|\leq s_{1}(A)\|B\|\leq\|A\|\|B\|$ for any
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$.

To prove the last assertion, suppose $\|\cdot\|$ is an algebra norm. Then for
any $c=(c_{1},\dots,c_{n})\in{\mathcal{S}}$,
$1=\|I\|\geq\|I\|_{c}=\sum_{j=1}^{n}c_{j}$. For any
$A\in{{\mathcal{B}}({\mathcal{H}})}$, there is a $c\in{\mathcal{S}}$ such that

$\|A\|=\|A\|_{c}=\sum_{j=1}^{n}c_{j}s_{j}(A)\leq s_{1}(A).$

But $\|\cdot\|$ is submultiplicative, so that $\|A\|\geq s_{1}(A)$. Thus,
$\|\cdot\|$ is the operator norm. The converse is clear. ∎

One may obtain similar inequalities for the triple product on
${{\mathcal{B}}({\mathcal{H}})}$ defined by

$A\circ B\circ C=\frac{1}{2}(AB^{*}C+CB^{*}A).$

######  Proposition 2.5.

Suppose $\|\cdot\|$ is a norm on ${{\mathcal{B}}({\mathcal{H}})}$ defined as
in (2.1). The following condition are equivalent.

  * (1)

$\|A\circ B\circ C\|\leq\|A\|\|B\|\|C\|$ for all
$A,B,C\in{{\mathcal{B}}({\mathcal{H}})}$.

  * (2)

$\|A\circ A\circ A\|\leq\|A\|^{3}$ for all
$A,\in{{\mathcal{B}}({\mathcal{H}})}$.

  * (3)

$f(e_{1})\geq 1$, equivalently, there is $(c_{1},\dots,c_{n})\in{\mathcal{S}}$
with $c_{1}\geq 1$.

Proof. If (1) holds, then (2) clearly holds. If (2) holds, consider $A=xy^{*}$
for unit vectors $x,y$. Clearly, $\|A\|=f(e_{1})$ and $A\circ A\circ A=A$.
Thus, the inequality in (2) becomes $f(e_{1})\leq f(e_{1})^{3}$, and hence
$f(e_{1})\geq 1$.

Let
$(s_{1},\dots,s_{n}),(a_{1},\dots,a_{n}),(b_{1},\dots,b_{n}),(c_{1},\dots,c_{n})$
be the vectors of the $n$ largest singular values of $A\circ B\circ C,A,B,C,$
respectively. Then
$\sum_{j=1}^{\ell}s_{j}\leq\sum_{j=1}^{\ell}a_{j}b_{j}c_{j}$ for
$\ell=1,\dots,n$. This ensures that

$f(s_{1},\dots,s_{n})\leq f(a_{1}b_{1}c_{1},\dots,a_{n}b_{n}c_{n}).$

If (3) holds, then for vectors $(x_{1},\dots,x_{n}),(y_{1},\dots,y_{n})$ with
nonnegative entries arranging in descending order, we have

$f(x_{1}y_{1},\dots,x_{n}y_{n})\leq f(x_{1},\dots,x_{n})f(y_{1},\dots,y_{n}).$

Consequently,

$\|A\circ B\circ C\|_{f}=f(s_{1},\dots,s_{n})\leq
f(a_{1}b_{1}c_{1},\dots,a_{n}b_{n}c_{n})$ $\leq
f(a_{1},\dots,a_{n})f(b_{1},\dots,b_{n})f(c_{1},\dots,c_{n})=\|A\|_{f}\|B\|_{f}\|C\|_{f}.$

Thus (1) holds. ∎

The next theorem extends the result for the finite dimensional case in [5]. In
fact, there was a gap in line 4 of the proof of [5, Theorem 2.4], which
asserted that $1\leq s_{1}(AB)/\|A\|\|B\|$ without justification. Our proof
fills the gap.

######  Theorem 2.6.

Suppose $\|\cdot\|$ on ${{\mathcal{B}}({\mathcal{H}})}$ defined as in (2.1) is
a submultiplicative norm. Then there exist nonzero
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$ such that $\|AB\|=\|A\|\|B\|$ if and
only if $\|\cdot\|$ is a cross norm.

In case $\|\cdot\|$ is a cross norm, the following conditions hold.

  * (1)

Two nonzero operators $A,B\in{{\mathcal{B}}({\mathcal{H}})}$ satisfy
$\|AB\|=\|A\|\|B\|$ if and only if $s_{1}(AB)=\|A\|\|B\|$.

  * (2)

Assume that $\\{T\in{{\mathcal{B}}({\mathcal{H}})}:\|T\|=s_{1}(T)\\}$ is a
subset of rank one matrices. Two nonzero operators
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$ satisfy $\|AB\|=\|A\|\|B\|$ if and only
if $A=xy^{*}$ and $B=y^{*}z$ for some nonzero $x,y,z\in{\mathcal{H}}$ with
${\langle}y,y{\rangle}=1$.

Proof. Suppose $\|\cdot\|$ is submultiplicative. If $\|\cdot\|$ is a cross
norm, then for any unit vector $x\in{\mathcal{H}}$ and $A=B=xx^{*}$, we have
$\|AB\|=\|A\|\|B\|=1.$

Suppose $\|\cdot\|$ is not a cross norm. Then $\|xy^{*}\|=d>1$ for any unit
vectors $x,y\in{\mathcal{H}}$. We will show that $\|A\|\|B\|>\|AB\|$ for any
nonzero $A,B\in{{\mathcal{B}}({\mathcal{H}})}$. For any
$\varepsilon\in(0,s_{1}(A))$, take a unit vector $x\in{\mathcal{H}}$ such that
$\|Ax\|>s_{1}(A)-\varepsilon$ and let $y=Ax/\|Ax\|$. Then $s_{1}(Axy^{*})\leq
s_{1}(A)$ and $s_{j}(Axy^{*})=0$ for all $j>1$. Consequently,
$\sum_{j=1}^{k}s_{j}(A)\geq\sum_{j=1}^{k}s_{j}(Axy^{*})$ for all
$k=1,\ldots,n$. It follows that
$\|A\|\geq\|Axy^{*}\|=\|Ax\|\|yy^{*}\|>(s_{1}(A)-\varepsilon)d$. As this is
true for all such $\varepsilon$, $\|A\|\geq s_{1}(A)d>s_{1}(A)$. Thus
$\|A\|\|B\|>s_{1}(A)\|B\|\geq\|AB\|$.

Now, assume that $\|\cdot\|$ is a cross norm.

(1) Note that $s_{1}(AB)\leq\|AB\|\leq\|A\|\|B\|$. If $s_{1}(AB)=\|A\|\|B\|$,
then $\|AB\|=\|A\|\|B\|$.

Conversely, if $\|AB\|=\|A\|\|B\|$, then by Proposition 2.3 (a) and
Proposition 2.4 (1), $s_{1}(A)=\|A\|$ and $s_{1}(B)=\|B\|$. Suppose
$s_{1}(AB)<\|A\|\|B\|=s_{1}(A)s_{1}(B)$. Also, recall that $s_{j}(AB)\leq
s_{j}(A)s_{1}(B)$ for $j=2,\dots,n$. Hence, for $k=1,\dots,n$,

$\sum_{j=1}^{k}s_{j}(AB)<\sum_{j=1}^{k}s_{j}(A)s_{1}(B)\quad\hbox{ so that
}\quad\sum_{j=1}^{k}s_{j}(AB)\leq
t\sum_{j=1}^{k}s_{j}(A)s_{1}(B)=\sum_{j=1}^{k}s_{j}(ts_{1}(B)A)$

for some $t\in(0,1)$. Consequently,

$\|AB\|\leq\|ts_{1}(B)A\|=t\|A\|\|B\|<\|A\|\|B\|,$

which is a contradiction.

(2) Direct verification. ∎

We remark that in Theorem 2.6 (2), the assumption that
$\\{T\in{{\mathcal{B}}({\mathcal{H}})}:\|T\|=s_{1}(T)\\}$ is a subset of rank
one matrices is equivalent to the set equality

$\\{T\in{{\mathcal{B}}({\mathcal{H}})}:\|T\|=s_{1}(T)=1\\}=\\{xy^{*}:x,y\in{\mathcal{H}}\hbox{
are unit vecotrs}\\}.$

By [5, Example 2.5], the result in Theorem 2.6 (2) is not true if the above
set equality fails. For instance, if $\|\cdot\|$ is the spectral norm, any
unitary $A,B$ will satisfy $\|AB\|=\|A\|\|B\|$.

Specializing the above results to the $c$-norm, we have the following.

