GENERAL MAZUR-ULAM TYPE THEOREMS AND SOME APPLICATIONS
LAJOS MOLNÁR
Dedicated to Professor Charles J.K. Batty on the occasion of his 60th birtday.
ABSTRACT. Recently we have presented several structural results on certain
isometries of spaces of positive definite matrices and on those of unitary
groups. The aim of this paper is to put those previous results into a common
perspective and extend them to the context of operator algebras, namely, to
that of von Neumann factors.
1. INTRODUCTION AND STATEMENT OF THE RESULTS
The famous Mazur-Ulam theorem states that every surjective isometry (i.e.,
surjective distance preserving map) from a normed real linear space onto an-
other one is automatically affine, in other words, it is necessarily an isomor-
phism with respect to the operation of convex combinations.
Recently we have extensively investigated how this fundamental theorem can
be generalized to more general settings. In [12] we have obtained results in
the context of groups (and some of their substructures) which state that un-
der certain conditions surjective distance preserving transformations between
such structures necessarily preserve locally the operation of the so-called in-
verted Jordan product. This means that also in that general setting the surjec-
tive isometries necessarily have a particular algebraic property. This property
opens the way for employing algebraic ideas, techniques and computations to
get more information about the isometries under considerations. In some of our
latter papers we have successfully used that approach to describe explicitly the
isometries of different non-linear structures of matrices and operators.
In [13] we have determined the surjective isometries of the unitary group over
a Hilbert space equipped with the metric of the operator norm. In [21] we de-
scribed the surjective isometries of the space of all positive definite operators on
a Hilbert space relative to the so-called Thompson part metric. In [14] we have
presented generalizations of the latter two results for the setting of C
∗
-algebras.
It has turned out that the corresponding surjective isometries are closely related
to Jordan *-isomorphisms between the underlying full algebras. In [25], [22] we
2010 Mathematics Subject Classification. Primary: 47B49. Secondary: 46L40.
Key words and phrases. Mazur-Ulam type theorems, generalized distance measures, positive
definite cone, unitary group, Jordan triple map, inverted Jordan triple map, operator algebras.
The author was supported by the "Lendület" Program (LP2012-46/2012) of the Hungarian
Academy of Sciences and by the Hungarian Scientific Research Fund (OTKA) Reg.No. K81166
NK81402.
1
2 LAJOS MOLNÁR
have proceeded further and described the structure of surjective isometries of
the unitary group with respect to complete symmetric norms (see the defini-
tion later) both in the infinite and in the finite dimensional cases. Furthermore,
in [22] we have also determined the isometries relative to the elements of a re-
cently introduced collection of metrics [8] (having connections to quantum in-
formation science) on the group of unitary matrices. In [23] we have revealed the
structure of surjective isometries of the space of positive definite matrices rela-
tive to certain metrics of differential geometric origin (they are common gener-
alizations of the Thompson part metric and the natural Riemannian metric on
positive definite matrices) as well as to a new metric obtained from the Jensen-
Shannon symmetrization of the important divergence called Stein’s loss. In [26]
we have made an important step toward further generality. Namely, we have de-
scribed the structure of those surjective maps on the space of all positive definite
matrices which leave invariant a given element of a large collection of certain
so-called generalized distance measures. In that way we could present a com-
mon generalization of the mentioned results in [23] and also provide structural
information on a large class of transformations preserving other particular im-
portant distance measures including Stein’s loss itself. We also mention that by
the help of appropriate modifications in our general results in [12] we have man-
aged to determine the surjective isometries of Grassmann spaces of projections
of a fixed rank on a Hilbert space relative to the gap metric [5].
In this paper we develop even further the ideas and approaches we have
worked out and used in the papers [22], [23], [26], and extend our previous re-
sults concerning distance measure preserving maps on matrix algebras for the
case of operator algebras, especially, von Neumann factors. We obtain results
which show that if the positive definite cones or the unitary groups in those al-
gebras equipped with a sort of very general distance measures are "isometric",
then the underlying full algebras are Jordan *-isomorphic (either *-isomorphic
or *-antiisomorphic).
We begin the presentation with the case of positive definite cones. As the
starting point of the route leading to our corresponding result we exhibit a
Mazur-Ulam type theorem for a certain very general structure called point-
reflection geometry equipped with a generalized distance measure. In fact, we
believe that with this result we have found in some sense the most general ver-
sion of Mazur-Ulam theorem that one can obtain using the approach followed
in [12] in the setting of groups. We point out that many of the arguments below
use ideas that have already appeared in our previous papers [22], [23], [26]. In
several cases only small changes need to be performed while in other cases we
really have to work to find solutions for particular problems that emerge from
the fact that instead of matrix algebras here we consider much more compli-
cated objects, namely operator algebras. In order to make the material readable
we present the results with complete proofs.
