Positive element

In mathematics, an element of a *-algebra is called '''positive''' if it is the sum of elements of the form {{nowrap|<math>a^*a</math>.{{sfn|Palmer|2001|p=798}}}}

== Definition == Let <math>\mathcal{A}</math> be a *-algebra. An element <math>a \in \mathcal{A}</math> is called positive if there are finitely many elements <math>a_k \in \mathcal{A} \; (k = 1,2,\ldots,n)</math>, so that <math display="inline">a = \sum_{k=1}^n a_k^*a_k</math> {{nowrap|holds.{{sfn|Palmer|2001|p=798}}}} This is also denoted by {{nowrap|<math>a \geq 0</math>.{{sfn|Blackadar|2006|p=63}}}}

The set of positive elements is denoted by {{nowrap|<math>\mathcal{A}_+</math>.}}

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

== Examples == * The unit element <math>e</math> of an unital *-algebra is positive. * For each element <math>a \in \mathcal{A}</math>, the elements <math>a^* a</math> and <math>aa^*</math> are positive by {{nowrap|definition.{{sfn|Palmer|2001|p=798}}}} In case <math>\mathcal{A}</math> is a C*-algebra, the following holds: * Let <math>a \in \mathcal{A}_N</math> be a normal element, then for every positive function <math>f \geq 0</math> which is continuous on the spectrum of <math>a</math> the continuous functional calculus defines a positive element {{nowrap|<math>f(a)</math>.{{sfn|Kadison|Ringrose|1983|p=271}}}} * Every projection, i.e. every element <math>a \in \mathcal{A}</math> for which <math>a = a^* = a^2</math> holds, is positive. For the spectrum <math>\sigma(a)</math> of such an idempotent element, <math>\sigma(a) \subseteq \{ 0, 1 \}</math> holds, as can be seen from the continuous functional {{nowrap|calculus.{{sfn|Kadison|Ringrose|1983|p=271}}}}

== Criteria == Let <math>\mathcal{A}</math> be a C*-algebra and {{nowrap|<math>a \in \mathcal{A}</math>.}} Then the following are equivalent:{{sfn|Kadison|Ringrose|1983|pages=247-248}}

* For the spectrum <math>\sigma(a) \subseteq [0, \infty)</math> holds and <math>a</math> is a normal element. * There exists an element <math>b \in \mathcal{A}</math>, such that {{nowrap|<math>a = bb^*</math>.}} * There exists a (unique) self-adjoint element <math>c \in \mathcal{A}_{sa}</math> such that {{nowrap|<math>a = c^2</math>.}}

If <math>\mathcal{A}</math> is a unital *-algebra with unit element <math>e</math>, then in addition the following statements are {{nowrap|equivalent:{{sfn|Kadison|Ringrose|1983|p=245}}}} * <math>\left\| te - a \right\| \leq t</math> for every <math>t \geq \left\| a \right\|</math> and <math>a</math> is a self-adjoint element. * <math>\left\| te - a \right\| \leq t</math> for some <math>t \geq \left\| a \right\|</math> and <math>a</math> is a self-adjoint element.

== Properties == === In *-algebras === Let <math>\mathcal{A}</math> be a *-algebra. Then:

* If <math>a \in \mathcal{A}_+</math> is a positive element, then <math>a</math> is self-adjoint.{{sfn|Palmer|2001|p=800}} * The set of positive elements <math>\mathcal{A}_+</math> is a convex cone in the real vector space of the self-adjoint elements {{nowrap|<math>\mathcal{A}_{sa}</math>.}} This means that <math>\alpha a, a+b \in \mathcal{A}_+</math> holds for all <math>a,b \in \mathcal{A}</math> and {{nowrap|<math>\alpha \in [0, \infty)</math>.{{sfn|Palmer|2001|p=800}}}} * If <math>a \in \mathcal{A}_+</math> is a positive element, then <math>b^*ab</math> is also positive for every element {{nowrap|<math>b \in \mathcal{A}</math>.{{sfn|Blackadar|2006|p=64}}}} * For the linear span of <math>\mathcal{A}_+</math> the following holds: <math>\langle \mathcal{A}_+ \rangle = \mathcal{A}^2</math> and {{nowrap|<math>\mathcal{A}_+ - \mathcal{A}_+ = \mathcal{A}_{sa} \cap \mathcal{A}^2</math>.{{sfn|Palmer|2001|p=802}}}}

=== In C*-algebras === Let <math>\mathcal{A}</math> be a C*-algebra. Then:

