Positive element

In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form $a^{*}a$ .^[1]

Definition

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 ${\textstyle a=\sum _{k=1}^{n}a_{k}^{*}a_{k}}$ holds.^[1] This is also denoted by $a\geq 0$ .^[2]

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

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.

Examples

The unit element $e$ of an unital *-algebra is positive.
For each element $a\in {\mathcal {A}}$ , the elements $a^{*}a$ and $aa^{*}$ are positive by definition.^[1]

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

Let $a\in {\mathcal {A}}_{N}$ 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 $f(a)$ .^[3]
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 calculus.^[3]

Criteria

Let ${\mathcal {A}}$ be a C*-algebra and $a\in {\mathcal {A}}$ . Then the following are equivalent^[4]:

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

If ${\mathcal {A}}$ is an unital *-algebra with unit element $e$ , then in addition the following statements are equivalent^[5]:

$\left\|te-a\right\|\leq t$ 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 ${\mathcal {A}}$ be a *-algebra. Then:

If $a\in {\mathcal {A}}_{+}$ is a positive element, then $a$ is self-adjoint.^[6]
The set of positive elements ${\mathcal {A}}_{+}$ is a convex cone in the real vector space of the self-adjoint elements ${\mathcal {A}}_{sa}$ . This means that $\alpha a,a+b\in {\mathcal {A}}_{+}$ holds for all $a,b\in {\mathcal {A}}$ and $\alpha \in [0,\infty )$ .^[6]
If $a\in {\mathcal {A}}_{+}$ is a positive element, then $b^{*}ab$ is also positive for every element $b\in {\mathcal {A}}$ .^[7]
For the linear span of ${\mathcal {A}}_{+}$ the following holds: $\langle {\mathcal {A}}_{+}\rangle ={\mathcal {A}}^{2}$ and ${\mathcal {A}}_{+}-{\mathcal {A}}_{+}={\mathcal {A}}_{sa}\cap {\mathcal {A}}^{2}$ .^[8]

In C*-algebras

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

Using the continuous functional calculus, for every $a\in {\mathcal {A}}_{+}$ and $n\in \mathbb {N}$ there is an uniquely determined $b\in {\mathcal {A}}_{+}$ that satisfies $b^{n}=a$ , i.e. an 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 an unique absolute value: ${\textstyle |b|=(b^{*}b)^{\frac {1}{2}}}$ .^[9]
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 $\beta \in [0,\infty )$ . The mapping $\alpha \mapsto a^{\alpha }$ is continuous. Negative values for $\alpha$ are also possible for invertible elements $a$ .^[7]
Products of commutative positive elements are also positive. So if $ab=ba$ holds for positive $a,b\in {\mathcal {A}}_{+}$ , then $ab\in {\mathcal {A}}_{+}$ .^[5]
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 calculus.^[10] For it holds that ${\mathcal {A}}_{sa}={\mathcal {A}}_{+}-{\mathcal {A}}_{+}$ , since ${\mathcal {A}}^{2}={\mathcal {A}}$ .^[8]
If both $a$ and $-a$ are positive $a=0$ holds.^[5]
If ${\mathcal {B}}$ is a C*-subalgebra of ${\mathcal {A}}$ , then ${\mathcal {B}}_{+}={\mathcal {B}}\cap {\mathcal {A}}_{+}$ .^[5]
If ${\mathcal {B}}$ is another C*-algebra and $\Phi$ is a *-homomorphism from ${\mathcal {A}}$ to ${\mathcal {B}}$ , then $\Phi ({\mathcal {A}}_{+})=\Phi ({\mathcal {A}})\cap {\mathcal {B}}_{+}$ holds.^[11]
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 $a\bot b$ .^[12]

Partial Order

Let ${\mathcal {A}}$ be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements ${\mathcal {A}}_{sa}$ . If $b-a\in {\mathcal {A}}_{+}$ holds for $a,b\in {\mathcal {A}}$ , one writes $a\leq b$ or $b\geq a$ .^[13]

This partial order fulfills the properties $ta\leq tb$ and $a+c\leq b+c$ for all $a,b,c\in {\mathcal {A}}_{sa}$ with $a\leq b$ and $t\in [0,\infty )$ .^[8]

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

If $a\leq b$ holds, then $c^{*}ac\leq c^{*}bc$ is true for every $c\in {\mathcal {A}}$ . For every $c\in {\mathcal {A}}_{+}$ that commutates with $a$ and $b$ even $ac\leq bc$ holds.^[14]
If $-b\leq a\leq b$ holds, then $\left\|a\right\|\leq \left\|b\right\|$ .^[15]
If $0\leq a\leq b$ holds, then ${\textstyle a^{\alpha }\leq b^{\alpha }}$ holds for all real numbers $0<\alpha \leq 1$ .^[16]
If $a$ is invertible and $0\leq a\leq b$ holds, then $b$ is invertible and for the inverses $b^{-1}\leq a^{-1}$ holds.^[15]

Citations

References

1 2 3 Palmer 1977, p. 798.
↑ Blackadar 2006, p. 63.
1 2 Kadison & Ringrose 1983, p. 271.
↑ Kadison & Ringrose 1983, pp. 247–248.
1 2 3 4 Kadison & Ringrose 1983, p. 245.
1 2 Palmer 1977, p. 800.
1 2 Blackadar 2006, p. 64.
1 2 3 Palmer 1977, p. 802.
↑ Blackadar 2006, pp. 63–65.
↑ Kadison & Ringrose 1983, p. 247.
↑ Dixmier 1977, p. 18.
↑ Blackadar 2006, p. 67.
↑ Palmer 1977, p. 799.
↑ Kadison & Ringrose 1983, p. 249.
1 2 Kadison & Ringrose 1983, p. 250.
↑ Blackadar 2006, p. 66.

Bibliography

Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9.
Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3.
Palmer, Theodore W. (1994). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0.

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[FOOTNOTEPalmer1977798-1] 1 2 3 Palmer 1977, p. 798.

[FOOTNOTEBlackadar200663-2] Blackadar 2006, p. 63.

[FOOTNOTEKadisonRingrose1983271-3] 1 2 Kadison & Ringrose 1983, p. 271.

[FOOTNOTEKadisonRingrose1983247–248-4] Kadison & Ringrose 1983, pp. 247–248.

[FOOTNOTEKadisonRingrose1983245-5] 1 2 3 4 Kadison & Ringrose 1983, p. 245.

[FOOTNOTEPalmer1977800-6] 1 2 Palmer 1977, p. 800.

[FOOTNOTEBlackadar200664-7] 1 2 Blackadar 2006, p. 64.

[FOOTNOTEPalmer1977802-8] 1 2 3 Palmer 1977, p. 802.

[FOOTNOTEBlackadar200663–65-9] Blackadar 2006, pp. 63–65.

[FOOTNOTEKadisonRingrose1983247-10] Kadison & Ringrose 1983, p. 247.

[FOOTNOTEDixmier197718-11] Dixmier 1977, p. 18.

[FOOTNOTEBlackadar200667-12] Blackadar 2006, p. 67.

[FOOTNOTEPalmer1977799-13] Palmer 1977, p. 799.

[FOOTNOTEKadisonRingrose1983249-14] Kadison & Ringrose 1983, p. 249.

[FOOTNOTEKadisonRingrose1983250-15] 1 2 Kadison & Ringrose 1983, p. 250.

[FOOTNOTEBlackadar200666-16] Blackadar 2006, p. 66.

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Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem