State (math)

A state is a mathematical term that is examined in functional analysis. It is about certain linear functionals on real or complex vector spaces that are normalized in a certain way. The definitions are often set up in such a way that the states with regard to an order structure are positive, that is, they map the positive elements of this order to non-negative real numbers. Furthermore, the state space , that is the set of states, forms a topologically or geometrically interesting space.

Involutive algebras

The most important case for applications is that of a state on an involutive algebra, which is explained as follows. Let it be a normalized algebra , with the field or standing for one on which an involution is also defined. ${\displaystyle A}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {}^{*}:A\rightarrow A,\,a\mapsto a^{*}}$

A state on is a continuous, linear functional with ${\displaystyle A}$ ${\displaystyle f:A\rightarrow \mathbb {K} }$

• ${\displaystyle f(a^{*}a)\geq 0}$ for all ${\displaystyle a\in A}$
• ${\displaystyle \|f\|=1}$.

The set of all states is called the state space and is often denoted by (S stands for the English word state ). Substituting the condition by , it is called a quasi state; the quasi-state space is the set of all quasi-states. If there is a single element , one also demands . ${\displaystyle S(A)}$ ${\displaystyle \|f\|=1}$ ${\displaystyle \|f\|\leq 1}$ ${\displaystyle Q(A)}$ ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle f(e)=1}$

Examples

Vector states

Let be an involutive subalgebra of , the algebra of bounded, linear operators on a Hilbert space with one element . If a vector of norm 1 is then defined by ${\displaystyle A}$ ${\displaystyle L(H)}$ ${\displaystyle H}$ ${\displaystyle e:=\mathrm {id} _{H}\in A}$ ${\displaystyle \xi \in H}$

${\displaystyle \omega _{\xi }(a):=\langle a\xi ,\xi \rangle }$ For ${\displaystyle a\in A}$

a state on the so-called by -defined vector state, because it applies to any ${\displaystyle \omega _{\xi }}$ ${\displaystyle A}$ ${\displaystyle \xi }$ ${\displaystyle a\in A}$

${\displaystyle \omega _{\xi }(a^{*}a)=\langle a^{*}a\xi ,\xi \rangle =\langle a\xi ,a\xi \rangle =\|a\xi \|^{2}\geq 0}$ and

${\displaystyle \|\omega _{\xi }\|=\sup _{a\in A,\|a\|\leq 1}|\omega _{\xi }(a)|=\sup _{a\in A,\|a\|\leq 1}|\langle a\xi ,\xi \rangle |\leq \sup _{a\in A,\|a\|\leq 1}\|a\|\|\xi \|^{2}=1}$

Equality applies here, because . Hence is a state. ${\displaystyle |\omega _{\xi }(e)|=\langle e\xi ,\xi \rangle =\|\xi \|^{2}=1}$ ${\displaystyle \omega _{\xi }}$

Is a number from the amount 1, a so-called especially physics phase factor, define so and the same state, because for is ${\displaystyle \lambda \in \mathbb {K} }$ ${\displaystyle \xi }$ ${\displaystyle \lambda \xi }$ ${\displaystyle a\in A}$

${\displaystyle \omega _{\lambda \xi }(a)=\langle a(\lambda \xi ),\lambda \xi \rangle =\lambda {\overline {\lambda }}\langle a\xi ,\xi \rangle =|\lambda |^{2}\langle a\xi ,\xi \rangle =\langle a\xi ,\xi \rangle =\omega _{\xi }(a)}$.

In quantum mechanics one identifies normalized to 1 Hilbert space vectors with quantum mechanical states, but really means that they define vector states, since the measurement of observables in the state is . This makes it clear that a Hilbert space vector uniquely defines a state except for a phase factor that occurs in the form . ${\displaystyle \xi }$ ${\displaystyle \langle a\xi ,\xi \rangle =\omega _{\xi }(a)}$ ${\displaystyle e^{i\omega t}}$

Spaces of measure

Let be the C * -algebra of continuous functions , the involution is defined by the complex conjugation . The dual space is well known, the space of the signed Borel measures , wherein the operation of such a measure on a continuous function by ${\displaystyle A=C[0,1]}$ ${\displaystyle [0,1]\rightarrow \mathbb {K} }$ ${\displaystyle \mu }$ ${\displaystyle f\in C[0,1]}$