######  Corollary 2.7.

Suppose $c=(c_{1},\dots,c_{n})$ has positive entries arranged in descending
order.

  * (a)

The norm $\|\cdot\|_{c}$ is submultiplicative if and only if $c_{1}\geq 1$.

  * (b)

The norm $\|\cdot\|_{c}$ is a cross norm if and only if $c_{1}=1$.

  * (c)

Suppose $c_{1}\geq 1$. There exist nonzero
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$ such that
$\|AB\|_{c}=\|A\|_{c}\|B\|_{c}$ if and only if $c_{1}=1$.

  * (d)

Suppose $c=(1,0,\dots,0)$. Two nonzero operators
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$ satisfy $\|AB\|_{c}=\|A\|_{c}\|B\|_{c}$
if and only only if $s_{1}(AB)=s_{1}(A)s_{1}(B)$.

  * (e)

Suppose $c_{1}=1$ and $c_{2}>0$. Two nonzero operators
$A,B\in{{\mathcal{B}}({\mathcal{H}})}$ satisfy $\|AB\|_{c}=\|A\|_{c}\|B\|_{c}$
if and only if $A=xy^{*}$ and $B=yz^{*}$ for some nonzero vectors
$x,y,z\in{\mathcal{H}}$.

##  3\. Distance preserving maps

In this section, we always assume that $\|\cdot\|$ is defined as in (2.1) for
$n>1$. So,

$\|A\|=f(s_{1}(A),\dots,s_{n}(A))\
(=\|A\|_{f})=\max\\{\|A\|_{c}:c\in{\mathcal{S}}\\},$

for a symmetric norm $f$ on ${\mathbb{R}}^{n}$ and a compact set
${\mathcal{S}}$ in ${\mathbb{R}}^{n}$. This property will be used later
without further reference. Our main result in this section is the following.

######  Theorem 3.1.

A surjective map
$L:{{\mathcal{B}}({\mathcal{H}})}\rightarrow{{\mathcal{B}}({\mathcal{H}})}$
satisfies

(3.3) $\|A-B\|=\|L(A)-L(B)\|\quad\hbox{ for all
}A,B\in{{\mathcal{B}}({\mathcal{H}})}$

if and only if there are unitary operators
$U,V\in{{\mathcal{B}}({\mathcal{H}})}$ such that $L$ has the form $A\mapsto
U^{*}\phi(A)V+R_{0}$, where $R_{0}\in{{\mathcal{B}}({\mathcal{H}})}$ and
$\phi$ is one of the maps

$A\mapsto A,\quad A\mapsto A^{t},\quad A\mapsto A^{*},\quad
A\mapsto(A^{*})^{t}.$

Here $A^{t}\in{{\mathcal{B}}({\mathcal{H}})}$ is the transpose of $A$ with
respect to a fixed orthonormal basis of ${\mathcal{H}}$.

If ${\mathcal{H}}$ is finite dimensional, then the result holds without the
assumption that $L$ is surjective. It is a special case of [2, Theorem 3.2].
We will therefore focus on the case when ${\mathcal{H}}$ has infinite
dimension. Note also that a characterization of complex isometric isomorphisms
on ${{\mathcal{B}}({\mathcal{H}})}$, when $\dim{\mathcal{H}}$ is infinite, is
given in [3] for a special class of unitarily invariant norms.

Let
$L:{{\mathcal{B}}({\mathcal{H}})}\rightarrow{{\mathcal{B}}({\mathcal{H}})}$ be
a surjective map satisfying the hypothesis of Theorem 3.1. Then by the Mazur-
Ulam Theorem, the map $\hat{L}:A\mapsto L(A)-L(0)$ is real linear, surjective,
and satisfies $\|\hat{L}(A)\|=\|A\|$ for all
$A\in{{\mathcal{B}}({\mathcal{H}})}$. So, $\hat{L}$ preserves
${\mathcal{B}}=\\{X\in{{\mathcal{B}}({\mathcal{H}})}:\|X\|\leq 1\\}$, the
closed unit ball for $\|\cdot\|$, and the extreme points of it. In the
following, we give some auxiliary results concerning the extreme points of
${\mathcal{B}}$. In particular, we show that $U/\|U\|$ is an extreme point of
${\mathcal{B}}$ for any maximal partial isometry $U$, i.e.,
$U\in{{\mathcal{B}}({\mathcal{H}})}$ such that $UU^{*}=I$ or $U^{*}U=I$.

######  Lemma 3.2.

Let $A\in{{\mathcal{B}}({\mathcal{H}})}$ be an extreme point of
${\mathcal{B}}$. Then $A=\sum_{j=1}^{n}s_{j}(A)x_{j}y_{j}^{*}+s_{n}(A)U$,
where $s_{1}\geq\cdots\geq s_{n}\geq 0$, $\\{x_{1},\ldots,x_{n}\\}$,
$\\{y_{1},\ldots,y_{n}\\}$ are orthonormal sets in ${\mathcal{H}}$ and $U$ is
a partial isometry from $\\{y_{1},\ldots,y_{n}\\}^{\perp}$ onto
$\\{x_{1},\ldots,x_{n}\\}^{\perp}$.

On the other hand, $U/\|U\|$ is an extreme point of ${\mathcal{B}}$ for every
maximal partial isometry $U\in{{\mathcal{B}}({\mathcal{H}})}$.

Proof. Let $A\in{{\mathcal{B}}({\mathcal{H}})}$ be an extreme point of
${\mathcal{B}}$. We shall show that $A$ is of the given form using the ideas
in [3]. First of all, we have $s_{k}(A)=s_{n}(A)$ for every $k>n$. Otherwise,
there is a $k>n$ such that $s_{k}(A)<s_{n}(A)$ and $s_{k}(A)$ is an eigenvalue
of $|A|$. Take a unit eigenvector $x$ of $|A|$ corresponding to $s_{k}(A)$ and
let $B=(Ax)x^{*}$. For sufficiently small $\varepsilon>0$, $A\pm\varepsilon B$
have the same $n$ largest singular values as $A$. So, $\|A\pm\varepsilon
B\|=1$, contradicting the choice of $A$. Next, we claim that
$\sigma(|A|)\subseteq\\{0,s_{1}(A),\ldots,s_{n}(A)\\}$. An argument as above
shows that it is true if every non-zero element of $\sigma(|A|)$ is an
eigenvalue. Otherwise we can use the construction in [3, Lemma 3.7] to get a
contradiction if the claim is not true. So, $A$ is of the form
$\sum_{j=1}^{n}s_{j}(A)x_{j}y_{j}^{*}+s_{n}(A)U$ for orthonormal sets
$\\{x_{1},\ldots,x_{n}\\}$, $\\{y_{1},\ldots,y_{n}\\}$ and a partial isometry
$U$. This $U$ must be a maximal partial isometry. Otherwise, a contradiction
can be obtained as above by considering $B=yx^{*}$ for unit vectors $x\in{\rm
Ker}\,U$ and $y\in{\rm Im}\,U$.

The last statement is a consequence of [3, Lemma 3.5] which asserts that if
$U$ is a maximal partial isometry, then $U/\|U\|_{c}$ is an extreme point of
${\mathcal{B}}_{c}$, the closed unit ball of $\|\cdot\|_{c}$, for every
$c\in{\mathcal{S}}$. So, take $c\in{\mathcal{S}}$ such that $\|U\|=\|U\|_{c}$.
Then $U/\|U\|=U/\|U\|_{c}$ is an extreme point of ${\mathcal{B}}_{c}$. It must
be an extreme point of ${\mathcal{B}}$ as
${\mathcal{B}}\subseteq{\mathcal{B}}_{c}$. The lemma is proved. ∎

To prove Lemma 3.4, we need the following result.

######  Lemma 3.3.

Suppose $c=(c_{1},\dots,c_{p})$ has all entries positive and $C={\rm
diag}\,(c_{1},\ldots,c_{p})$. If $R\in M_{p}$ satisfies $\|R\|_{c}={\rm
tr}\,(CR),$ then $R$ is positive semidefinite.

Proof. It is known that $R\in M_{p}$ $R$ is positive semidefinite if and only
if ${\rm tr}\,R=\sum_{j=1}^{p}s_{j}(R)$; e.g., see [12, Corollary 3.2]. In the
following, we will show that ${\rm tr}\,R=\sum_{j=1}^{p}s_{j}(R)$ if
$\|R\|_{c}={\rm tr}\,(CR)$.

