1. Introduction
The concept of means plays an important role in mathematics in general. In matrix theory and operator theory, the study of means represents a very active research field with wide spearing applications in various areas of pure and applied mathematics ( [1] - [6]). There are many different approaches to matrix or operator means and the Kubo-Ando theory [7] is the mean we want to consider in this paper ( [8] - [14]). Means are originally rather algebraic objects and they have close connection with the geometric features of the underlying structures. For example, the weighted arithmetic means
,
of two elements A and B in a Euclidean space form the unique geodesic between A and B. Also the weighted geometric means
of two given positive definite matrices
form the unique geodesic in a Riemannian structure on the positive definite cone of matrices which has many applications (see, e.g., Chapter 6 in [1]).
Throughout this paper, we always assume that
is an unital C*-algebra with unit I. Let
and
We say
and
are positive semidefinite cone and positive definite cone of C*-algebra
respectively. For basic of C*-algebras and von Neumann algebras, we refer to [15]. For a complex Hilbert space
, we let
be the set of all bounded linear operators on
and
be the positive semidefinite cone of
. There are several kinds of means defined on the positive definite cone
of a C*-algebra
. The arithmetic mean, the harmonic mean and the geometric means are defined by
,
and
respectively. These three means are special case of the Kubo-Ando means [7].
Definition 1.1. A binary operation
on
is a Kubo-Ando mean if
1)
;
2) If
, then
;
3)
;
4) If
in strong operator topology, then
in strong operator topology (here
means monotone decreasing convergent in usual order on
and all operators appeared are assumed in
).
Suppose
is a C*-algebra. Let
be the set of all positive invertible elements in
. We use
to denote that
. The spectral geometric mean is the operation defined by
where
is the geometric mean of A and B.
In [8], the authors studied the maps preserving the spectral geometric mean and many interesting results are obtained. There are many interesting and important results related to the norms of means (see [3] [9] [11] [13] and references therein).
In this paper, we give some results on norms related to spectral geometric means and geometric means. We first give a norm inequality related to the spectral geometric mean and the geometric mean. We give a condition for an operator to be a scalar using norm equality between the spectral geometric mean and the geometric mean. We also show that a C*-algebra is commutative under certain conditions.
2. Main Results
Suppose
is a C*-algebra. For
,
(i.e., A and B are positive and invertible),
is unitary equivalent to
. Note that since
is unitary equivalent to
, then we have that
Proposition 2.1. Suppose
is a C*-algebra. For
,
, we have
.
Proof. If
, that is,
and then
, this shows that
. Then
, that is,
.
This implies that
, and this is equivalent to
that is,
. Therefore,
. □
Proposition 2.2. Let H be a Hilbert space and
be the set of all bounded linear operators on H. Let
. If
then A is a scalar.
Proof. For any projection P, since
, we have that
In particular, if P is a rank-one projection (written as
), we have that
where
is the strength of A along P. This implies that
Above equation is true for all
. Note that
for all
with
. Hence
for every unit vector
. Then one can derive that
Put
in the above equation, we can see that
that is,
for all
. Then
for some
. For any
with
, if
for some
, then we have that
and hence
. If
for any
, it follows from
that
Therefore, we have that
for some
. □
Proposition 2.3. Let
be a C*-algebra. Suppose for any
, we have that
Then
is commutative.
Proof. Let
and
. It follows that
Put
, this shows that
that is,
where
is the Thompson metric. Then
is a non-isometric dilation, this forces
is commutative (see [14], Theorem 18).
3. Conclusion
Mean is an important concept in mathematics. There are many interesting results from studying operator means. In this paper, we give some results on norms related to spectral geometric means and geometric means. We first give a norm inequality related to the spectral geometric mean and the geometric mean. We give a condition for an operator to be a scalar using norm equality between the spectral geometric mean and the geometric mean. We also show that a C*-algebra is commutative under certain conditions.
Acknowledgements
The author would like to thank the anonymous referee for constructive criticisms and valuable comments.
Funding
Partially supported by NFS of China (11871303, 11971463) and NSF of Shandong Province (ZR2019MA039 and ZR2020MA008).