Unitary element

Let \mathcal{A} be a *-algebra with unit An element a \in \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 \mathcal{A}_U or {{nowrap|U(\mathcal{A}).}} A special case from particular importance is the case where \mathcal{A} is a complete normed *-algebra. This algebra satisfies the C*-identity (\left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A}) and is called a C*-algebra. == Criteria ==

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

• The unit e is unitary. Let \mathcal{A} be a unital C*-algebra, then: • Every projection, i.e. every element a \in \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 \in \mathcal{A}_{N} is a normal element of a C*-algebra \mathcal{A}, then for every continuous function f on the spectrum \sigma(a) the continuous functional calculus defines an unitary element f(a), if {{nowrap|f(\sigma(a)) \subseteq \mathbb{T}.}} == Properties ==

Let \mathcal{A} be a unital *-algebra and {{nowrap|a,b \in \mathcal{A}_U.}} Then: • The element ab is unitary, since {{nowrap|((ab)^*)^{-1} = (b^*a^*)^{-1} = (a^*)^{-1} (b^*)^{-1} = ab.}} In particular, \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 \mathcal{A} is a C*-algebra, a has norm 1, i.e. == See also ==