1. Introduction
The theory of operator spaces is very recent. It was developed after Ruan’s thesis (1988) by Effros and Ruan and Blecher and Paulsen and Pisier.
Theorem (Ruan) in [1] suppose that 
is a vector space, and that for each 
we are given a norm 
on
. Then 
is completely isometrically linearly isomorphic to a subspace of
, for some Hilbert space
, if and only if conditions (R1) and (R2) above hold.
R1:
R2:
.
In this new category, the objects remain Banach spaces but the morphisms become the completely bounded maps. In fact, the completely bounded maps appeared in the early 1980s following Stinespring’s pioneering work (1955) and Arveson’s fundamental results (1969) on completely positive maps see [1] -[4] .
Definition 1 Let 
and 
be operator spaces and that 
be a linear map. For a positive integer
, we write 
for the associated map 
from 
to
. This is often called the (nth) amplification of
, and may also be thought of as the map 
on
. A map 
is completely bounded (in short c.b.) if
We denote by 
the Banach space of all c.b. maps from 
into 
equipped with the c.b. norm. If 
is c.b. map, we have
and
The notion of isomorphism is replaced by that of “complete isomorphism”. If 
is a completely bounded linear bijection, and if its inverse is completely bounded too, then we say that 
is complete isomorphism. In this case, we say that 
and 
are completely isomorphic and we write “
as operator spaces”. Similarly, if 
is an isometry for any
, then we say that 
is complete isometry. We often identify two operator spaces 
and 
if they are completely isometrically isomorphic. In this case we often write “
as operator spaces”.
In 1932 [5] , Mazur and Ulam got the famous theorem that a surjective isometry between two real normed spaces must be linear up to translation, which started the study of the theory of isometry. Many researchers have tried to generalize it. In 1972 Mankiewicz [6] considered the extension problem for isometries and every surjective isometry between the open connected subsets of two normed spaces can be extended to a surjective affine isometry on the whole space. This result can be seen as local Mazur-Ulam Theorem. From this result, we get easily that a map’s property on unit ball of a normed space determines the relationship between distance preserving and linearity. In 1987, D.Tingley raised the following problem (the isometric extension problem) in [7] :
Problem. Suppose that 
is a surjective isometric mapping, does there exist a linear isometric mapping 
such that
?
D. Tingley, who is the first one to study the problem, gave some results in [7] under the condition that 
and 
are finite dimensional spaces. More precisely, the answer can be formulated as follows: Suppose that 
is a surjective isometric mapping, then
, i.e. 
is an odd mapping. In the complex spaces, the answer to Tingleys problem is negative. For example, we take 
(complex plane) and
. Some affirmative results have been obtained between classical real Banach spaces, which had been shown in [8] -[12] . The recent development on this problem you can find in [13] . We ask can we discuss the isometric extension problem in operator space category? Through example, we give some results on it.
2. Some Examples about Isometric Extension Problem on Operator Spaces
Theorem Let 
be 
(resp.
, 
, 
, 
, 
, etc.). Let 
be a 
-Lip operator. Then 
can be extended to a linearly complete isometry on
.
Proof. By [11] , 
can be linearly extended to the whole Hilbert space. Since 
are homogeneous operator space, the the result can be got directly.
How about the result on non-homogeneous operator spaces?
Theorem There exists an isometry 
such that there is no complete isometry 
on the whole space 
satisfying
.
Proof. Let 
be the basis of
. To express conveniently, we denote 
to be the basis of the n-dimensional column Hilbert operator space 
and 
be the basis of the n-dimensional row Hilbert operator space
. Then 
constructs a basis of
. Give the 
elements an order defined as
Let
, 
and 
maps other elements 
of the basis to
. So 
forms a permutation among the basis of
. By the isometric theory on Hilbert space, 
is an isometry on the Banach space
. Of course let 
is an isometry between unit sphere of
. But 
is not complete isometry on
.
Indeed, for 
where
Then
It is easy to calculate that 
but
. So 
is not complete isometry.
3. On Stability of (-Isometries of Operator Spaces
The study on the stability of ò-isometry is from Banach Space Theory.
Let 
be two Banach spaces and
. A mapping 
is said to be an ò-isometry provided
In 1945, Hyers and Ulam proposed the following question: whether for every surjective ò-isometry 
with 
there exist a surjective linear isometry 
and 
such that 
After many efforts of a number of mathematicians (see, for instance [14] -[19] ) Omladi 
and 
emrl gave the sharp estimate that 
in 1995 (see [20] ).
In this section, we ask whether we can discuss the the stability of ò-isometry in operator space category? We try to propose a question and give a little answer.
Firstly, we get the following result.
Theorem Let 
and 
be operator spaces. If a mapping 
satisfies that for any
,
then 
is an isometry.
Proof. For any
, 
Let
. Specially, we put
, then 
and we have
Since
we get 
So 
holds for any
. 
is an isometry.
Acknowledgements
A part of this work was completed when I visited the University of Illinois, Urbana. I would like to thank Professor Zhong-Jin Ruan for the invitation and great help.
The author also expresses his appreciation to Dr. Liu Rui for many very helpful comments regarding the stability of ò-isometry in operator space category.
Funding
This work is supported by the NSFC grant 11101220, 11271199 and the Fundamental Research Funds for the Central Universities.