involutory ringInvolutaryRing2006-02-16 01:39:302007-06-22 17:33:02Definitioninvolutionadjoint elementself-adjoint elementprojection elementnorm elementtrace elementskew symmetric element*-homomorphismnormal elementunitary element\usepackage{amssymb,amscd}
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% define commands here\subsubsection*{General Definition of a Ring with Involution}
Let $R$ be a ring. An \emph{\PMlinkescapetext{involution}} $*$ on $R$ is an anti-endomorphism whose square is the identity map. In other words, for $a,b\in R$:
\begin{enumerate}
\item $(a+b)^*=a^*+b^*$,
\item $(ab)^*=b^*a^*$,
\item $a^{**}=a$
\end{enumerate}
A ring admitting an involution is called an \emph{involutory ring}. $a^*$ is called the \emph{adjoint} of $a$. By (3), $a$ is the adjoint of $a^*$, so that every element of $R$ is an adjoint.
\textbf{Remark}. Note that the traditional definition of an \PMlinkname{involution}{Involution} on a vector space is different from the one given here. Clearly, $*$ is bijective, so that it is an anti-automorphism. If $*$ is the identity on $R$, then $R$ is commutative.
\textbf{Examples}. Involutory rings occur most often in rings of endomorphisms over a module. When $V$ is a finite dimensional vector space over a field $k$ with a given basis $\boldsymbol{b}$, any linear transformation over $T$ (to itself) can be represented by a square matrix $M$ over $k$ via $\boldsymbol{b}$. The map taking $M$ to its transpose $M^T$ is an involution. If $k$ is $\mathbb{C}$, then the map taking $M$ to its conjugate transpose $\overline{M}^T$ is also an involution. In general, the composition of an isomorphism and an involution is an involution, and the composition of two involutions is an isomorphism.
\subsubsection*{*-Homomorphisms} Let $R$ and $S$ be involutory rings with involutions $*_R$ and $*_S$. A \emph{*-homomorphism} $\phi:R\to S$ is a ring homomorphism which respects involutions. More precisely,
$$\phi(a^{*_R})=\phi(a)^{*_S},\quad\mbox{ for any }a\in R.$$
By abuse of notation, if we use $*$ to denote both $*_R$ and $*_S$, then we see that any *-homomorphism $\phi$ commutes with $*$: $\phi*=*\phi$.
\subsubsection*{Special Elements} An element $a\in R$ such that $a=a^*$ is called a \emph{self-adjoint}. A ring with involution is usually associated with a ring of square matrices over a field, as such, a self-adjoint element is sometimes called a \emph{Hermitian element}, or a \emph{symmetric element}. For example, for any element $a\in R$,
\begin{enumerate}
\item $aa^*$ and $a^*a$ are both self-adjoint, the first of which is called the \emph{norm} of $a$. A \emph{norm element} $b$ is simply an element expressible in the form $aa^*$ for some $a\in R$, and we write $b=\operatorname{n}(a)$. If $aa^*=a^*a$, then $a$ is called a \emph{normal element}. If $a^*$ is the multiplicative inverse of $a$, then $a$ is a \emph{unitary element}. If $a$ is unitary, then it is normal.
\item With respect to addition, we can also form self-adjoint elements $a+a^*=a^*+a$, called the \emph{trace} of $a$, for any $a\in R$. A \emph{trace element} $b$ is an element expressible as $a+a^*$ for some $a\in R$, and written $b=\operatorname{tr}(a)$.
\end{enumerate}
Let $S$ be a subset of $R$, write $S^*:=\lbrace a^*\mid a\in S\rbrace$. Then $S$ is said to be \emph{self-adjoint} if $S=S^*$.
A self-adjoint that is also an idempotent in $R$ is called a \emph{projection}. If $e$ and $f$ are two projections in $R$ such that $eR=fR$ (principal ideals generated by $e$ and $f$ are equal), then $e=f$. For if $ea=ff=f$ for some $a\in R$, then $f=ea=eea=ef$. Similarly, $e=fe$. Therefore, $e=e^*=(fe)^*=e^*f^*=ef=f$.
If the characteristic of $R$ is not 2, we also have a companion concept to self-adjointness, that of skew symmetry. An element $a$ in $R$ is skew symmetric if $a=-a^*$. Again, the name of this is borrowed from linear algebra.