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

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, " algebras" , North-Holland (1977) (Translated from French)
[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)