positive element

Let $\backslash mathcal\{A\}$ be a *-algebra. An element $a\; \backslash in\; \backslash mathcal\{A\}$ is called positive if there are finitely many elements $a\_k\; \backslash in\; \backslash mathcal\{A\}\; \backslash ;\; (k\; =\; 1,2,\backslash ldots,n)$, so that $a\; =\; \backslash sum\_\{k=1\}^n\; a\_k^*a\_k$ {{nowrap|holds.{{sfn|Palmer|2001|p=798}}}} This is also denoted by {{nowrap|$a\; \backslash geq\; 0$.{{sfn|Blackadar|2006|p=63}}}}

The set of positive elements is denoted by {{nowrap|$\backslash mathcal\{A\}\_+$.}}

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

• The unit element $e$ of an unital *-algebra is positive.
• For each element $a\; \backslash in\; \backslash mathcal\{A\}$, the elements $a^*\; a$ and $aa^*$ are positive by {{nowrap|definition.{{sfn|Palmer|2001|p=798}}}}

In case $\backslash mathcal\{A\}$ is a C*-algebra, the following holds:

• Let $a\; \backslash in\; \backslash mathcal\{A\}\_N$ be a normal element, then for every positive function $f\; \backslash geq\; 0$ which is continuous on the spectrum of $a$ the continuous functional calculus defines a positive element {{nowrap|$f(a)$.{{sfn|Kadison|Ringrose|1983|p=271}}}}
• Every projection, i.e. every element $a\; \backslash in\; \backslash mathcal\{A\}$ for which $a\; =\; a^*\; =\; a^2$ holds, is positive. For the spectrum $\backslash sigma(a)$ of such an idempotent element, $\backslash sigma(a)\; \backslash subseteq\; \backslash \{\; 0,\; 1\; \backslash \}$ holds, as can be seen from the continuous functional {{nowrap|calculus.{{sfn|Kadison|Ringrose|1983|p=271}}}}

Let $\backslash mathcal\{A\}$ be a C*-algebra and {{nowrap|$a\; \backslash in\; \backslash mathcal\{A\}$.}} Then the following are equivalent:{{sfn|Kadison|Ringrose|1983|pages=247-248}}

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

If $\backslash mathcal\{A\}$ is a unital *-algebra with unit element $e$, then in addition the following statements are {{nowrap|equivalent:{{sfn|Kadison|Ringrose|1983|p=245}}}}

• $\backslash left\backslash |\; te\; -\; a\; \backslash right\backslash |\; \backslash leq\; t$ for every $t\; \backslash geq\; \backslash left\backslash |\; a\; \backslash right\backslash |$ and $a$ is a self-adjoint element.
• $\backslash left\backslash |\; te\; -\; a\; \backslash right\backslash |\; \backslash leq\; t$ for some $t\; \backslash geq\; \backslash left\backslash |\; a\; \backslash right\backslash |$ and $a$ is a self-adjoint element.

Let $\backslash mathcal\{A\}$ be a *-algebra. Then:

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

Let $\backslash mathcal\{A\}$ be a C*-algebra. Then:

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

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

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

If $\backslash mathcal\{A\}$ is a C*-algebra, the partial order also has the following properties for $a,b\; \backslash in\; \backslash mathcal\{A\}$:

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

• Nonnegative matrix
• Positive operator (Hilbert space)

{{reflist}}

{{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 |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}}

{{SpectralTheory}}

Category:Abstract algebra

Category:C*-algebras