In mathematics , an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.
Definition
Let
A
{\displaystyle {\mathcal {A}}}
be a *-algebra with unit
e
{\displaystyle e}
. An element
a
∈ ∈ -->
A
{\displaystyle a\in {\mathcal {A}}}
is called unitary if
a
a
∗ ∗ -->
=
a
∗ ∗ -->
a
=
e
{\displaystyle aa^{*}=a^{*}a=e}
. In other words, if
a
{\displaystyle a}
is invertible and
a
− − -->
1
=
a
∗ ∗ -->
{\displaystyle a^{-1}=a^{*}}
holds, then
a
{\displaystyle a}
is unitary.
The set of unitary elements is denoted by
A
U
{\displaystyle {\mathcal {A}}_{U}}
or
U
(
A
)
{\displaystyle U({\mathcal {A}})}
.
A special case from particular importance is the case where
A
{\displaystyle {\mathcal {A}}}
is a complete normed *-algebra . This algebra satisfies the C*-identity (
‖
a
∗ ∗ -->
a
‖
=
‖
a
‖
2
∀ ∀ -->
a
∈ ∈ -->
A
{\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}
) and is called a C*-algebra .
Criteria
Let
A
{\displaystyle {\mathcal {A}}}
be a unital C*-algebra and
a
∈ ∈ -->
A
N
{\displaystyle a\in {\mathcal {A}}_{N}}
a normal element. Then,
a
{\displaystyle a}
is unitary if the spectrum
σ σ -->
(
a
)
{\displaystyle \sigma (a)}
consists only of elements of the circle group
T
{\displaystyle \mathbb {T} }
, i.e.
σ σ -->
(
a
)
⊆ ⊆ -->
T
=
{
λ λ -->
∈ ∈ -->
C
∣ ∣ -->
|
λ λ -->
|
=
1
}
{\displaystyle \sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}}
.
Examples
The unit
e
{\displaystyle e}
is unitary.
Let
A
{\displaystyle {\mathcal {A}}}
be a unital C*-algebra, then:
Every projection , i.e. every element
a
∈ ∈ -->
A
{\displaystyle a\in {\mathcal {A}}}
with
a
=
a
∗ ∗ -->
=
a
2
{\displaystyle a=a^{*}=a^{2}}
, is unitary. For the spectrum of a projection consists of at most
0
{\displaystyle 0}
and
1
{\displaystyle 1}
, as follows from the continuous functional calculus .
If
a
∈ ∈ -->
A
N
{\displaystyle a\in {\mathcal {A}}_{N}}
is a normal element of a C*-algebra
A
{\displaystyle {\mathcal {A}}}
, then for every continuous function
f
{\displaystyle f}
on the spectrum
σ σ -->
(
a
)
{\displaystyle \sigma (a)}
the continuous functional calculus defines an unitary element
f
(
a
)
{\displaystyle f(a)}
, if
f
(
σ σ -->
(
a
)
)
⊆ ⊆ -->
T
{\displaystyle f(\sigma (a))\subseteq \mathbb {T} }
.
Properties
Let
A
{\displaystyle {\mathcal {A}}}
be a unital *-algebra and
a
,
b
∈ ∈ -->
A
U
{\displaystyle a,b\in {\mathcal {A}}_{U}}
. Then:
The element
a
b
{\displaystyle ab}
is unitary, since
(
(
a
b
)
∗ ∗ -->
)
− − -->
1
=
(
b
∗ ∗ -->
a
∗ ∗ -->
)
− − -->
1
=
(
a
∗ ∗ -->
)
− − -->
1
(
b
∗ ∗ -->
)
− − -->
1
=
a
b
{\textstyle ((ab)^{*})^{-1}=(b^{*}a^{*})^{-1}=(a^{*})^{-1}(b^{*})^{-1}=ab}
. In particular,
A
U
{\displaystyle {\mathcal {A}}_{U}}
forms a multiplicative group .
The element
a
{\displaystyle a}
is normal.
The adjoint element
a
∗ ∗ -->
{\displaystyle a^{*}}
is also unitary, since
a
=
(
a
∗ ∗ -->
)
∗ ∗ -->
{\displaystyle a=(a^{*})^{*}}
holds for the involution *.
If
A
{\displaystyle {\mathcal {A}}}
is a C*-algebra,
a
{\displaystyle a}
has norm 1, i.e.
‖
a
‖
=
1
{\displaystyle \left\|a\right\|=1}
.
See also
Notes
References
Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras . Berlin/Heidelberg: Springer. pp. 57, 63. ISBN 3-540-28486-9 .
Dixmier, Jacques (1977). C*-algebras . Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1 . English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras . Volume 1 Elementary Theory . New York/London: Academic Press. ISBN 0-12-393301-3 .
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