Positive element

Let \mathcal{A} be a *-algebra. An element a \in \mathcal{A} is called positive if there are finitely many elements a_k \in \mathcal{A} \; (k = 1,2,\ldots,n), so that a = \sum_{k=1}^n a_k^*a_k This is also denoted by The set of positive elements is denoted by {{nowrap|\mathcal{A}_+.}} A special case from particular importance is the case where \mathcal{A} is a complete normed *-algebra, that satisfies the C*-identity (\left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A}), which is called a C*-algebra. == Examples ==

• The unit element e of an unital *-algebra is positive. • For each element a \in \mathcal{A}, the elements a^* a and aa^* are positive by In case \mathcal{A} is a C*-algebra, the following holds: • Let a \in \mathcal{A}_N be a normal element, then for every positive function f \geq 0 which is continuous on the spectrum of a the continuous functional calculus defines a positive element • Every projection, i.e. every element a \in \mathcal{A} for which a = a^* = a^2 holds, is positive. For the spectrum \sigma(a) of such an idempotent element, \sigma(a) \subseteq \{ 0, 1 \} holds, as can be seen from the continuous functional == Criteria ==

Let \mathcal{A} be a C*-algebra and {{nowrap|a \in \mathcal{A}.}} Then the following are equivalent: • For the spectrum \sigma(a) \subseteq [0, \infty) holds and a is a normal element. • There exists an element b \in \mathcal{A}, such that • There exists a (unique) self-adjoint element c \in \mathcal{A}_{sa} such that If \mathcal{A} is a unital *-algebra with unit element e, then in addition the following statements are • \left\| te - a \right\| \leq t for every t \geq \left\| a \right\| and a is a self-adjoint element. • \left\| te - a \right\| \leq t for some t \geq \left\| a \right\| and a is a self-adjoint element. == Properties ==

In *-algebras Let \mathcal{A} be a *-algebra. Then: • If a \in \mathcal{A}_+ is a positive element, then a is self-adjoint. • The set of positive elements \mathcal{A}_+ is a convex cone in the real vector space of the self-adjoint elements {{nowrap|\mathcal{A}_{sa}.}} This means that \alpha a, a+b \in \mathcal{A}_+ holds for all a,b \in \mathcal{A} and • If a \in \mathcal{A}_+ is a positive element, then b^*ab is also positive for every element {{nowrap|b \in \mathcal{A}.}} • For the linear span of \mathcal{A}_+ the following holds: \langle \mathcal{A}_+ \rangle = \mathcal{A}^2 and {{nowrap|\mathcal{A}_+ - \mathcal{A}_+ = \mathcal{A}_{sa} \cap \mathcal{A}^2.}} In C*-algebras Let \mathcal{A} be a C*-algebra. Then: • Using the continuous functional calculus, for every a \in \mathcal{A}_+ and n \in \mathbb{N} there is a uniquely determined b \in \mathcal{A}_+ that satisfies b^n = a, i.e. a unique n-th root. In particular, a square root exists for every positive element. Since for every b \in \mathcal{A} the element b^*b is positive, this allows the definition of a unique absolute value: {{nowrap||b| = (b^*b)^\frac{1}{2}.}} • For every real number \alpha \geq 0 there is a positive element a^\alpha \in \mathcal{A}_+ for which a^\alpha a^\beta = a^{\alpha + \beta} holds for all The mapping \alpha \mapsto a^\alpha is continuous. Negative values for \alpha are also possible for invertible elements • Products of positive commutative elements are also positive. So if ab = ba holds for positive a,b \in \mathcal{A}_+, then {{nowrap|ab \in \mathcal{A}_+.}} • Each element a \in \mathcal{A} can be uniquely represented as a linear combination of four positive elements. To do this, a is first decomposed into the self-adjoint real and imaginary parts and these are then decomposed into positive and negative parts using the continuous functional For it holds that \mathcal{A}_{sa} = \mathcal{A}_+ - \mathcal{A}_+, since {{nowrap|\mathcal{A}^2 = \mathcal{A}.}} • If both a and -a are positive a = 0 • If \mathcal{B} is a C*-subalgebra of \mathcal{A}, then {{nowrap|\mathcal{B}_+ = \mathcal{B} \cap \mathcal{A}_+.}} • If \mathcal{B} is another C*-algebra and \Phi is a *-homomorphism from \mathcal{A} to \mathcal{B}, then \Phi(\mathcal{A}_+) = \Phi(\mathcal{A}) \cap \mathcal{B}_+ • If a,b \in \mathcal{A}_+ are positive elements for which ab = 0, they commutate and \left\| a + b \right\| = \max(\left\| a \right\|, \left\| b \right\|) holds. Such elements are called orthogonal and one writes == Partial order ==

Let \mathcal{A} be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements {{nowrap|\mathcal{A}_{sa}.}} If b - a \in \mathcal{A}_+ holds for a,b \in \mathcal{A}, one writes a \leq b or This partial order fulfills the properties ta \leq tb and a + c \leq b + c for all a,b,c \in \mathcal{A}_{sa} with If \mathcal{A} is a C*-algebra, the partial order also has the following properties for a,b \in \mathcal{A}: • If a \leq b holds, then c^*ac \leq c^*bc is true for every {{nowrap|c \in \mathcal{A}.}} For every c \in \mathcal{A}_+ that commutes with a and b even ac \leq bc • If -b \leq a \leq b holds, then • If 0 \leq a \leq b holds, then a^\alpha \leq b^\alpha holds for all real numbers • If a is invertible and 0 \leq a \leq b holds, then b is invertible and for the inverses b^{-1} \leq a^{-1} == See also ==