${\displaystyle \mu (f):=\int f\mathrm {d} \mu }$

given is. There

${\displaystyle \mu ({\overline {f}}f)=\mu (|f|^{2})=\int |f|^{2}\mathrm {d} \mu }$,

are the states to exactly the positive Borel measures with the total variation norm . These considerations can be generalized to any algebras of C ₀ functions . ${\displaystyle C[0,1]}$ ${\displaystyle \|\mu \|=1}$

Local compact groups

Let it be the group algebra of a locally compact group , that is, the convolutionalgebra of the functions that can be integrated with regard to the left hair measure . The dual space is well known , that is, the space of essentially limited functions. A function operates on by definition ${\displaystyle L^{1}(G)}$ ${\displaystyle L^{\infty }(G)}$ ${\displaystyle L^{\infty }(G)}$ ${\displaystyle \varphi }$ ${\displaystyle L^{1}(G)}$

${\displaystyle \varphi (f):=\int \varphi \cdot f\mathrm {d} \mu }$,

where the Haarsche measure is. is a state on if and only if ${\displaystyle \mu }$ ${\displaystyle \varphi }$ ${\displaystyle L^{1}(G)}$

• ${\displaystyle \|\varphi \|=1}$
• ${\displaystyle \varphi }$almost everywhere agrees with a continuous, positive-definite function .

A function is called positive-definite if the matrix is positive definite for every finite number of elements . ${\displaystyle \varphi }$ ${\displaystyle (\varphi (s_{i}^{-1}s_{j}))_{i,j}}$ ${\displaystyle s_{1},\ldots ,s_{n}\in G}$

Meaning, GNS construction

A Hilbert space representation of an involutive Banach algebra is a * -homomorphism into the algebra of bounded linear operators on a Hilbert space . For the sake of simplicity, let's assume have a one element and it is . (If there is no unity element, one can adjoint one or consider algebras with an approximation of one .) If a vector state is on and , then a state is on , because ${\displaystyle \pi :A\rightarrow L(H)}$ ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle \pi (e)=\mathrm {id} _{H}}$ ${\displaystyle A}$ ${\displaystyle \omega _{\xi }}$ ${\displaystyle L(H)}$ ${\displaystyle \|\pi \|\leq 1}$ ${\displaystyle \omega _{\xi }\circ \pi }$ ${\displaystyle A}$

${\displaystyle (\omega _{\xi }\circ \pi )(a^{*}a)=\langle \pi (a^{*}a)\xi ,\xi \rangle =\langle \pi (a^{*})\pi (a)\xi ,\xi \rangle =\langle \pi (a)\xi ,\pi (a)\xi \rangle =\|\pi (a)\xi \|^{2}\geq 0}$.

The essential meaning of the states results from the fact that one can reverse this consideration, that is, one can come from a state to a Hilbert space representation and a vector , so that ${\displaystyle f}$ ${\displaystyle \pi _{f}:A\rightarrow L(H_{f})}$ ${\displaystyle \xi _{f}\in H_{f}}$

${\displaystyle f(a)=\langle \pi _{f}(a)\xi _{f},\xi _{f}\rangle }$for everyone . ${\displaystyle a\in A}$

For the construction, which is also called GNS construction according to Gelfand , Neumark and Segal , the left ideal is first formed for the state ${\displaystyle f}$

${\displaystyle N_{f}:=\{a\in A|f(a^{*}a)=0\}}$.

On the factor space is given by the formula ${\displaystyle A/N_{f}}$

${\displaystyle \langle a+N_{f},b+N_{f}\rangle _{f}:=f(b^{*}a)}$

defines a scalar product that turns into a Prähilbert space , the completion of which is a Hilbert space . Using the left ideal property of , one can show that each defines a continuous, linear mapping that clearly continues to a continuous, linear mapping . The mapping defined by this is a Hilbert space representation and with the definition ${\displaystyle \langle \cdot ,\cdot \rangle _{f}}$ ${\displaystyle A/N_{f}}$ ${\displaystyle H_{f}}$ ${\displaystyle N_{f}}$ ${\displaystyle a\in A}$ ${\displaystyle A/N_{f}\rightarrow A/N_{f},\,b+N_{f}\mapsto ab+N_{f}}$ ${\displaystyle \pi _{f}(a):H_{f}\rightarrow H_{f}}$ ${\displaystyle \pi _{f}:A\rightarrow L(H_{f})}$