As some of the entries in $c$ may be equal, write
$C=\xi_{1}I_{p_{1}}\oplus\cdots\oplus\xi_{u}I_{p_{u}}$ with
$\xi_{1}>\cdots>\xi_{u}>0$ and $R=(R_{ij})\in M_{p}$ with $R_{jj}\in
M_{p_{j}}$ for $j=1,\dots,u$. It is known if $R$ has diagonal entries
$d_{1},\dots,d_{p}$, then
$|\sum_{j=1}^{\ell}d_{j}|\leq\sum_{j=1}^{\ell}s_{j}(R)$; e.g., see [12]. So,

${\rm tr}\,(CR)=\sum_{j=1}^{u}\xi_{j}{\rm
tr}\,R_{jj}=\sum_{j=1}^{u-1}(\xi_{j}-\xi_{j+1}){\rm
tr}\,(R_{1}+\cdots+R_{j})+\xi_{u}{\rm tr}\,R,$

and

$\left|\sum_{j=1}^{u-1}(\xi_{j}-\xi_{j+1}){\rm
tr}\,(R_{1}+\cdots+R_{j})+\xi_{u}{\rm tr}\,R\right|\hskip 36.135pt\ $ $\
\hskip
21.68121pt\leq\sum_{j=1}^{u-1}(\xi_{j}-\xi_{j+1})[s_{1}(R)+\cdots+s_{p_{1}+\cdots+p_{j}}(R)]+\xi_{u}\sum_{j=1}^{p}s_{j}(R)=\|R\|_{c}.$

Thus, ${\rm tr}\,(CR)=\|R\|_{c}$ ensures ${\rm tr}\,R=\sum_{j=1}^{p}s_{j}(R)$,
which is the desired equality mentioned at the beginning of the proof. Hence
$R$ is positive semidefinite. ∎

######  Lemma 3.4.

Let $A=a_{1}I_{r_{1}}\oplus\cdots\oplus
a_{m}I_{r_{m}}+a_{m+1}U\in{{\mathcal{B}}({\mathcal{H}})}$ with
$a_{1}>\cdots>a_{m+1}\geq 0$ and $U$ a maximal partial isometry from
$[\ker(A-a_{1}I)\oplus\cdots\oplus\ker(A-a_{m}I)]^{\perp}$ into itself.
Suppose $A=B+D$ and they satisfy

(3.4) $\|\mu B+\nu D\|=\max\\{|\mu|,|\nu|\\}\quad\mbox{for
any\quad$\mu,\nu\in{\mathbb{R}}$.}$

Then there is a vector $c\in{\mathcal{S}}$ such that $\|A+B\|_{c}=\|A+B\|=2$.
Let $B_{1}$ be the compression of $B$ onto ${\mathcal{K}}=\ker(A-a_{1}I)$. If
$c$ has $p$ nonzero entries, then one of the following holds.

  * (a)

$p<r_{1}$ and there is a linear functional
$\varphi\in{{\mathcal{B}}({\mathcal{K}})}^{*}$, the dual space of
${{\mathcal{B}}({\mathcal{K}})}$, such that $\varphi(B_{1})=1$. Moreover, if
$Y\in{{\mathcal{B}}({\mathcal{H}})}$ satisfies $\|\mu B+\nu
Y\|=\max\\{|\mu|,|\nu|\\}$ for any $\mu,\nu\in{\mathbb{R}}$ and $Y_{1}$ is the
compression of $Y$ onto ${\mathcal{K}}$, then $\varphi(Y_{1})=0$.

  * (b)

$p>r_{1}$ and $B_{1}=(a_{1}/2)I_{r_{1}}$.

Proof. Assume that $A,B$ and $D$ satisfy the assumptions of the lemma. Then
$\|A\|=\|B\|=\|D\|=1$. Take $c\in{\mathcal{S}}$ such that

$\|A+B\|_{c}=\|A+B\|=\|2B+D\|=2,$

where ${\mathcal{S}}$ is the set in (2.2). Then

$2=\|A+B\|_{c}\leq\|A\|_{c}+\|B\|_{c}\leq\|A\|+\|B\|=2$

so that $\|A+B\|_{c}=\|A\|_{c}+\|B\|_{c}$. If $c$ has $p$ nonzero values
$c_{1}\geq\cdots\geq c_{p}$, then

$\sum_{j=1}^{p}c_{j}s_{j}(A+B)=\sum_{j=1}^{p}c_{j}(s_{j}(A)+s_{j}(B)).$

Since

$\displaystyle\sum_{j=1}^{p}c_{j}s_{j}(A+B)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{p-1}(c_{k}-c_{k+1})\sum_{j=1}^{k}s_{j}(A+B)+c_{p}\sum_{j=1}^{p}s_{j}(A+B)$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{p-1}(c_{k}-c_{k+1})\sum_{j=1}^{k}[s_{j}(A)+s_{j}(B)]+c_{p}\sum_{j=1}^{p}[s_{j}(A)+s_{j}(B)]$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{p}c_{j}(s_{j}(A)+s_{j}(B)),$

we see that $\sum_{j=1}^{p}s_{j}(A+B)=\sum_{j=1}^{p}[s_{j}(A)+s_{j}(B)]$. By
[3, Proposition 2.3], there are orthonormal sequences
$\\{x_{1}^{(m)},\ldots,x_{p}^{(m)}\\}$, $\\{y_{1}^{(m)},\ldots,y_{p}^{(m)}\\}$
such that

$\displaystyle\|A\|_{c}+\|B\|_{c}=\|A+B\|_{c}=\lim_{m}\sum_{j=1}^{p}c_{j}{\langle}(A+B)x_{j}^{(m)},y_{j}^{(m)}{\rangle}$
$\displaystyle\leq\
\lim_{m}\left|\sum_{j=1}^{p}c_{j}{\langle}Ax_{j}^{(m)},y_{j}^{(m)}{\rangle}\right|+\lim_{m}\left|\sum_{j=1}^{p}c_{j}{\langle}Bx_{j}^{(m)},y_{j}^{(m)}{\rangle}\right|\leq\|A\|_{c}+\|B\|_{c}.$

Thus,
$\lim_{m}\sum_{j=1}^{p}c_{j}{\langle}Ax_{j}^{(m)},y_{j}^{(m)}{\rangle}=\|A\|_{c}=\sum_{j=1}^{p}c_{j}s_{j}(A)$
and similarly for $B$. Let $w=\min\\{p,r_{1}\\}$. We may assume that for
$j=1,\ldots,w$, ${\langle}Ax_{j}^{(m)},y_{j}^{(m)}{\rangle}\to
s_{j}(A)=a_{1}$. Let $P$ be the orthogonal projection of ${\mathcal{H}}$ onto
${\mathcal{K}}$. As ${\mathcal{K}}$ is finite dimensional, we may further
assume that $Px_{j}^{(m)}\to x_{j}$ and $Py_{j}^{(m)}\to y_{j}$ for
$x_{j},y_{j}\in{\mathcal{K}}$. Then it is not difficult to show that
$x_{j}^{(m)}\to x_{j}$, $y_{j}^{(m)}\to y_{j}$ and $x_{j}=y_{j}$. Indeed, from

$\displaystyle\big{|}{\langle}Ax_{j}^{(m)},y_{j}^{(m)}{\rangle}\big{|}=\big{|}{\langle}APx_{j}^{(m)},Py_{j}^{(m)}{\rangle}+{\langle}A(x_{j}^{(m)}-Px_{j}^{(m)}),y_{j}^{(m)}-Py_{j}^{(m)}{\rangle}\big{|}$
$\displaystyle\leq$ $\displaystyle
a_{1}\|Px_{j}^{(m)}\|\|Py_{j}^{(m)}\|+a_{2}\|x_{j}^{(m)}-Px_{j}^{(m)}\|\|y_{j}^{(m)}-Py_{j}^{(m)}\|$
$\displaystyle\leq$
$\displaystyle\big{[}a_{1}\|Px_{j}^{(m)}\|^{2}+a_{2}\|x_{j}^{(m)}-Px_{j}^{(m)}\|^{2}\big{]}/2+\big{[}a_{1}\|Py_{j}^{(m)}\|^{2}+a_{2}\|y_{j}^{(m)}-Py_{j}^{(m)}\|^{2}\big{]}/2,$

