# Involution algebra

algebra with involution

An algebra $E$ over the field of complex numbers endowed with an involution $x \mapsto x ^ {*}$, $x \in E$. 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 $x ^ {*} \in E$ is called the conjugate, or adjoint, of $x$. An element $x \in E$ is called self-adjoint, or Hermitian, if $x ^ {*} = x$, and normal if $x ^ {*} x = x x ^ {*}$. If $E$ contains a unit element 1, then an element $x \in E$ such that $x ^ {*} x = x x ^ {*} = 1$ is called unitary. The set $E _ {h}$ of Hermitian elements of $E$ is a real vector subspace of $E$, and any $x \in E$ can be uniquely written in the form $x = x _ {1} + i x _ {2}$, where $x _ {1} , x _ {2} \in E _ {h}$. In this case $x \in E$ is normal if and only if $x _ {1}$ and $x _ {2}$ commute. Every element of the form $x ^ {*} x$ is Hermitian, and so is the unit element. If $x$ is invertible, then so is $x ^ {*}$, and $( x ^ {*} ) ^ {-} 1 = ( x ^ {-} 1 ) ^ {*}$. The spectrum of any Hermitian element (cf. Spectrum of an element) is symmetric about the real axis. An involution algebra $E$ is called a total involution algebra if the spectrum of any element of the form $x ^ {*} x$, $x \in E$, 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 $E$ is a total involution algebra if and only if all its maximal ideals are symmetric, or if and only if all characters of $E$ are Hermitian. Every $C ^ {*}$- algebra is a total involution algebra.

A subset $M$ of an involution algebra $E$ is called an involution set if $x ^ {*} \in M$ for all $x \in M$. A mapping $\phi : E \rightarrow F$ of involution algebras is called an involution mapping if $\phi ( x) ^ {*} = \phi ( x ^ {*} )$ for all $x \in E$. 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 $F$ of an involution algebra $E$ is an involution algebra. Let $\widetilde{E}$ be the direct sum of an involution algebra $E$ and the field $\mathbf C$, in which the linear operations and the involution are defined componentwise and the multiplication is given by

$$\{ x , \lambda \} \{ y , \mu \} = \ \{ x y + \lambda y + \mu x , \lambda \mu \}$$

for all $\lambda , \mu \in \mathbf C$, $x , y \in E$. Then $\widetilde{E}$ is an involution algebra with a unit element.

A linear functional $f$ on an involution algebra is called Hermitian if $f ( x ^ {*} ) = \overline{ {f ( x) }}\;$ for all $x \in E$, and positive if $f ( x ^ {*} x ) \geq 0$ for all $x \in E$. The set $E _ {h} ^ \prime$ of Hermitian linear functionals on $E$ is a real vector subspace of $E ^ \prime$, the dual of $E$, and $E ^ \prime$ is the direct sum of the subspaces $E _ {h} ^ \prime$ and $i E _ {h} ^ \prime$. If $E$ has a unit 1, then every positive functional $f$ on $E$ is Hermitian and $| f ( x) | ^ {2} \leq f ( 1) f ( x ^ {*} x )$ for all $x \in E$. If $f$ is a positive functional on an involution algebra $E$, then $f ( y ^ {*} x ) = f ( x ^ {*} y )$ and $| f ( y ^ {*} x ) | ^ {2} \leq f ( y ^ {*} y) f( x ^ {*} x )$ for all $x , y \in E$.

Let $E$ be an involution algebra equipped with a norm making $E$ into a normed algebra and satisfying the condition $\| x ^ {*} \| = \| x \|$ for all $x \in E$. Then $E$ is called a normed algebra with involution. If $E$ is complete with respect to this norm, then $E$ is called a Banach algebra with involution. Every normed algebra with involution $E$ can be imbedded in a Banach algebra with involution $\overline{E}\;$ containing $E$ as a dense involution subalgebra. $\overline{E}\;$ is uniquely defined up to an isometric involution isomorphism. $\overline{E}\;$ is called the completion of $E$. If $E$ is a Banach algebra with involution having an approximate identity, then every positive linear functional $f$ on $E$ is continuous and can be extended to a positive linear functional on $\overline{E}\;$. If $E$ has a unit 1 and $\| 1 \| = 1$, then for any positive linear functional $f$ on $E$, $\| f \| = f ( 1)$ and $f ( x ^ {*} x ) \leq f ( 1) r ( x ^ {*} x )$, where $r ( x ^ {*} x )$ is the spectral radius of $x ^ {*} x$( see Banach algebra).

A Hermitian element of a total involution algebra has a real spectrum. For any maximal closed left ideal $I$ in a total involution algebra $E$ with a unit there is a positive linear functional on $f$ on $E$ such that $I = \{ {x \in E } : {f ( x ^ {*} x ) = 0 } \}$. An element $x$ in a total involution algebra $E$ is left-invertible in $E$ if and only if $f ( x ^ {*} x ) > 0$ for all non-zero positive functionals $f$ on $E$. The radical of a total involution algebra $E$ coincides with the set of elements $x \in E$ such that $f ( x ^ {*} x ) = 0$ for all positive linear functionals $f$ on $E$. A Banach algebra with involution $E$ with a unit 1 is a total involution algebra if and only if $r ( x ^ {*} x ) = \sup f ( x ^ {*} x )$, where the supremum is taken over the set of positive linear functionals $f$ on $E$ for which $f ( 1) = 1$.

#### 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)

A (Banach, normed) algebra with involution is also called an involutive (Banach, normed) algebra. If in an involutive Banach algebra $A$ one has $\| x x ^ {*} \| = \| x \| ^ {2}$ for all $x \in A$, then $A$ is called a $B ^ {*}$- algebra.

Let $A$ be a Banach algebra. A left-approximate identity in $A$ is a net $\{ e _ {i} \} _ {i \in I }$ of elements of $A$( cf. Net (directed set)) such that $\lim\limits \| e _ {i} x - x \| = 0$ for all $x \in A$. A right-approximate identity is similarly defined, using $\| x e _ {i} - x \|$. A left- and right-approximate identity is simply called an approximate identity. Every $B ^ {*}$- algebra has an approximate identity.

An algebra with involution $E$ 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 $\phi : E \rightarrow F$ is called a symmetric homomorphism if $\phi ( x ^ {*} ) = \phi ( x ) ^ {*}$ for all $x \in E$. Unfortunately, the term symmetric algebra is also sometimes used to mean a total involution algebra.

A symmetric ideal of $E$ is an ideal $M$ such that $M ^ {*} = \{ {m ^ {*} } : {m \in M } \} = M$.

#### References

 [a1] W. Rudin, "Functional analysis" , McGraw-Hill (1979)
How to Cite This Entry:
Involution algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involution_algebra&oldid=47429
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article