unitary element

Let $\backslash mathcal\{A\}$ be a *-algebra with unit {{nowrap|$e$.}} An element $a\; \backslash in\; \backslash mathcal\{A\}$ is called unitary if {{nowrap|$aa^*\; =\; a^*a\; =\; e$.}} In other words, if $a$ is invertible and $a^\{-1\}\; =\; a^*$ holds, then $a$ is unitary.{{sfn|Dixmier|1977|p=5}}

The set of unitary elements is denoted by $\backslash mathcal\{A\}\_U$ or {{nowrap|$U(\backslash mathcal\{A\})$.}}

A special case from particular importance is the case where $\backslash mathcal\{A\}$ is a complete normed *-algebra. This algebra satisfies the C*-identity ($\backslash left\backslash |\; a^*a\; \backslash right\backslash |\; =\; \backslash left\backslash |\; a\; \backslash right\backslash |^2\; \backslash \; \backslash forall\; a\; \backslash in\; \backslash mathcal\{A\}$) and is called a C*-algebra.

• Let $\backslash mathcal\{A\}$ be a unital C*-algebra and $a\; \backslash in\; \backslash mathcal\{A\}\_N$ a normal element. Then, $a$ is unitary if the spectrum $\backslash sigma(a)$ consists only of elements of the circle group $\backslash mathbb\{T\}$, i.e. {{nowrap|$\backslash sigma(a)\; \backslash subseteq\; \backslash mathbb\{T\}\; =\; \backslash \{\; \backslash lambda\; \backslash in\; \backslash Complex\; \backslash mid\; |\; \backslash lambda\; |\; =\; 1\; \backslash \}$.{{sfn|Kadison|Ringrose|1983|p=271}}}}

• The unit $e$ is unitary.{{sfn|Dixmier|1977|pages=4-5}}

Let $\backslash mathcal\{A\}$ be a unital C*-algebra, then:

• Every projection, i.e. every element $a\; \backslash in\; \backslash mathcal\{A\}$ with $a\; =\; a^*\; =\; a^2$, is unitary. For the spectrum of a projection consists of at most $0$ and $1$, as follows from the {{nowrap|continuous functional calculus.{{sfn|Blackadar|2006|pages=57,63}}}}
• If $a\; \backslash in\; \backslash mathcal\{A\}\_\{N\}$ is a normal element of a C*-algebra $\backslash mathcal\{A\}$, then for every continuous function $f$ on the spectrum $\backslash sigma(a)$ the continuous functional calculus defines an unitary element $f(a)$, if {{nowrap|$f(\backslash sigma(a))\; \backslash subseteq\; \backslash mathbb\{T\}$.{{sfn|Kadison|Ringrose|1983|p=271}}}}

Let $\backslash mathcal\{A\}$ be a unital *-algebra and {{nowrap|$a,b\; \backslash in\; \backslash mathcal\{A\}\_U$.}} Then:

• The element $ab$ is unitary, since {{nowrap|$((ab)^*)^\{-1\}\; =\; (b^*a^*)^\{-1\}\; =\; (a^*)^\{-1\}\; (b^*)^\{-1\}\; =\; ab$.}} In particular, $\backslash mathcal\{A\}\_U$ forms a {{nowrap|multiplicative group.{{sfn|Dixmier|1977|p=5}}}}
• The element $a$ is normal.{{sfn|Dixmier|1977|pages=4-5}}
• The adjoint element $a^*$ is also unitary, since $a\; =\; (a^*)^*$ holds for the involution {{nowrap|*.{{sfn|Dixmier|1977|p=5}}}}
• If $\backslash mathcal\{A\}$ is a C*-algebra, $a$ has norm 1, i.e. {{nowrap|$\backslash left\backslash |\; a\; \backslash right\; \backslash |\; =\; 1$.{{sfn|Dixmier|1977|p=9}}}}

• Unitary matrix
• Unitary operator

{{reflist}}

• {{cite book |last=Blackadar|first=Bruce |title=Operator Algebras. Theory of C*-Algebras and von Neumann Algebras |publisher=Springer |location=Berlin/Heidelberg |year=2006 |isbn=3-540-28486-9 |pages=57, 63 }}
• {{cite book |last=Dixmier |first=Jacques |title=C*-algebras |publisher=North-Holland |location=Amsterdam/New York/Oxford |year=1977 |isbn=0-7204-0762-1 |translator-last=Jellett |translator-first=Francis }} English translation of {{cite book |display-authors=0 |last=Dixmier |first=Jacques |title=Les C*-algèbres et leurs représentations |language=fr |publisher=Gauthier-Villars |year=1969 }}
• {{cite book |last1=Kadison |first1=Richard V. |last2=Ringrose |first2=John R. |title=Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. |publisher=Academic Press |location=New York/London |year=1983 |isbn=0-12-393301-3}}

{{SpectralTheory}}

Category:Abstract algebra

Category:C*-algebras