we get $\|x_{j}^{(m)}-Px_{j}^{(m)}\|,\|y_{j}^{(m)}-Py_{j}^{(m)}\|\to 0$ and so
$x_{j}^{(m)}\to x_{j}$ and $y_{j}^{(m)}\to y_{j}$. Then $x_{j}=y_{j}$ follows
from
$a_{1}{\langle}x_{j},y_{j}{\rangle}={\langle}Ax_{j},y_{j}{\rangle}=a_{1}$.
Moreover, the set $\\{x_{1},\ldots,x_{w}\\}$ is orthonormal as
$\\{x_{1}^{(m)},\ldots,x_{p}^{(m)}\\}$ is orthonormal for each $m$. Consider
the $p\times p$ matrices $({\langle}Ax_{j}^{(m)},y_{k}^{(m)}{\rangle})$ and
$({\langle}Bx_{j}^{(m)},y_{k}^{(m)}{\rangle})$. Passing to a subsequence if
needed, $({\langle}Ax_{j}^{(m)},y_{k}^{(m)}{\rangle})\rightarrow\hat{A}$ and
$({\langle}Bx_{j}^{(m)},y_{k}^{(m)}{\rangle})\rightarrow\hat{B}$ for $p\times
p$ matrices $\hat{A}$ and $\hat{B}$. Note that if $p\geq r_{1}$, the upper
left $r_{1}\times r_{1}$ principal submatrices of $\hat{A}$ and $\hat{B}$ are
$({\langle}Ax_{j},x_{k}{\rangle})$ and $({\langle}Bx_{j},x_{k}{\rangle})$
respectively. They are the compressions of $A$ and $B$ onto ${\mathcal{K}}$
relative to the orthonormal basis $\\{x_{1},\ldots,x_{r_{1}}\\}$. If
$p<r_{1}$, $\hat{A}$ and $\hat{B}$ are the compressions of $A$ and $B$ onto
the $p$-dimensional subspace of ${\mathcal{K}}$ spanned by
$\\{x_{1},\ldots,x_{p}\\}$. Let $C={\rm diag}\,(c_{1},\dots,c_{p})$. Then

$1=\|A\|_{c}={\rm tr}\,C\hat{A}\quad\hbox{ and }\quad 1=\|B\|_{c}={\rm
tr}\,C\hat{B}.$

We have the following situations.

  * (a)

$p<r_{1}$. Let $C_{1}=C\oplus 0_{r_{1}-p}$ and $\varphi$ be the functional on
${{\mathcal{B}}({\mathcal{K}})}$ given by $X\mapsto{\rm tr}\,C_{1}X$. Then
$\varphi(B_{1})={\rm tr}\,C_{1}B_{1}=1$. If
$Y\in{{\mathcal{B}}({\mathcal{H}})}$ satisfies the said condition, then

$|1\pm{\rm tr}\,C_{1}\hat{Y}_{1}|=|{\rm
tr}\,C_{1}(\hat{B}_{1}\pm\hat{Y}_{1})|\leq\|\hat{B}_{1}\pm\hat{Y}_{1}\|_{c}\leq\|B\pm
Y\|_{c}\leq\|B\pm Y\|=1.$

So, $\varphi(Y_{1})={\rm tr}\,(C_{1}\hat{Y}_{1})=0$.

  * (b)

$p\geq r_{1}$. Let $\hat{D}=\hat{A}-\hat{B}$. Then

${\rm tr}\,C\hat{D}={\rm tr}\,C(\hat{A}-\hat{B})={\rm tr}\,C\hat{A}-{\rm
tr}\,C\hat{B}=\|A\|_{c}-\|B\|_{c}=0.$

It follows that

$1={\rm
tr}\,C(\hat{B}-\hat{D})\leq\|\hat{B}-\hat{D}\|_{c}\leq\|B-D\|_{c}\leq\|B-D\|=1.$

By Lemma 3.3, $\hat{B}-\hat{D}$ is positive semidefinite. As the upper left
$r_{1}\times r_{1}$ principal submatrix of $\hat{B}-\hat{D}$ is the
compression of $B-D$ on ${\mathcal{K}}$, $B_{1}-D_{1}\geq 0$, where $D_{1}$
denotes the compression of $D$ onto ${\mathcal{K}}$. Now, apply the above
argument to $D$ starting with $\|A+D\|=2$, we get a $d\in{\mathcal{S}}$ such
that $\|A+C\|_{d}=\|A\|_{d}+\|C\|_{d}$. Suppose $d$ has $q$ nonzero entries.
If $d<r_{1}$, the compression of $D-B$ onto a $d$ dimensional subspace of
${\mathcal{K}}$ is positive semidefinite with norm one. There is a unit vector
$v\in{\mathcal{K}}$ such that ${\langle}Cv,v{\rangle}>{\langle}Bv,v{\rangle}$.
This contradicts our observation $B_{1}-C_{1}\geq 0$. So, $d\geq r_{1}$ and we
have $C_{1}-B_{1}\geq 0$. It follows that $B_{1}=C_{1}=(a_{1}/2)I_{r_{1}}$.

Note that $p=r_{1}$ cannot happen. Otherwise we will get

$\|A\|=\sum_{j=1}^{p}c_{j}a_{1}=1\quad\mbox{and}\quad\|B\|=\sum_{j=1}^{p}c_{j}(a_{1}/2)=1,$

which is absurd. ∎

Conditions (a) and (b) in the above lemma can indeed happen as the following
examples show.

######  Example 3.5.

Let ${\mathcal{H}}=\ell_{2}$ and ${{\mathcal{B}}({\mathcal{H}})}$ under two
different unitarily invariant norms defined below.

  1. (1)

Consider the norm on ${{\mathcal{B}}({\mathcal{H}})}$ defined by

$\|A\|=\max\left\\{5s_{1}(A)/2,s_{1}(A)+s_{2}(A)+s_{3}(A)\right\\}.$

Let $A=(2/5)I_{2}\oplus(1/5)I$. Then it is not hard to check that $A$ is an
extreme point of the unit ball ${\mathcal{B}}$. We have $A=B+D$ with
$B=(2/5){\rm diag}(1,0)\oplus(1/10)I$ and $D=(2/5){\rm
diag}(0,1)\oplus(1/10)I$. Moreover, $\mu B+\nu D=(2/5){\rm
diag}(\mu,\nu)\oplus[(\mu+\nu)/10]I$. A direct computation shows that

$\|\mu B+\nu D\|=\max\\{|\mu|,|\nu|\\}.$

We have $\|A+B\|_{\rm sp}=\|A+B\|=2$. So, $(r_{1},p)=(2,1)=2$ in this example.

  2. (2)

Consider ${{\mathcal{B}}({\mathcal{H}})}$ under the Ky Fan 2-norm
$\|A\|_{2}=s_{1}(A)+s_{2}(A)$. Then $A={\rm diag}(1,0,\ldots)$ is an extreme
point of ${\mathcal{B}}$ by [3, Theorem 3.2]. Then $A=B+D$ with $B={\rm
diag}(1/2,1/2,0,\ldots)$ and $D={\rm diag}(1/2,-1/2,0,\ldots)$. Moreover,

$\|\mu B+\nu D\|_{2}=(1/2)(|\mu+\nu|+|\mu-\nu|)=\max\\{|\mu|,|\nu|\\}$

for all $\mu,\nu\in\mathbb{R}$. We have $\|A+B\|_{2}=2$ and so
$(r_{1},p)=(1,2)$.∎

Suppose $U$ is a maximal partial isometry. Then $A=U/\|U\|$ is an extreme
point of ${\mathcal{B}}$. It can be written as $A=B+D$ for which (3.4) is
satisfied. However, the example above shows that it is not a characteristic of
this type of extreme points. Indeed, for every positive integer $N$, we can
write $A=A_{1}+\cdots+A_{N}$ such that

(3.5) $\big{\|}\mbox{$\sum_{j=1}^{N}a_{j}A_{j}$}\big{\|}=\max\\{|a_{j}|:1\leq
j\leq N\\}.$

For other extreme points of ${\mathcal{B}}$, the number of operators in such a
decomposition is limited by the multiplicity of its largest singular value.

######  Theorem 3.6.

Let $A$ be an extreme point of the norm ball ${\mathcal{B}}$. Then $A$ is a
scalar multiple of a maximal partial isometry if and only if
$A=A_{1}+\cdots+A_{N}$ for $N=n^{2}+1$ such that (3.5) is satisfied.

Proof. For the forward implication, assume that $\|I\|=f(1,\ldots,1)=1$. Let
$A$ be a maximal partial isometry. Then $\|A\|=1$. Take orthogonal projections
$P_{1},\dots,P_{N}$ each having rank at least $n$ such that
$P_{1}+\cdots+P_{N}=I$ and put $A_{j}=AP_{j}$ or $A_{j}=P_{j}A$ for
$j=1,\dots,N$, depending on $A^{*}A=I$ or $AA^{*}=I$. Then
$A=\sum_{j=1}^{N}A_{j}$. Moreover, for any $a_{1},\ldots,a_{N}\in\mathbb{R}$,
$\sum_{j=1}^{N}a_{j}A_{j}$ has the $n$ largest singular values equal to
$|a_{\ell}|,\dots,|a_{\ell}|$ if $|a_{\ell}|=\max\\{|a_{j}|:j=1,\dots,N\\}$.
Thus, $\|\sum_{j=1}^{N}a_{j}A_{j}\|=|a_{\ell}|f(1,\dots,1)=f(I)=|a_{\ell}|$.

