Positive element explained

In mathematics, an element of a

• -algebra

is called positive if it is the sum of elements of the form

Definition

Let

l{A} be a *-algebra. An element a \in l{A} is called positive if there are finitely many elements a _{k} \in l{A} (k = 1 , 2 ,\ldots,n) , so that $a\; =\; \backslash sum\_^n\; a\_k^*a\_k$ This is also denoted by

The set of positive elements is denoted by

A special case from particular importance is the case where

l{A} is a complete normed *-algebra, that satisfies the C*-identity (\left\| a ^{*a} \right\| = \left\| a \right\| ^{2} \forall a \in l{A} ), which is called a C*-algebra.

Examples

• The unit element

e of an unital *-algebra is positive.

• For each element

a \in l{A} , the elements a ^{*} a and aa ^{*} are positive by In case l{A} is a C*-algebra, the following holds:

• Let

a \in l{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 l{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

l{A} be a C*-algebra and Then the following are equivalent:

• For the spectrum

\sigma(a) \subseteq [ 0 , inf ty) holds and a is a normal element.

• There exists an element

b \in l{A} , such that c \in l{A} _{sa} such that

If

l{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

l{A} be a *-algebra. Then:

• If

a \in l{A} _{+} is a positive element, then a is self-adjoint.

• The set of positive elements

l{A} _{+} is a convex cone in the real vector space of the self-adjoint elements This means that \alpha a, a + b \in l{A} _{+} holds for all a,b \in l{A} and

• If

a \in l{A} _{+} is a positive element, then b ^{*ab} is also positive for every element

• For the linear span of

l{A} _{+} the following holds: \langle l{A} _{+} \rangle = l{A} ^{2} and

In C*-algebras

Let

l{A} be a C*-algebra. Then:

• Using the continuous functional calculus, for every

a \in l{A} _{+} and n \in N there is a uniquely determined b \in l{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 l{A} the element b ^{*b} is positive, this allows the definition of a unique absolute value:

• Products of commutative positive elements are also positive. So if

ab = ba holds for positive a,b \in l{A} _{+} , then

• Each element

a \in l{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 l{A} _{sa} = l{A} _{+} - l{A} _{+} , since

• If both

a and - a are positive a = 0

• If

l{B} is a C*-subalgebra of l{A} , then

• If

l{B} is another C*-algebra and \Phi is a *-homomorphism from l{A} to l{B} , then \Phi( l{A} _{+)} = \Phi( l{A}) \cap l{B} _{+}

• If

a,b \in l{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

l{A} be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements If b - a \in l{A} _{+} holds for a,b \in l{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 l{A} _{sa} with

If

l{A} is a C*-algebra, the partial order also has the following properties for a,b \in l{A} :

• If

a \leq b holds, then c ^{*ac} \leq c ^{*bc} is true for every For every c \in l{A} _{+} that commutates with a and b even ac \leq bc

• If

- b \leq a \leq b holds, then

• If

0 \leq a \leq b holds, then $a^\backslash alpha\; \backslash leq\; b^\backslash alpha$ 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

• Nonnegative matrix
• Positive operator (Hilbert space)

Citations

Bibliography

• Book: Blackadar, Bruce . Operator Algebras. Theory of C*-Algebras and von Neumann Algebras . Springer . Berlin/Heidelberg . 2006 . 3-540-28486-9 .
• Book: Dixmier, Jacques . C*-algebras . North-Holland . Amsterdam/New York/Oxford . 1977 . 0-7204-0762-1 . Jellett . Francis . English translation of Book: Dixmier, Jacques . 0 . Les C*-algèbres et leurs représentations . fr . Gauthier-Villars . 1969 .
• Book: Kadison . Richard V. . Ringrose . John R. . Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. . Academic Press . New York/London . 1983 . 0-12-393301-3.
• Book: Palmer, Theodore W. . Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. . Cambridge university press . 2001 . 0-521-36638-0 .