Positive element

In mathematics, especially functional analysis, a self-adjoint (or Hermitian) element of a C*-algebra is called positive if its spectrum consists of non-negative real numbers. Moreover, an element of a C*-algebra is positive if and only if there is some in such that . A positive element is self-adjoint and thus normal.

If is a bounded linear operator on a complex Hilbert space , then this notion coincides with the condition that is non-negative for every vector in . Note that is real for every in if and only if is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint (and a self-adjoint everywhere defined operator on a Hilbert space is always bounded because of the Hellinger-Toeplitz theorem).

The set of positive elements of a C*-algebra forms a convex cone.

Positive and positive definite operators

A bounded linear operator on an inner product space is said to be positive (or positive semidefinite) if for some bounded operator on , and is said to be positive definite if is also non-singular.

(I) The following conditions for a bounded operator on to be positive semidefinite are equivalent:

• for some bounded operator on ,
• for some self-adjoint operator on ,

(II) The following conditions for a bounded operator on to be positive definite are equivalent:

• for some non-singular bounded operator on ,
• for some non-singular self-adjoint operator on ,
• is self adjoint and in .

(III) A complex matrix represents a positive (semi)definite operator if and only if is Hermitian (or self-adjoint) and , and are (strictly) positive real numbers.

Let the Banach spaces and be ordered vector spaces and let be a linear operator. The operator is called positive if for all in . For a positive operator we write .

A positive operator maps the positive cone of onto a subset of the positive cone of . If is the field of then is called a positive linear functional.

Many important operators are positive. For example:

• the Laplace operators and are positive,
• the limit and Banach limit functionals are positive,
• the identity and absolute value operators are positive,
• the integral operator with a positive measure is positive.

The Laplace operator is an example of an unbounded positive linear operator. Hence, by the Hellinger-Toeplitz theorem it cannot be everywhere defined.

Examples

• The following matrix is not positive definite since . However, is positive semidefinite since , and are non-negative.

Partial ordering using positivity

By introducing the convention

for self-adjoint elements in a C*-algebra , one obtains a partial order on the set of self-adjoint elements in . Note that according to this convention, we have if and only if is positive, which is convenient.

This partial order is analoguous to the natural order on the real numbers, but only to some extent. For example, it respects multiplication by positive reals and addition of self-adjoint elements, but need not hold for positive elements with and .