positive element

In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form

In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form 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 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. 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 For the linear span of \mathcal{A}_+ the following holds: \langle \mathcal{A}_+ \rangle = \mathcal{A}^2 and 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: 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 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 If both a and -a are positive a = 0 If \mathcal{B} is a C*-subalgebra of \mathcal{A} , then 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 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}

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