For our new general Mazur-Ulam type result we need the following concept
that has been defined by Manara and Marchi in [19] (also see [16], [18]).
3
Definition 1. Let X be a set equipped with a binary operation ¦ which satisfies
the following conditions:
(a1) a ¦a = a holds for every a ∈ X ;
(a2) a ¦(a ¦b) =b holds for any a,b ∈ X ;
(a3) the equation x ¦a =b has a unique solution x ∈ X for any given a,b ∈ X .
In this case the pair (X ,¦) (or X itself) is called a point-reflection geometry.
Observe that from the properties (a1)-(a2) above we easily obtain that the
equation a ¦x =b also has unique solution x ∈ X for any given a,b ∈ X .
As for our present purposes, the most important example of such a structure
is given as follows. In the rest of the paper by a C
∗
-algebra we always mean
a unital C
∗
-algebra with unit I . Let A be such an algebra. We denote by A
s
the self-adjoint part of A and A
+
stands for the cone of all positive elements of
A (self-adjoint elements with non-negative spectrum). The set of all invertible
elements in A
+
is denoted by A
−1
+
. Sometimes A
−1
+
is called positive definite
cone and its elements are said positive definite. For any A,B ∈ A
−1
+
define A ¦
B = AB
−1
A. In that way A
−1
+
becomes a point-reflection geometry. Indeed, the
conditions (a1), (a2) above are trivial to check. Concerning (a3) we recall that for
any given A,B ∈ A
−1
+
, the so-called Ricatti equation X A
−1
X = B has a unique
solution X = A#B which is just the geometric mean of A and B defined by
A#B = A
1/2
(A
−1/2
B A
−1/2
)
1/2
A
1/2
.
This assertion is usually termed as Anderson-Trapp theorem (for the original
source see [1]).
In our general Mazur-Ulam type theorem that we are going to present we do
not need to confine the considerations to true metrics, the theorem works also
for so-called generalized distance measures.
Definition 2. Given an arbitrary set X , the function d : X × X → [0,∞[ is called
a generalized distance measure if it has the property that for an arbitrary pair
x, y ∈ X we have d(x, y) =0 if and only if x = y.
Hence, in the definition above we require only the definiteness property of
a metric but neither the symmetry nor the triangle inequality is assumed. Our
new general Mazur-Ulam type theorem reads as follows.
Theorem 3. Let X ,Y be sets equipped with binary operations ¦,?, respectively,
with which they form point-reflection geometries. Let d : X ×X → [0,∞[, ρ : Y ×
Y →[0,∞[ be generalized distance measures. Pick a,b ∈ X , set
L
a,b
={x ∈ X : d(a,x) =d(x,b ¦a) =d(a,b)}
and assume the following:
(b1) d(b ¦x,b ¦x
0
) =d(x
0
,x) holds for all x,x
0
∈ X ;
(b2) sup{d(x,b) : x ∈L
a,b
} <∞;
(b3) there exists a constant K >1 such that d(x,b¦x) ≥K d(x,b) holds for every
x ∈L
a,b
.
4 LAJOS MOLNÁR
Let φ : X →Y be a surjective map such that
ρ(φ(x),φ(x
0
)) =d(x, x
0
), x,x
0
∈ X
and also assume that
(b4) for the element c ∈ Y with c ? φ(a) = φ(b ¦a) we have ρ(c ? y,c ? y
0
) =
ρ(y
0
, y) for all y, y
0
∈Y .
Then we have
φ(b ¦a) =φ(b) ? φ(a).
The maps φ appearing in the theorem may be called "generalized isometries".
Moreover, observe that the above result trivially includes the original Mazur-
Ulam theorem. To see this, take normed real linear spaces X ,Y and a surjective
isometry φ : X → Y . Define the operation ¦ by x ¦x
0
= 2x −x
0
, x,x
0
∈ X and the
operation ? similarly. Let d ,ρ be the metrics corresponding to the norms on X
and Y . Selecting any pair a,b of points in X , it is apparent that all conditions in
the theorem are fulfilled and hence we have φ(2b−a) =2φ(b)−φ(a). It easily im-
plies that φ respects the operation of the arithmetic mean from which it follows
that φ respects all dyadic convex combinations and finally, by the continuity of
φ, we conclude that φ is affine.
The above result shows that maps which conserve the "distances " with re-
spect to a pair of generalized distance measures respect a pair of algebraic op-
erations in some sense. We emphasize that in the result above as well as in our
other general Mazur-Ulam type results that appeared in [12], the isometries re-
spect or, in other words, preserve algebraic operations only locally, for certain
pairs a,b of elements. In fact, in that generality nothing more can be expected.
To see this, one may refer to groups equipped with the discrete metrics: any
bijection between them is a surjective isometry but clearly not necessarily an
isomorphism in any adequate sense. Nonetheless, even if only locally, surjec-
tive distance measure preserver transformations appearing in the above theo-
rem do have a certain algebraic property. And in the cases what we consider
in the present paper it turns out that they in fact have this property globally.
Therefore, the problem of describing those distance measure preserver trans-
formations can be transformed to the problem of describing certain algebraic
isomorphisms. This is exactly the strategy we are going to follow below.
Let us proceed toward the first group of our results which concern transfor-
mations between the positive definite cones of C
∗
-algebras. Before presenting
the results we need to make some preparations. By a symmetric norm on a C
∗
-
algebra A we mean a norm N for which N (AX B) ≤ kAkN(X )kBk holds for all
A, X ,B ∈A . Here and in what follows k.kstands for the original norm on A what
we sometimes call operator norm. Whenever we speak about topological prop-
erties (convergence, continuity, etc.) without specifying the topology we always
mean the norm topology of k.k. We call a norm N on A unitarily invariant if
N(U AV ) = N(A) holds for all A ∈ A and unitary U,V ∈A . Furthermore, a norm
N on A is said to be unitary similarity invariant if we have N(U AU
∗
) = N(A) for
5
all A ∈ A and unitary U ∈A . It is easy to see that any symmetric norm is unitar-
ily invariant and it is trivial that every unitarily invariant norm is unitary similar-
ity invariant. For several examples of complete symmetric norms on B(H ), the
algebra of all bounded linear operators on a complex Hilbert space H , we refer
to [7]. They include the so-called (c,p)-norms and, in particular, the Ky Fan k-
norms. We mention that in that paper the authors use the expression "uniform
norm" for symmetric norms. Apparently, the above examples provide examples
of complete symmetric norms on any C
∗
-subalgebra of B(H) and hence on von
Neumann algebras, too.
Now, we recall that in [23] we have described the structure of isometries of
the space P
n
of all positive definite n ×n complex matrices with respect to the
metric defined by
(1) d
N
(A,B) =N(log A
−1/2
B A
−1/2
), A,B ∈P
n
,
where N is a unitarily invariant norm on M
n
. (It is a well-known fact that on
matrix algebras a norm is symmetric if and only if it is unitarily invariant, see
Proposition IV.2.4 in [3]). The importance of that metric comes from its dif-
ferential geometric background (it is a shortest path distance in a Finsler-type
structure on P
n
which generalizes its fundamental natural Riemann structure,
for references see [23]). In the recent paper [26] we have presented a substan-
tial extension of that result for the case where the logarithmic function in (1) is
replaced by any continuous function f : ]0,∞[→R that satisfies
(c1) f (y) =0 holds if and only if y =1;
(c2) there exists a number K >1 such that
|f (y
2
)|≥K |f (y)|, y ∈]0,∞[.
We must point out that with this replacement we usually get not a true metric,
only a generalized distance measure. However in that way we cover the cases of
many important concepts of matrix divergences whose preserver transforma-
tions could hence be explicitly described, for details see [26].
We now define that new class of generalized distance measures in the context
of C
∗
-algebras. Let A be a C
∗
-algebra, N a norm on A , f : ]0,∞[→ R a given
continuous function with property (c1). Define d
N,f
: A
−1
+
×A
−1
+
→[0,∞[ by
(2) d
N,f
(A,B) =N(f (A
1/2
B
−1
A
1/2
)), A,B ∈A
−1
+
.
It is apparent that d
N,f
is a generalized distance measure.
We also need the following notions. If A is a C
∗
-algebra and A,B ∈ A , then
AB A is called the Jordan triple product of A and B while AB
−1
A is said to be their
inverted Jordan triple product. If B is another C
∗
-algebra and φ : A
−1
+
→B
−1
+
is
a map which satisfies
φ(AB A) =φ(A)φ(B)φ(A), A,B ∈A
−1
+
,
then it is called a Jordan triple map. If φ : A
−1
+
→B
−1
+
fulfills
φ(AB
−1
A) =φ(A)φ(B)
−1
φ(A), A,B ∈A
−1
+
,