For the converse, suppose $A$ is an extreme point of ${\mathcal{B}}$ but not a
scalar multiple of a maximal partial isometry. By Lemma 3.2,
$A=\sum_{j=1}^{n}s_{j}(A)x_{j}y_{j}^{*}+s_{n}(A)U$, for
$s_{j}(A),x_{j},y_{j}$’s and $U$ as in the lemma. Let $V_{1}$ be given by
$x_{j}\mapsto y_{j}$ for $j=1,\ldots,n$ and $V_{2}$ any unitary operator from
$\\{x_{1},\ldots,x_{n}\\}^{\perp}$ onto $\\{y_{1},\ldots,y_{n}\\}^{\perp}$.
Then $V=V_{1}\oplus V_{2}$ is unitary and $VA$ has the form in Lemma 3.4.

Suppose $A=A_{1}+\cdots+A_{N}$ satisfies the said property. As $V$ is unitary,
the singular values of $B$ and $VB$ are the same for any
$B\in{{\mathcal{B}}({\mathcal{H}})}$. So, $VA=VA_{1}+\cdots+VA_{N}$ also
satisfies the same property. For simplicity, let us assume that $A$ is already
in that nice form. Let $B_{1},\dots,B_{N}$ be the compression of $A_{j}$ onto
${\mathcal{K}}=\ker(A-a_{1}I)$. Note that by Lemma 3.2, $\dim{\mathcal{K}}<n$.
Now, $A=B+D$ for $B=A_{j}$ and $D=(A-A_{j})$, and (3.4) holds. We must have
either (a) or (b) of Lemma 3.4.

Case 1. Suppose for each $j$, there is a
$\varphi_{j}\in{{\mathcal{B}}({\mathcal{K}})}^{*}$ such that
$\varphi(B_{j})=1$ and that $\varphi(Y_{1})=0$ whenever
$Y\in{{\mathcal{B}}({\mathcal{H}})}$ satisfies $\|\mu A_{j}+\nu
Y\|=\max\\{|\mu|,|\nu|\\}$ for any $\mu,\nu\in{\mathbb{R}}$ and $Y_{1}$ is the
compression of $Y$ onto ${\mathcal{K}}$. The condition is clearly satisfied by
$Y=A_{k}$ for $k\neq j$. So, the bi-orthogonal condition
$\varphi_{j}(B_{k})=\delta_{jk}$ ensures $B_{1},\dots,B_{N}$ are linearly
independent. However, $\dim{\mathcal{K}}\leq n-1$. There cannot be so many
linearly independent matrices. This case is impossible.

Case 2. Suppose for some $A_{j}$, say $A_{1}$, there is a $c\in{\mathcal{S}}$
with nonzero entries $c_{1},\ldots,c_{p}$ for $p\geq r_{1}$ such that
$\|A+A_{1}\|_{c}=\|A+A_{1}\|=2$. By Lemma 3.4, the compression of $A_{1}$ to
${\mathcal{K}}$ is $(a_{1}/2)I_{r_{1}}$.

For $j>1$, we construct the $p\times p$ matrices $\hat{A}$, $\hat{A}_{1}$ and
$\hat{A}_{j}$ as in the proof of Lemma 3.4. Then for $C={\rm
diag}(c_{1},\ldots,c_{p})$,

$|1\pm{\rm tr}\,C\hat{A}_{j}|=|{\rm
tr}\,C(\hat{A}_{1}\pm\hat{A}_{j})|\leq\|\hat{A}_{1}\pm\hat{A}_{j}\|_{c}\leq\|A_{1}\pm
A_{j}\|=1.$

Thus ${\rm tr}\,(C\hat{A}_{j})=0$. It follows that

$2={\rm
tr}\,C[A+(\hat{A}_{1}+\hat{A}_{j})]\leq\|\hat{A}+(\hat{A}_{1}+\hat{A}_{j})\|_{c}\leq\|A+(A_{1}+A_{j})\|_{c}\leq\|A+(A_{1}+A_{j})\|=2.$

We have $\|A+(A_{1}+A_{j})\|_{c}=\|A+(A_{1}+A_{j})\|=2$ for $c$ with $p\geq
r_{1}$ nonzero entries. So, Lemma 3.4 applied to $B=A_{1}+A_{j}$ and $D=A-B$
gives us $B_{1}+B_{j}=(a_{1}/2)I_{r_{1}}$. (Here, note that
$\dim{\mathcal{K}}<n$ ensures $n>1$ so that $n^{2}+1\geq 5$ and $A-B$ is
nontrivial.) But we already have $B_{1}=(a_{1}/2)I_{r_{1}}$. So,
$B_{j}=0_{r_{1}}$ for all $j>1$. Then the compression of $A$ onto
${\mathcal{K}}$ is $B_{1}+\cdots+B_{N}=(a_{1}/2)I_{r_{1}}$, which contradicts
the form of $A$. This case is also impossible. ∎

Proof of Theorem 3.1. The sufficiency is clear. To prove the necessity,
suppose $L:{{\mathcal{B}}({\mathcal{H}})}\to{{\mathcal{B}}({\mathcal{H}})}$
satisfies the hypothesis of the theorem. Observe that by the Mazur-Ulam
Theorem, the map $\hat{L}:A\mapsto L(A)-L(0)$ is real linear and satisfies
$\|\hat{L}(A)\|=\|A\|$ for all $A\in{{\mathcal{B}}({\mathcal{H}})}$. So,
$\hat{L}$ preserves the extreme points of ${\mathcal{B}}$. By Theorem 3.6,
$\hat{L}$ sends maximal isometries to maximal isometries. It follows from [9,
Solution to Problem 107] (see also [20, Lemma 3]) that $\hat{L}$ is indeed an
isometry of the spectral norm. Using the characterization [4, Corollary 3.3],
we conclude that $L$ is of the asserted form. ∎

## Acknowledgement

Li is an affiliate member of the Institute for Quantum Computing, University
of Waterloo; his research was partially supported by the Simons Foundation
Grant 851334. The authors would like to thank Professor Ngai-Ching Wong for
some inspiring discussions and comments, and also the anonymous referee for
the helpful comments.

## References

  * [1] J. Arazy, The isometries of $C_{p}$, Israel J. Math. 22 (1975), 247-256.
  * [2] J.T. Chan, C.K. Li and N.S. Sze, Isometries for unitarily invariant norms, Linear Algebra Appl. 399 (2005), 53-70.
  * [3] J.T. Chan, C.K. Li and N.C. Tu, A Class of Unitarily Invariant Norms on $B(H)$, Proc. Amer. Math. Soc. 129 (2001), 1065-1076.
  * [4] T. Dang, Real Isometries between $JB^{*}$-Triples, Proc. Amer. Math. Soc. 114 (1992), 971-980.
  * [5] W. Ding, C.K. Li and Y. Li, A note on unitarily invariant matrix norms, Linear Algebra Appl. 607 (2020), 341-346.
  * [6] L. Fialkow and R. Loebl, Elementary mappings into ideals of operators, Illinois J. Math. 28 (1984), 555-578.
  * [7] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, Amer. Math. Soc., Providence, R.I., 1969.
  * [8] R. Grone and M. Marcus, Isometries of matrix algebras, J. Algebra 47 (1977), 180-189.
  * [9] P.R. Halmos, A Hilbert Space Problem Book, $2^{\rm nd}$ ed., Graduate Texts in Mathematics, 19, Springer-Verlag, New-York, 1982.
  * [10] C.R. Johnson and C.K. Li, Inequalities relating unitarily invariant norms and the numerical radius, Linear and Multilinear Algebra 23 (1988), 183-191.
  * [11] R.V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325-338.
  * [12] C.K. Li, Matrices with some extremal properties, Linear Algebra Appl. 101 (1988), 255-267.
  * [13] C.K. Li, Some aspects of the theory of norms, Linear Algebra Appl. 219 (1994), 93-110.
  * [14] C.K. Li and N.K. Tsing, On unitarily invariant norms and related results, Linear and Multilinear Algebra 20 (1987), 107-119.
  * [15] C.K. Li and N.K. Tsing, Linear operators preserving unitarily invariant norms of matrices, Linear and Multilinear Algebra 26 (1990), 119-132.
  * [16] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, Orlando, 1979.
  * [17] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford Ser. (2) 11 (1960), 50-59.
  * [18] J. von Neumann, Some matrix-inequalities and metrization of matrix- space, Tomsk Univ. Rev. 1 (1937), 286-300.
  * [19] A. Pietsch, Eigenvalues and $s$-numbers, Cambridge Studies in Advanced Mathematics 13, Cambridge University Press, Cambridge, 1985.
  * [20] M. Rais, The Unitary group preserving maps (the infinite dimensional case), Linear and Multilinear Algebra 20 (1987), 337-345.
  * [21] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, Berlin, 1960.
  * [22] A.R. Sourour, Isometries of norm ideals of compact operators, J. Funct. Anal. 43 (1981), 69-77.


