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Characterization of a-Birkhoff–James orthogonality in C∗-algebras and its applications
Let A be a unital C ∗ -algebra with unit 1 A and let a ∈ A be a positive and invertible element. Suppose that S ( A ) is the set of all states on A and let S a ( A ) = f f ( a ) : f ∈ S ( A ) , f ( a ) ≠ 0 . The norm ‖ x ‖ a for every x ∈ A is defined by ‖ x ‖ a = sup φ ∈ S a ( A ) φ ( x ∗ a x ) . I...
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| Published in: | Annals of functional analysis 2024-04, Vol.15 (2), Article 36 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | Let
A
be a unital
C
∗
-algebra with unit
1
A
and let
a
∈
A
be a positive and invertible element. Suppose that
S
(
A
)
is the set of all states on
A
and let
S
a
(
A
)
=
f
f
(
a
)
:
f
∈
S
(
A
)
,
f
(
a
)
≠
0
.
The norm
‖
x
‖
a
for every
x
∈
A
is defined by
‖
x
‖
a
=
sup
φ
∈
S
a
(
A
)
φ
(
x
∗
a
x
)
.
In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm
‖
·
‖
a
in
A
,
namely
a
-Birkhoff–James orthogonality. The characterization of
a
-Birkhoff–James orthogonality in
A
by means of the elements of generalized state space
S
a
(
A
)
is provided. As an application, a characterization for the best approximation to elements of
A
in a subspace
B
with respect to
‖
·
‖
a
is obtained. Moreover, a formula for the distance of an element of
A
to the subspace
B
=
C
1
A
is given. We also study the strong version of
a
-Birkhoff–James orthogonality in
A
.
The classes of
C
∗
-algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong
a
-Birkhoff–James orthogonality and
A
-valued inner product orthogonality in
A
implies that the center of
A
is trivial. Finally, we show that if the (strong)
a
-Birkhoff–James orthogonality is right-additive (left-additive) in
A
,
then the center of
A
is trivial. |
|---|---|
| ISSN: | 2639-7390 2008-8752 |
| DOI: | 10.1007/s43034-024-00339-8 |