Self-adjoint explained
In mathematics, an element of a
is called self-adjoint if it is the same as its adjoint (i.e.
).
Definition
Let
be a *-algebra. An element
is called self-adjoint if
The set of self-adjoint elements is referred to as
that is
closed under the
involution *, i.e.
, is called
A special case of particular importance is the case where
is a complete normed *-algebra, that satisfies the C*-identity (
\left\|a*a\right\|=\left\|a\right\|2 \foralla\inl{A}
), which is called a
C*-algebra.
Especially in the older literature on *-algebras and C*-algebras, such elements are often called Because of that the notations
,
or
for the set of self-adjoint elements are also sometimes used, even in the more recent literature.
Examples
of a *-algebra, the elements
and
are self-adjoint, since * is an
of a *-algebra, the real and imaginary parts
and
are self-adjoint, where
denotes the
is a
normal element of a C*-algebra
, then for every
real-valued function
, which is
continuous on the spectrum of
, the
continuous functional calculus defines a self-adjoint element
Criteria
Let
be a *-algebra. Then:
, then
is self-adjoint, since
. A similarly calculation yields that
is also
be the
product of two self-adjoint elements Then
is self-adjoint if
and
commutate, since
always
is a C*-algebra, then a normal element
is self-adjoint if and only if its spectrum is real, i.e.
Properties
In *-algebras
Let
be a *-algebra. Then:
can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements
, so that
holds. Where
and
- The set of self-adjoint elements
is a
real linear subspace of From the previous property, it follows that
is the
direct sum of two real linear subspaces, i.e.
is self-adjoint, then
is
is called a hermitian *-algebra if every self-adjoint element
has a real spectrum
In C*-algebras
Let
be a C*-algebra and
. Then:
\left\|a\right\|\in\sigma(a)
or
-\left\|a\right\|\in\sigma(a)
holds, since
is real and
holds for the spectral radius, because
is
- According to the continuous functional calculus, there exist uniquely determined positive elements
, such that
with For the norm,
\left\|a\right\|=max(\left\|a+\right\|,\left\|a-\right\|)
holds. The elements
and
are also referred to as the
positive and negative parts. In addition,
holds for the absolute value defined for every element
and odd
, there exists a uniquely determined
that satisfies
, i.e. a unique
-th root, as can be shown with the continuous functional
See also
References
- Book: Blackadar, Bruce . Operator Algebras. Theory of C*-Algebras and von Neumann Algebras . Springer . Berlin/Heidelberg . 2006 . 3-540-28486-9 . 63 .
- Book: Dixmier, Jacques . C*-algebras . North-Holland . Amsterdam/New York/Oxford . 1977 . 0-7204-0762-1 . Jellett . Francis . English translation of Book: Dixmier, Jacques . 0 . Les C*-algèbres et leurs représentations . fr . Gauthier-Villars . 1969 .
- Book: Kadison . Richard V. . Ringrose . John R. . Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. . Academic Press . New York/London . 1983 . 0-12-393301-3.
- Book: Palmer, Theodore W. . Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. . Cambridge university press . 2001 . 0-521-36638-0 .