Unitary element

 ( Abstract Algebra ) 

In mathematics, an element of a *-algebra is called unitary if it is Inverse element and its inverse element is the same as its adjoint element.

---

Definition Let $\backslash mathcal\{A\}$ be a *-algebra with Identity element An element $a\; \backslash in\; \backslash mathcal\{A\}$ is called unitary if In other words, if $a$ is invertible and $a^\{-1\}\; =\; a^*$ holds, then $a$ is unitary.

The set of unitary elements is denoted by $\backslash mathcal\{A\}\_U$ or

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.

---

Criteria

• Let $\backslash mathcal\{A\}$ be a unital C*-algebra and $a\; \backslash in\; \backslash mathcal\{A\}\_N$ a Normal element element. Then, $a$ is unitary if the spectrum $\backslash sigma(a)$ consists only of elements of the circle group $\backslash mathbb\{T\}$, i.e.

---

Examples

• The unit $e$ is unitary.

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
• 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

---

Properties Let $\backslash mathcal\{A\}$ be a unital *-algebra and Then:

• The element $ab$ is unitary, since In particular, $\backslash mathcal\{A\}\_U$ forms a
• The element $a$ is normal.
• The adjoint element $a^*$ is also unitary, since $a\; =\; (a^*)^*$ holds for the involution
• If $\backslash mathcal\{A\}$ is a C*-algebra, $a$ has norm 1, i.e.

---

See also

• Unitary matrix
• Unitary operator

---

Notes

• Bruce Blackadar (2025). 9783540284864, Springer. ISBN 9783540284864
• Jacques Dixmier (1977). 9780720407624, North-Holland. ISBN 9780720407624
  English translation of
• (1983). 9780123933010, Academic Press. ISBN 9780123933010

Post Comment