Unitary element

In mathematics, an element of a *-algebra is called '''unitary''' if it is invertible and its inverse element is the same as its adjoint element.{{sfn|Dixmier|1977|p=5}}

== Definition ==

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

The set of unitary elements is denoted by <math>\mathcal{A}_U</math> or {{nowrap|<math>U(\mathcal{A})</math>.}}

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

== Criteria ==

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

== Examples ==

* The unit <math>e</math> is unitary.{{sfn|Dixmier|1977|pages=4-5}}

Let <math>\mathcal{A}</math> be a unital C*-algebra, then:

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

== Properties ==

Let <math>\mathcal{A}</math> be a unital *-algebra and {{nowrap|<math>a,b \in \mathcal{A}_U</math>.}} Then:

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

== See also ==

* Unitary matrix * Unitary operator

== Notes == {{reflist}}

== References == * {{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 |author-link=Jacques Dixmier |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}}

Category:Abstract algebra Category:C*-algebras