${\displaystyle \xi _{f}:=e+N_{f}\in A/N_{f}\subset H_{f}}$

follows the desired relationship because for is . ${\displaystyle a\in A}$ ${\displaystyle \langle \pi _{f}(a)\xi _{f},\xi _{f}\rangle _{f}=\langle \pi _{f}(a)(e+N_{f}),e+N_{f}\rangle _{f}=\langle ae+N_{f},e+N_{f}\rangle _{f}=f(e^{*}ae)=f(a)}$

Each state can therefore be written as a vector state using a Hilbert space representation.

properties

C * algebras

For C * algebras with one element one can define states without reference to the involution. For the state space of such an algebra we have ${\displaystyle A}$

${\displaystyle S(A)=\{f\in A'|\,\|f\|=f(e)=1\}}$,

where denotes the dual space of . The property follows automatically. It is even more general for C * algebras without a unit element: ${\displaystyle A'}$ ${\displaystyle A}$ ${\displaystyle f(a^{*}a)\geq 0}$

If is a continuous, linear functional and holds for any 1-bounded approximation of the one of , then is a state. ${\displaystyle f\in A'}$ ${\displaystyle \textstyle \|f\|=\lim _{i\in I}f(e_{i})}$ ${\displaystyle (e_{i})_{i\in I}}$ ${\displaystyle A}$ ${\displaystyle f}$

Convex hull of the spectrum

Since the state space of a C * -algebra with one element is convex and weak - * - compact, and since for each the mapping is linear and weak - * - continuous, is also ${\displaystyle S(A)}$ ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle A'\rightarrow \mathbb {C} ,f\mapsto f(a)}$

${\displaystyle \{f(a);f\in S(A)\}\subset \mathbb {C} }$

convex and compact. One can show that the spectrum of is always contained in this set, that is, it is true ${\displaystyle \sigma (a)}$ ${\displaystyle a}$

${\displaystyle \mathrm {conv} (\sigma (a))\subset \{f(a);f\in S(A)\}}$,

where stands for the convex hull of a set. For normal elements of equality applies, in general, the inclusion is but really, as the example shows. The spectrum of this nilpotent element is so wrong with the convex hull own match, but the unit vector is not included in the convex hull of the spectrum. ${\displaystyle \mathrm {conv} }$ ${\displaystyle a={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\in L(\mathbb {C} ^{2})}$ ${\displaystyle a}$ ${\displaystyle \{0\}}$ ${\displaystyle \textstyle \xi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}}$ ${\displaystyle \textstyle \omega _{\xi }(a)={\frac {1}{2}}}$

Special conditions

Normal conditions

→ Main article : Normal condition

On Von Neumann algebras one has other operator topologies in addition to the standard topology and it is therefore of interest which states with regard to these topologies are continuous. The ultra- weak-steady states are called normal, they are exactly those that can be written as the countable sum of multiples of vector states. They can be characterized in different ways and play an important role in the theory of Von Neumann algebras, especially because the GNS construction leads to a homomorphism between Von Neumann algebras.

Faithful states

A state is called loyal if it already follows. In this case the left ideal from the GNS construction is equal to the zero ideal and the construction is considerably simplified, the constructed representation is true, that is, injective. There are always faithful states on separable C * algebras. The existence of faithful, normal states characterizes the σ-finite Von Neumann algebras . ${\displaystyle f}$ ${\displaystyle f(a^{*}a)=0}$ ${\displaystyle a=0}$ ${\displaystyle N_{f}}$

Pure states

The quasi-state space is convex and weak- * - compact , so it has many extremal points according to the Kerin-Milman theorem . The extremal points of the quasi-state space different from 0 are states and are called pure states , since they cannot be mixtures, i.e. convex combinations , of other states.