V^{\natural}[1/p,e(1/2p)]\otimes_{\mathbb{Z}[1/p,e(1/2p)]}\mathbb{Z}[1/pq,e(1/2pq)]\cong
V^{\natural}[1/pq,e(1/2pq)],$ $\displaystyle
V^{\natural}[1/p,e(1/2p)]\otimes_{\mathbb{Z}[1/p,e(1/2p)]}\mathbb{Z}[1/pr,e(1/2pr)]\cong
V^{\natural}[1/pr,e(1/2pr)],$ $\displaystyle
V^{\natural}[1/p,e(1/2p)]\otimes_{\mathbb{Z}[1/p,e(1/2p)]}\mathbb{Z}[1/p\ell,e(1/2p\ell)]\cong
V^{\natural}[1/p\ell,e(1/2p\ell)].$

Furthermore, this form has monster symmetry.

###### Proof.

We apply Lemma 3.19 for the triples $(p,q,r)$, $(p,q,\ell)$ and $(p,r,\ell)$
to obtain unique self-dual forms over $\mathbb{Z}[1/pqr,e(1/2p)]$,
$\mathbb{Z}[1/pq\ell,e(1/2p)]$, and $\mathbb{Z}[1/pr\ell,e(1/2p)]$ with
monster symmetry, and they are all isomorphic to each other when base-changed
to $\mathbb{Z}[1/pqr\ell,e(1/2p)]$. For each pair of these forms, we apply
Zariski descent (following Lemma 2.30) to obtain unique self-dual forms
$V^{\natural}[1/pq,e(1/2p)]$ over $\mathbb{Z}[1/pq,e(1/2p)]$,
$V^{\natural}[1/pr,e(1/2p)]$ over $\mathbb{Z}[1/pr,e(1/2p)]$, and
$V^{\natural}[1/p\ell,e(1/2p)]$ over $\mathbb{Z}[1/p\ell,e(1/2p)]$, satisfying

$\displaystyle
V^{\natural}[1/pq,e(1/2p)]\otimes_{\mathbb{Z}[1/pq,e(1/2p)]}\mathbb{Z}[1/pq,e(1/2pq)]\cong
V^{\natural}[1/pq,e(1/2pq)],$ $\displaystyle
V^{\natural}[1/pr,e(1/2p)]\otimes_{\mathbb{Z}[1/pr,e(1/2p)]}\mathbb{Z}[1/pr,e(1/2pr)]\cong
V^{\natural}[1/pr,e(1/2pr)],$ $\displaystyle
V^{\natural}[1/p\ell,e(1/2p)]\otimes_{\mathbb{Z}[1/p\ell,e(1/2p)]}\mathbb{Z}[1/p\ell,e(1/2p\ell)]\cong
V^{\natural}[1/p\ell,e(1/2p\ell)].$

Applying Zariski descent to any pair of these forms yields a self-dual
$\mathbb{Z}[1/p,e(1/2p)]$-form with monster symmetry, and the uniqueness claim
in Lemma 3.18 implies any pair yields an isomorphic object. ∎

######  Lemma 3.21.

Let $p$, $q$, $r$ be distinct elements of $P_{0}$, such that $pq$, $qr$, and
$pr$ are not in $\\{65,91\\}$. Then, there exists a unique self-dual
$\mathbb{Z}[1/pqr]$-form $V^{\natural}[1/pqr]$ of $V^{\natural}$ such that

$\displaystyle V^{\natural}[1/pqr]\otimes\mathbb{Z}[e(1/2pq)]\cong
V^{\natural}[1/pq,e(1/2pq)]\otimes_{\mathbb{Z}[1/pq,e(1/2pq)]}\mathbb{Z}[1/pqr,e(1/2pq)],$
$\displaystyle V^{\natural}[1/pqr]\otimes\mathbb{Z}[e(1/2pr)]\cong
V^{\natural}[1/pr,e(1/2pr)]\otimes_{\mathbb{Z}[1/pr,e(1/2pr)]}\mathbb{Z}[1/pqr,e(1/2pr)],$
$\displaystyle V^{\natural}[1/pqr]\otimes\mathbb{Z}[e(1/2qr)]\cong
V^{\natural}[1/qr,e(1/2qr)]\otimes_{\mathbb{Z}[1/qr,e(1/2qr)]}\mathbb{Z}[1/pqr,e(1/2qr)].$

Furthermore, $V^{\natural}[1/pqr]$ has monster symmetry.

###### Proof.

We apply Lemma 3.19 for the triples $(p,q,r)$, $(q,r,p)$, and $(r,p,q)$ to
obtain unique self-dual forms over $\mathbb{Z}[1/pqr,e(1/2p)]$,
$\mathbb{Z}[1/pqr,e(1/2q)]$, and $\mathbb{Z}[1/pqr,e(1/2r)]$. For any pair of
these forms, applying faithfully flat gluing as in Lemma 2.31 yields a self-
dual $\mathbb{Z}[1/pqr]$-form satisfying the expected conditions, and
uniqueness and monster symmetry follow from Lemma 3.18. ∎

We now come to the main theorem.

######  Theorem 3.22.

There exists a unique self-dual $\mathbb{Z}$-form $V^{\natural}_{\mathbb{Z}}$
of the vertex operator algebra $V^{\natural}$ such that for any distinct
$p,q\in P_{0}$ satisfying $pq\not\in\\{65,91\\}$, we have

$\displaystyle
V^{\natural}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}[1/pq,e(1/2pq)]\cong
V^{\natural}[1/pq,e(1/2pq)].$

Furthermore, this integral form has monster symmetry.

###### Proof.

If we let $(p,q,r)$ range over triples in $\\{2,3,5,7\\}$, Lemma 3.21 yields
$\mathbb{Z}[1/pqr]$-forms $V^{\natural}[1/pqr]$. Since the four rings
$\mathbb{Z}[1/pqr]$ form a Zariski cover of $\operatorname{Spec}\mathbb{Z}$,
we obtain a self-dual $\mathbb{Z}$-form $V^{\natural}_{\mathbb{Z}}$ by
pairwise gluing, and Lemma 3.18 implies monster symmetry and uniqueness with
respect to isomorphisms

$\displaystyle
V^{\natural}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}[1/pq,e(1/2pq)]\cong
V^{\natural}[1/pq,e(1/2pq)]$

as $p$, $q$ range over distinct elements of $\\{2,3,5,7\\}$.

We now consider the remaining cases, both of which involve the prime $13$. Let
$p\in\\{2,3\\}$. Recall from Lemma 3.20 that for any distinct $q$, $r$, $\ell$
in $P_{0}\setminus\\{p\\}$, including the case $q=13$, we have a
$\mathbb{Z}[1/p,e(1/2p)]$-form $V^{\natural}[1/p,e(1/2p)]$ satisfying

$\displaystyle
V^{\natural}[1/p,e(1/2p)]\otimes_{\mathbb{Z}[1/p,e(1/2p)]}\mathbb{Z}[1/pq,e(1/2pq)]\cong
V^{\natural}[1/pq,e(1/2pq)].$

We claim that this form is independent of the choice of the primes $q$, $r$,
$\ell$. Indeed, applying Lemma 3.19 to all possible triples $(p,q,r)$ yields a
collection of forms over the various rings $\mathbb{Z}[1/pqr,e(1/2p)]$. By the
tensor product compatibilities, these forms are isomorphic on Zariski
intersections, so all of the
$\operatorname{Spec}\mathbb{Z}[1/p,e(1/2p)]$-forms obtained by gluing are
isomorphic by uniqueness.

Thus, it suffices to show that
$V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/p,e(1/2p)]\cong
V^{\natural}[1/p,e(1/2p)]$. We have established that the isomorphism type of
$V^{\natural}[1/p,e(1/2p)]$ is independent of the choice of $q$, $r$, $\ell$,
so we may choose $q,r,\ell\in\\{2,3,5,7\\}\setminus\\{p\\}$, and then the
isomorphism follows from the uniqueness of our construction of
$V^{\natural}_{\mathbb{Z}}$ in the first paragraph. ∎

######  Corollary 3.23.

The inner product on $V^{\natural}_{\mathbb{Z}}$ is positive definite.

###### Proof.

The $\mathbb{Z}[1/2]$-form of $V^{\natural}$ that was constructed in [6] is
positive definite, because base-change to $\mathbb{Q}$ yields the positive-
definite $\mathbb{Q}$-form constructed in [27].

It therefore suffices to show that
$V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/2]$ is isomorphic to the
$\mathbb{Z}[1/2]$-form of $V^{\natural}$ constructed in [6]. This follows from
the same method as Proposition 3.14: we decompose both forms under the action
of a 2B-pure 4-group $H_{2}$, and obtain identifications with the 2a-fixed
point vertex operator subalgebra of $(V_{\Lambda})_{\mathbb{Z}[1/2]}$. In
fact, we don’t need the full isomorphism to show positive-definiteness, since
it suffices to show that the corresponding $H_{2}$-eigenspaces are isomorphic,
hence positive-definite. ∎

R. Griess has informed me that the following result is new, even without the
positive definite condition.

######  Corollary 3.24.

There exists a $196884$-dimensional positive-definite unimodular lattice with
a faithful monster action by orthogonal transformations.

###### Proof.

The weight 2 subspace of $V^{\natural}_{\mathbb{Z}}$ satisfies the required
properties. ∎

######  Corollary 3.25 (newer modular moonshine).

The stronger version of the modular moonshine conjecture, as stated in [6] and
[4], is unconditionally true. That is, there exists a self-dual integral form
$V$ of $V^{\natural}$ with $\mathbb{M}$ symmetry, such that for each element
$g$ of prime order $p$, the graded Brauer character of any $p$-regular element
$h\in C_{\mathbb{M}}(g)$ on the Tate cohomology $\hat{H}^{*}(g,V)$ is given by

$\displaystyle\operatorname{Tr}(h|\hat{H}^{0}(g,V))=\begin{cases}T_{gh}(\tau),&g\in
p\text{A},3\text{C},\vspace{1mm}\\\
\dfrac{T_{gh}(\tau)+T_{gh}(\tau+1/2)}{2},&g\in 2\text{B},\vspace{1mm}\\\
\dfrac{T_{gh}(\tau)+T_{gh\sigma}(\tau)}{2},&g\in
p\text{B},2|(p-1),\end{cases}$
$\displaystyle\operatorname{Tr}(h|\hat{H}^{1}(g,V))=\begin{cases}0,&g\in
p\text{A},3\text{C},\vspace{1mm}\\\
\dfrac{T_{gh}(\tau)-T_{gh}(\tau+1/2)}{2},&g\in 2\text{B},\vspace{1mm}\\\
\dfrac{T_{gh}(\tau)-T_{gh\sigma}(\tau)}{2},&g\in
p\text{B},2|(p-1),\end{cases}$

where the element $\sigma$ is the unique involution in
$C_{\mathbb{M}}(g)/O_{p}(C_{\mathbb{M}}(g))$ that acts as $1$ on
$\hat{H}^{0}(g,V)$ and $-1$ on $\hat{H}^{1}(g,V)$, when $p\in\\{3,5,7,13\\}$.

###### Proof.

The conditional proof given in [6] and [4] only requires the assumptions that
$V^{\natural}_{\mathbb{Z}}$ exist (with a small technical condition at $p=2$),
and that a statement about $\mathbb{Z}_{p}$-forms of the monster Lie algebra
hold. The existence of $V^{\natural}_{\mathbb{Z}}$ is given in Theorem 3.22,
and the assumption about $\mathbb{Z}_{p}$-forms was shown to hold in [5]. The
technical condition at $p=2$ appears in Section 5 of [6], where Assumption 5.1
asserts that $V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/3]$ decomposes into
graded $2.M_{12}.2$-modules that are isomorphic to submodules of
$(V_{\Lambda})_{\mathbb{Z}[1/3]}$. We will not prove this assumption, but we
note that it is only used to transfer the vanishing properties of Tate
cohomology from the form of $V_{\Lambda}$ to the form of $V^{\natural}$. The
proofs of vanishing only use the existence of involutions in classes 2A and 2B
instead of a $2.M_{12}.2$ action. The vanishing of Tate cohomology is
preserved and reflected by faithfully flat base change and also insensitive to
inverting irrelevant primes, so we can check it after adding roots of unity
and inverting odd primes greater than 3. For any distinct
$q\in\\{2,5,7,13\\}$, Lemma 3.6 gives an isomorphism from the 3B-fixed vectors
in $V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/3q,e(1/6q)]$ to a submodule
of $(V_{\Lambda})_{\mathbb{Z}[1/3q,e(1/6q)]}$, and the order 9 3B-pure group
$H_{3}$ given in Lemma 3.13 gives a decomposition of
$V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/3q,e(1/6q)]$ into pieces that
embed equivariantly (with respect to $C_{\mathbb{M}}(3B)\cap
N_{\mathbb{M}}(H_{3})$) into the 3B-fixed vectors. When $q$ is odd, this is
sufficient to transfer the Tate cohomology vanishing properties that we need
from $V_{\Lambda}$. ∎

We showed in Lemma 3.6 that some prime order orbifold constructions of
$V^{\natural}$ can be defined over rather small rings. In particular, if
$p,q\in\\{2,3,5,7,13\\}$ and $pq\not\in\\{65,91\\}$, then we have isomorphisms

$\displaystyle(V_{\Lambda})_{\mathbb{Z}[1/pq,e(1/pq)]}^{\sigma}\overset{\sim}{\to}(V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/pq,e(1/pq)])^{g}$

for $\sigma$ an order $p$ lift of a fixed-point free automorphism
$\bar{\sigma}$ of $\Lambda$, and $g\in\mathbb{M}$ in class $pB$. Here, we show
that these isomorphisms can be defined over $\mathbb{Z}[1/p,e(1/p)]$. For
$p=3$, this refinement was conjectured in [6], in the hope that such a
construction could be used to construct a self-dual integral form. Thus, we
are approaching this question in a somewhat backward way.

######  Corollary 3.26.

Let $p\in\\{2,3,5,7\\}$. Then there exists:

  1. $1)$

an order $p$ automorphism $\sigma$ of $(V_{\Lambda})_{\mathbb{Z}[1/p,e(1/p)]}$
lifting a fixed-point free automorphism $\bar{\sigma}$ of $\Lambda$ of order
$p$,

  2. $2)$

an integral form $V$ of $V^{\natural}$ with invariant bilinear form and
monster symmetry, admitting an order $p$ automorphism $g$ in class $pB$,

  3. $3)$

an isomorphism
$(V_{\Lambda})_{\mathbb{Z}[1/p,e(1/p)]}^{\sigma}\overset{\sim}{\to}(V\otimes\mathbb{Z}[1/p,e(1/p)])^{g}$
of $\mathbb{Z}[1/p,e(1/p)]$-vertex algebras preserving inner products.

###### Proof.

By Lemma 3.6, we have isomorphisms

$\displaystyle(V_{\Lambda})_{\mathbb{Z}[1/pq,e(1/2pq)]}^{\sigma}\overset{\sim}{\to}(V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/pq,e(1/2pq)])^{g}$

for all $q\in\\{2,3,5,7\\}\setminus\\{p\\}$. We apply a descent argument
following the same lines as in Theorem 3.22: Faithfully flat gluing for

$\displaystyle\mathbb{Z}[1/pqr,e(1/2pq)]\to\mathbb{Z}[1/pqr,e(1/2pqr)]\leftarrow\mathbb{Z}[1/pqr,e(1/2pr)]$

yields isomorphic $\mathbb{Z}[1/pqr,e(1/2p)]$-forms, as $q$ and $r$ vary over
the set $\\{2,3,5,7\\}\setminus\\{p\\}$. Zariski gluing then yields isomorphic
$\mathbb{Z}[1/p,e(1/2p)]$-forms. When $p$ is odd, we have
$\mathbb{Z}[1/p,e(1/2p)]=\mathbb{Z}[1/p,e(1/p)]$, and when $p=2$, the
isomorphism of $\mathbb{Z}[1/2]$-forms was established in [6]. ∎

It seems reasonable to expect that one can improve this result by removing the
roots of unity.

######  Conjecture 3.27.

For any $p\in\\{2,3,5,7,13\\}$, there is an isomorphism

$\displaystyle(V_{\Lambda})_{\mathbb{Z}[1/p]}^{\sigma}\overset{\sim}{\to}\big{(}V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/p]\big{)}^{g}$

of vertex operator algebras over $\mathbb{Z}[1/p]$, where $\sigma$ is an order
$p$ lift of a fixed-point free automorphism of $\Lambda$, and $g$ is an
element of the monster in conjugacy class $p$B.