* Using the continuous functional calculus, for every <math>a \in \mathcal{A}_+</math> and <math>n \in \mathbb{N}</math> there is a uniquely determined <math>b \in \mathcal{A}_+</math> that satisfies <math>b^n = a</math>, i.e. a unique <math>n</math>-th root. In particular, a square root exists for every positive element. Since for every <math>b \in \mathcal{A}</math> the element <math>b^*b</math> is positive, this allows the definition of a unique absolute value: {{nowrap|<math display="inline">|b| = (b^*b)^\frac{1}{2}</math>.{{sfn|Blackadar|2006|pages=63-65}}}} * For every real number <math>\alpha \geq 0</math> there is a positive element <math>a^\alpha \in \mathcal{A}_+</math> for which <math>a^\alpha a^\beta = a^{\alpha + \beta}</math> holds for all {{nowrap|<math>\beta \in [0, \infty)</math>.}} The mapping <math>\alpha \mapsto a^\alpha</math> is continuous. Negative values for <math>\alpha</math> are also possible for invertible elements {{nowrap|<math>a</math>.{{sfn|Blackadar|2006|p=64}}}} * Products of positive commutative elements are also positive. So if <math>ab = ba</math> holds for positive <math>a,b \in \mathcal{A}_+</math>, then {{nowrap|<math>ab \in \mathcal{A}_+</math>.{{sfn|Kadison|Ringrose|1983|p=245}}}} *Each element <math>a \in \mathcal{A}</math> can be uniquely represented as a linear combination of four positive elements. To do this, <math>a</math> 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 {{nowrap|calculus.{{sfn|Kadison|Ringrose|1983|p=247}}}} For it holds that <math>\mathcal{A}_{sa} = \mathcal{A}_+ - \mathcal{A}_+</math>, since {{nowrap|<math>\mathcal{A}^2 = \mathcal{A}</math>.{{sfn|Palmer|2001|p=802}}}} * If both <math>a</math> and <math>-a</math> are positive <math>a = 0</math> {{nowrap|holds.{{sfn|Kadison|Ringrose|1983|p=245}}}} *If <math>\mathcal{B}</math> is a C*-subalgebra of <math>\mathcal{A}</math>, then {{nowrap|<math>\mathcal{B}_+ = \mathcal{B} \cap \mathcal{A}_+</math>.{{sfn|Kadison|Ringrose|1983|p=245}}}} *If <math>\mathcal{B}</math> is another C*-algebra and <math>\Phi</math> is a *-homomorphism from <math>\mathcal{A}</math> to <math>\mathcal{B}</math>, then <math>\Phi(\mathcal{A}_+) = \Phi(\mathcal{A}) \cap \mathcal{B}_+</math> {{nowrap|holds.{{sfn|Dixmier|1977|p=18}}}} *If <math>a,b \in \mathcal{A}_+</math> are positive elements for which <math>ab = 0</math>, they commutate and <math>\left\| a + b \right\| = \max(\left\| a \right\|, \left\| b \right\|)</math> holds. Such elements are called orthogonal and one writes {{nowrap|<math>a \bot b</math>.{{sfn|Blackadar|2006|p=67}}}}

== Partial order == Let <math>\mathcal{A}</math> be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements {{nowrap|<math>\mathcal{A}_{sa}</math>.}} If <math>b - a \in \mathcal{A}_+</math> holds for <math>a,b \in \mathcal{A}</math>, one writes <math>a \leq b</math> or {{nowrap|<math>b \geq a</math>.{{sfn|Palmer|2001|p=799}}}}

This partial order fulfills the properties <math>ta \leq tb</math> and <math>a + c \leq b + c</math> for all <math>a,b,c \in \mathcal{A}_{sa}</math> with {{nowrap|<math>a \leq b</math> and <math>t \in [0, \infty)</math>.}}{{sfn|Palmer|2001|p=802}}

If <math>\mathcal{A}</math> is a C*-algebra, the partial order also has the following properties for <math>a,b \in \mathcal{A}</math>:

* If <math>a \leq b</math> holds, then <math>c^*ac \leq c^*bc</math> is true for every {{nowrap|<math>c \in \mathcal{A}</math>.}} For every <math>c \in \mathcal{A}_+</math> that commutes with <math>a</math> and <math>b</math> even <math>ac \leq bc</math> {{nowrap|holds.{{sfn|Kadison|Ringrose|1983|p=249}}}} * If <math>-b \leq a \leq b</math> holds, then {{nowrap|<math>\left\| a \right\| \leq \left\| b \right\|</math>.{{sfn|Kadison|Ringrose|1983|p=250}}}} * If <math>0 \leq a \leq b</math> holds, then <math display="inline">a^\alpha \leq b^\alpha</math> holds for all real numbers {{nowrap|<math>0 < \alpha \leq 1</math>.{{sfn|Blackadar|2006|p=66}}}} * If <math>a</math> is invertible and <math>0 \leq a \leq b</math> holds, then <math>b</math> is invertible and for the inverses <math>b^{-1} \leq a^{-1}</math> {{nowrap|holds.{{sfn|Kadison|Ringrose|1983|p=250}}}}

==Citations== ===References=== {{reflist}}

=== Bibliography === {{refbegin|30em}} * {{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 }} * {{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}} * {{cite book |last=Palmer|first=Theodore W. |title=Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. |publisher=Cambridge university press |year=2001 |isbn=0-521-36638-0 }} {{refend}}

Category:Abstract algebra Category:C*-algebras