In the case of commutative C * algebras , the pure states are exactly the * homomorphisms . In the case of non-commutative C * algebras, the pure states are precisely those whose GNS construction leads to irreducible representations . ${\displaystyle A}$ ${\displaystyle A\rightarrow \mathbb {C} }$

Banach algebras

The characterization of the states on a C * -algebra with one element as such continuous, linear functionals, for which applies, can be transferred to any Banach algebra with one element. One defines ${\displaystyle \|f\|=f(e)=1}$ ${\displaystyle A}$

${\displaystyle S(A):=\{f\in A'|\,\|f\|=f(e)=1\}}$,

${\displaystyle V(a):=\{f(a)|\,f\in S(A)\}\subset \mathbb {C} }$ for a ${\displaystyle a\in A}$

${\displaystyle S(A)}$is called state space, numerical range of values . As in the case of the C * algebras described above, it is a convex, compact subset of the complex plane that encompasses the spectrum of . This concept formation has many applications in the theory of Banach algebras, in particular it leads to characterizations of the C * algebras among all Banach algebras ( Vidav-Palmer theorem ). ${\displaystyle V(a)}$ ${\displaystyle V(a)}$ ${\displaystyle a}$

Ordered vector spaces

If there is an ordered vector space with a unit of order , a linear functional is called a state, if and for all . The state space, i.e. the set of all states, is convex, the extreme points of this set are called pure states. A state is pure if and only if for every linear functional with for all it already follows that . ${\displaystyle V}$ ${\displaystyle e}$ ${\displaystyle f:V\rightarrow \mathbb {R} }$ ${\displaystyle f(e)=1}$ ${\displaystyle f(v)\geq 0}$ ${\displaystyle v\in V,v\geq 0}$ ${\displaystyle g:V\rightarrow \mathbb {R} }$ ${\displaystyle 0\leq g(v)\leq f(v)}$ ${\displaystyle v\in V,v\geq 0}$ ${\displaystyle g=g(e)\cdot f}$

If one takes as the space of the self-adjoint elements of a C * -algebra with one element , it also functions as a unit of order. One is thus in the situation of the states on C * algebras described above. ${\displaystyle V}$ ${\displaystyle e}$ ${\displaystyle e}$

Ordered groups

The notion of state can even be generalized to ordered Abelian groups . If such a group has a positive semigroup and a scale is marked in it, a figure is called a state if the following applies: ${\displaystyle G}$ ${\displaystyle G^{+}}$ ${\displaystyle \Gamma (G)\subset G^{+}}$ ${\displaystyle f:G\rightarrow \mathbb {R} }$

• ${\displaystyle f}$is a group homomorphism into the additive group of real numbers,
• ${\displaystyle f(G^{+})\subset \mathbb {R} _{0}^{+}}$,
• ${\displaystyle f(\Gamma (G))=f(G^{+})\cap [0,1]}$.

The application case that is important for the C * algebra theory is the K ₀ group of a C * algebra, in particular of AF-C * algebras . States of the K ₀ group belong to traces on the AF-C * algebras.

See also

• Weight (functional analysis)
• State (quantum mechanics)
• Kadison-Singer problem

Individual evidence

1. ^ J. Dixmier: C * -algebras and their representations , North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , definition 2.1.1
2. ^ J. Dixmier: C * -algebras and their representations , North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , Theorem 13.4.5
3. ^ J. Dixmier: C * -algebras and their representations. North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , set 2.4.4.
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc. (1979), ISBN 0125494505 , Chapter 3.3 The Gelfand-Naimark-Segal construction.
5. ^ J. Dixmier: C * -algebras and their representations , North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , sentence 2.1.9
6. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Lemma I.9.9
7. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras I , 1983, ISBN 0-12-393301-3 , Theorem 4.3.3
8. ^ FF Bonsall , J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §38, Lemma 3, Lemma 4
9. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 3.7.2 - 3.7.4
10. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras I , 1983, ISBN 0-12-393301-3 , Theorem 3.4.7
11. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Chapter 3.13 Pure states and irreducible representations
12. ^ J. Dixmier: C * -algebras and their representations , North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , Chapter 2.5: Pure forms and irreducible representations
13. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §10, definition 1
14. ^ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Cambridge University Press (1971), ISBN 0-521-07988-8
15. ↑ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras I , 1983, ISBN 0-12-393301-3 , Definition 3.4.5 + Lemma 3.4.6
16. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Chapter IV.5, page 114
17. ^ B. Blackadar: K-Theory for Operator-Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , definition 6.8.1
18. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Theorem IV.5.3