Section 5 of [6] describes how to produce a $\mathbb{Z}[1/3]$-form of
$V^{\natural}$ using a $\mathbb{Z}[1/3,e(1/3)]$-form, by transporting a Galois
automorphism through an ${\rm SL}_{2}(\mathbb{F}_{3})$-action on a 3B-pure
elementary subgroup. This method works for $p=5$ and $p=7$, as well, but I do
not know how to show that the Galois actions on
$\big{(}V^{\natural}_{\mathbb{Z}}\otimes\mathbb{Z}[1/p,e(1/p)]\big{)}^{g}$ and
$(V_{\Lambda})_{\mathbb{Z}[1/p,e(1/p)]}^{\sigma}$ have matching fixed-point
submodules. For $p=13$, one may need to do more explicit work.

### Acknowledgements

I would like to thank Toshiyuki Abe for describing the constructions in [1] in
detail at the “VOA and related topics” workshop at Osaka University in March
2017. I would also like to thank the anonymous referees for many helpful
comments, and one referee in particular for their help with the proof of Lemma
2.13. This research was partly funded by JSPS Kakenhi Grant-in-Aid for Young
Scientists (B) 17K14152.

## References

  * [1] Abe T., Lam C.H., Yamada H., On ${\mathbb{Z}}_{p}$-orbifold constructions of the moonshine vertex operator algebra, Math. Z. 290 (2018), 683–697, arXiv:1705.09022.
  * [2] Borcherds R.E., Vertex algebras, Kac–Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068–3071.
  * [3] Borcherds R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.
  * [4] Borcherds R.E., Modular moonshine. III, Duke Math. J. 93 (1998), 129–154, arXiv:math.QA/9801101.
  * [5] Borcherds R.E., The fake Monster formal group, Duke Math. J. 100 (1999), 139–165, arXiv:math.QA/9805123.
  * [6] Borcherds R.E., Ryba A.J.E., Modular moonshine. II, Duke Math. J. 83 (1996), 435–459.
  * [7] GAP – Groups, Algorithms, and Programming, Version 4.5.5, 2012, http://www.gap-system.org.
  * [8] Carnahan S., Generalized moonshine IV: Monstrous Lie algebras, arXiv:1208.6254.
  * [9] Carnahan S., 51 constructions of the Moonshine module, Commun. Number Theory Phys. 12 (2018), 305–334, arXiv:1707.02954.
  * [10] Chen H.-Y., Lam C.H., Shimakura H., ${\mathbb{Z}}_{3}$-orbifold construction of the Moonshine vertex operator algebra and some maximal 3-local subgroups of the Monster, Math. Z. 288 (2018), 75–100, arXiv:1606.05961.
  * [11] Conway J.H., A simple construction for the Fischer–Griess monster group, Invent. Math. 79 (1985), 513–540.
  * [12] Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, Oxford University Press, Eynsham, 1985.
  * [13] Conway J.H., Norton S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308–339.
  * [14] Curtis C.W., Reiner I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. 11, Interscience Publishers, New York – London, 1962.
  * [15] Demazure M., Grothendieck A., Schémas en groupes. I–III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Math., Vol. 151–153, Springer-Verlag, Berlin – New York, 1970.
  * [16] Diederichsen F.-E., Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz, Abh. Math. Sem. Hansischen Univ. 13 (1940), 357–412.
  * [17] Dong C., Griess Jr. R.L., Automorphism groups and derivation algebras of finitely generated vertex operator algebras, Michigan Math. J. 50 (2002), 227–239, arXiv:math.QA/0106051.
  * [18] Dong C., Griess Jr. R.L., Integral forms in vertex operator algebras which are invariant under finite groups, J. Algebra 365 (2012), 184–198, arXiv:1201.3411.
  * [19] Dong C., Lepowsky J., Generalized vertex algebras and relative vertex operators, Progress in Mathematics, Vol. 112, Birkhäuser Boston, Inc., Boston, MA, 1993.
  * [20] Dong C., Li H., Mason G., Vertex operator algebras and associative algebras, J. Algebra 206 (1998), 67–96, arXiv:q-alg/9612010.
  * [21] Dong C., Li H., Mason G., Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), 1–56, arXiv:q-alg/9703016.
  * [22] Dong C., Mason G., The construction of the moonshine module as a $Z_{p}$-orbifold, in Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, MA, 1992), Contemp. Math., Vol. 175, Amer. Math. Soc., Providence, RI, 1994, 37–52.
  * [23] Dong C., Mason G., On quantum Galois theory, Duke Math. J. 86 (1997), 305–321, arXiv:hep-th/9412037.
  * [24] Dong C., Nagatomo K., Automorphism groups and twisted modules for lattice vertex operator algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 117–133, arXiv:math.QA/9808088.
  * [25] Dong C., Ren L., Vertex operator algebras associated to the Virasoro algebra over an arbitrary field, Trans. Amer. Math. Soc. 368 (2016), 5177–5196.
  * [26] Frenkel I.B., Lepowsky J., Meurman A., A moonshine module for the Monster, in Vertex Operators in Mathematics and Physics (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., Vol. 3, Springer, New York, 1985, 231–273.
  * [27] Frenkel I.B., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
  * [28] Gao Y., Li H., Generalized vertex algebras generated by parafermion-like vertex operators, J. Algebra 240 (2001), 771–807, arXiv:math.QA/0006104.
  * [29] Griess Jr. R.L., The friendly giant, Invent. Math. 69 (1982), 1–102.
  * [30] Grothendieck A., Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, in Séminaire Bourbaki, Vol. 5, Soc. Math. France, Paris, 1960, Exp. No. 190, 299–327.
  * [31] Grothendieck A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5–223.
  * [32] Lam C.H., Shimakura H., On orbifold constructions associated with the Leech lattice vertex operator algebra, Math. Proc. Cambridge Phil. Soc., to appear, arXiv:1705.01281.
  * [33] Li H., The regular representation, Zhu’s $A(V)$-theory, and induced modules, J. Algebra 238 (2001), 159–193, arXiv:math.QA/9909007.
  * [34] Li W., Abelian intertwining algebras and modules related to rational lattices, J. Algebra 214 (1999), 356–384.
  * [35] Mac Lane S., Cohomology theory of abelian groups, in Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, Vol. 2, Amer. Math. Soc., Providence, R. I., 1952, 8–14.
  * [36] Mason G., Vertex rings and their Pierce bundles, in Vertex Algebras and Geometry, Contemp. Math., Vol. 711, Amer. Math. Soc., Providence, RI, 2018, 45–104, arXiv:1708.00328.
  * [37] Matsumura H., Commutative algebra, 2nd ed., Mathematics Lecture Note Series, Vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980\.
  * [38] McRae R., On integral forms for vertex algebras associated with affine Lie algebras and lattices, J. Pure Appl. Algebra 219 (2015), 1236–1257, arXiv:1401.2505.
  * [39] Mesablishvili B., Pure morphisms of commutative rings are effective descent morphisms for modules – a new proof, Theory Appl. Categ. 7 (2000), no. 3, 38–42.
  * [40] Montague P.S., Third and higher order NFPA twisted constructions of conformal field theories from lattices, Nuclear Phys. B 441 (1995), 337–382, arXiv:hep-th/9502138.
  * [41] Olivier J.-P., Descente par morphismes purs, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A821–A823.
  * [42] Ryba A.J.E., Modular Moonshine?, in Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Contemp. Math., Vol. 193, Amer. Math. Soc., Providence, RI, 1996, 307–336.
  * [43] Stacks Project, https://stacks.math.columbia.edu.
  * [44] Thompson J.G., Some numerology between the Fischer–Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), 352–353.
  * [45] van Ekeren J., Möller S., Scheithauer N.R., Construction and classification of vertex operator algebras, J. Reine Angew. Math., to appear, arXiv:1507.08142.
  * [46] Vistoli A., Notes on Grothendieck topologies, fibered categories, and descent theory, Mathematical Surveys and Monographs, Vol. 123, Amer. Math. Soc., Providence, RI, 2005, arXiv:math.AG/0412512.
  * [47] Wilson R.A., The odd-local subgroups of the Monster, J. Austral. Math. Soc. Ser. A 44 (1988), 1–16.
  * [48] Wilson R.A., Maximal subgroups of sporadic groups, in Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemp. Math., Vol. 694, Amer. Math. Soc., Providence, RI, 2017, 57–72, arXiv:1701.02095.