Involution algebra
algebra with involution
An algebra over the field of complex numbers endowed with an involution
,
. Some examples are: the algebra of continuous functions on a compact set, in which the involution sends any function to its complex conjugate; the algebra of bounded linear operators on a Hilbert space, in which the involution sends any operator to its adjoint; the group algebra of a locally compact group; and the algebra of measures on a locally compact group. The element
is called the conjugate, or adjoint, of
. An element
is called self-adjoint, or Hermitian, if
, and normal if
. If
contains a unit element 1, then an element
such that
is called unitary. The set
of Hermitian elements of
is a real vector subspace of
, and any
can be uniquely written in the form
, where
. In this case
is normal if and only if
and
commute. Every element of the form
is Hermitian, and so is the unit element. If
is invertible, then so is
, and
. The spectrum of any Hermitian element (cf. Spectrum of an element) is symmetric about the real axis. An involution algebra
is called a total involution algebra if the spectrum of any element of the form
,
, is contained in the set of non-negative real numbers. Examples of total involution algebras are: the involution algebra of continuous functions on a compact set; the involution algebra of bounded linear operators on a Hilbert space; and group algebras of compact and commutative locally compact groups. The group algebras of non-compact semi-simple Lie groups are not total involution algebras. A commutative involution algebra
is a total involution algebra if and only if all its maximal ideals are symmetric, or if and only if all characters of
are Hermitian. Every
-algebra is a total involution algebra.
A subset of an involution algebra
is called an involution set if
for all
. A mapping
of involution algebras is called an involution mapping if
for all
. The kernel of an involution homomorphism of involution algebras is a symmetric two-sided ideal. Every symmetric one-sided ideal is two-sided and the quotient algebra of an involution algebra by a symmetric ideal admits the structure of an involution algebra in a natural way. The radical (cf. Radical of rings and algebras) of an involution algebra is a symmetric ideal. An involution subalgebra
of an involution algebra
is an involution algebra. Let
be the direct sum of an involution algebra
and the field
, in which the linear operations and the involution are defined componentwise and the multiplication is given by
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for all ,
. Then
is an involution algebra with a unit element.
A linear functional on an involution algebra is called Hermitian if
for all
, and positive if
for all
. The set
of Hermitian linear functionals on
is a real vector subspace of
, the dual of
, and
is the direct sum of the subspaces
and
. If
has a unit 1, then every positive functional
on
is Hermitian and
for all
. If
is a positive functional on an involution algebra
, then
and
for all
.
Let be an involution algebra equipped with a norm making
into a normed algebra and satisfying the condition
for all
. Then
is called a normed algebra with involution. If
is complete with respect to this norm, then
is called a Banach algebra with involution. Every normed algebra with involution
can be imbedded in a Banach algebra with involution
containing
as a dense involution subalgebra.
is uniquely defined up to an isometric involution isomorphism.
is called the completion of
. If
is a Banach algebra with involution having an approximate identity, then every positive linear functional
on
is continuous and can be extended to a positive linear functional on
. If
has a unit 1 and
, then for any positive linear functional
on
,
and
, where
is the spectral radius of
(see Banach algebra).
A Hermitian element of a total involution algebra has a real spectrum. For any maximal closed left ideal in a total involution algebra
with a unit there is a positive linear functional on
on
such that
. An element
in a total involution algebra
is left-invertible in
if and only if
for all non-zero positive functionals
on
. The radical of a total involution algebra
coincides with the set of elements
such that
for all positive linear functionals
on
. A Banach algebra with involution
with a unit 1 is a total involution algebra if and only if
, where the supremum is taken over the set of positive linear functionals
on
for which
.
References
[1] | M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian) |
[2] | J. Dixmier, "![]() |
[3] | V. Pták, "Banach algebras with involution" Manuscripta Math. , 6 : 3 (1972) pp. 245–290 |
[4] | E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1979) |
Comments
A (Banach, normed) algebra with involution is also called an involutive (Banach, normed) algebra. If in an involutive Banach algebra one has
for all
, then
is called a
-algebra.
Let be a Banach algebra. A left-approximate identity in
is a net
of elements of
(cf. Net (directed set)) such that
for all
. A right-approximate identity is similarly defined, using
. A left- and right-approximate identity is simply called an approximate identity. Every
-algebra has an approximate identity.
An algebra with involution is also termed a symmetric algebra, and a total involution algebra is also called a completely symmetric algebra. Correspondingly, a homomorphism of algebras with involution
is called a symmetric homomorphism if
for all
. Unfortunately, the term symmetric algebra is also sometimes used to mean a total involution algebra.
A symmetric ideal of is an ideal
such that
.
References
[a1] | W. Rudin, "Functional analysis" , McGraw-Hill (1979) |
Involution algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involution_algebra&oldid=18429