C*-algebra - Citizendia

C*-algebras are an important area of research in functional analysis, a branch of mathematics. For functional analysis as used in psychology see the Functional analysis (psychology article Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The prototypical example of a C*-algebra is a complex algebra, A, of linear operators on a complex Hilbert space with two additional properties:

A is a topologically closed set in the norm topology of operators. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This article assumes some familiarity with Analytic geometry and the concept of a limit. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of

A is closed under the operation of taking adjoints of operators. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator.

It is generally believed that C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical This line of research began in an extremely rudimentary form with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 Pascual Jordan (b October 18, 1902 in Hanover, Germany; d July 31, 1980 in Hamburg, Federal Republic Year 1933 ( MCMXXXIII) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar. Subsequently John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras. In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the

Around 1943, the work of Israel Gelfand, Mark Naimark and Irving Segal yielded an abstract characterisation of C*-algebras making no reference to operators. Year 1943 ( MCMXLIII) was a Common year starting on Friday (the link will display full 1943 calendar of the Gregorian calendar. Israïl Moiseevich Gelfand (Израиль Моисеевич Гельфанд ישראל געלפֿאַנד (born on) is a Mathematician who has contributed substantially Mark Aronovich Naimark (Марк Аронович Наймарк ( December 5, 1909 - December 30, 1978) was a Soviet Mathematician Irving Ezra Segal ( September 13, 1918 – December 24, 1998) was a mathematician known for work on theoretical Quantum mechanics.

C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.

1 Abstract characterization
2 Examples
3 Properties of C*-algebras
4 Type for C*-algebras
5 C*-algebras and quantum field theory
6 References
7 See also

Abstract characterization

We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gel'fand and Naimark.

A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map, * : A → A, called involution. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics and related technical fields the term map or mapping is often a Synonym for function. The image of an element x of A under involution is written x*. Involution has the following properties:

For all x, y in A:

$(x + y)^* = x^* + y^* \quad$

$(x y)^* = y^* x^*. \quad$

For every λ in C and every x in A:

$(\lambda x)^* = \overline{\lambda} x^*.$

For all x in A

$(x^*)^* = x. \quad$

The C*–identity holds for all x in A:

$\|x^* x \| = \|x\|^2.$

Note that the C* identity is equivalent to: for all x in A:

$\|x x^* \| = \|x\|^2.$

Any C*-algebra is automatically a B*-algebra, since the C* condition implies that

$\|x \| = \|x^*\|$

for all x in A. B*-algebras are mathematical structures studied in Functional analysis. However, not every B*-algebra is a C*-algebra.

The C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies the C*-norm is unique. In Mathematics, the spectral radius of a matrix or a Bounded linear operator is the Supremum among the Absolute values of the elements

A bounded linear map, π : A → B, between B*-algebras A and B is called a *-homomorphism if

For x and y in A

$\pi(x y) = \pi(x) \pi(y). \quad$

For x in A

$\pi(x^*) = \pi(x)^*. \quad$

In the case of C*-algebras, any *-homomorphism π between C*-algebras is non-expansive, i. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In the mathematical theory of Metric spaces a metric map or short map is a Continuous function between metric spaces that does not increase any e. bounded with norm ≤ 1. Furthermore, a *-homomorphism between C*-algebras is isometry. These are consequences of the C*-identity.

A bijective *-homomorphism π is called a C*-isomorphism, in which case A and B are said to be isomorphic.

Examples

Finite-dimensional C*-algebras

The algebra M_n(C) of n-by-n matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, Cⁿ, and use the operator norm ||. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of || on matrices. The involution is given by the conjugate transpose. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with More generally, one can consider finite direct sums of matrix algebras. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In fact, all finite dimensional C*-algebras are of this form. The self-adjoint requirement means finite-dimensional C*-algebras are semisimple, from which fact one can deduce the following theorem of Artin–Wedderburn type:

Theorem. In mathematics the term semisimple is used in a number of related ways within different subjects In Abstract algebra, the Artin–Wedderburn theorem is a Classification theorem for semisimple rings. A finite-dimensional C*-algebra, A, is canonically isomorphic to a finite direct sum

$A = \bigoplus_{e \in \min A } A e$

where min A is the set of minimal nonzero self-adjoint central projections of A. Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from

Each C*-algebra, Ae, is isomorphic (in a noncanonical way) to the full matrix algebra M_dim(e)(C). The finite family indexed on min A given by {dim(e)}_e is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra.

C*-algebras of operators

The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H → H. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. In fact, every C*-algebra, A, is *-isomorphic to a norm-closed adjoint closed subalgebra of B(H) for a suitable Hilbert space, H; this is the content of the Gelfand–Naimark theorem. In Mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of Bounded operators

Commutative C*-algebras

Let X be a locally compact Hausdorff space. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks The space C₀(X) of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*-algebra C₀(X) under pointwise multiplication and addition. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks The involution is pointwise conjugation. C₀(X) has a multiplicative unit element if and only if X is compact. As does any C*-algebra, C₀(X) has an approximate identity. In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda In the case of C₀(X) this is immediate: consider the directed set of compact subsets of X, and for each compact K let f_K be a function of compact support which is identically 1 on K. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces. In Topology, the Tietze extension theorem states that if X is a Normal topological space and f: A &rarr R {f_K}_K is an approximate identity.

The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra C₀(X), where X is the space of characters equipped with the weak* topology. In Mathematics, the Gelfand representation in Functional analysis (named after I In Mathematics, a character is (most commonly a special kind of function from a group to a field (such as the Complex numbers) In Mathematics, weak topology is an alternative term for Initial topology. Furthermore if C₀(X) is isomorphic to C₀(Y) as C*-algebras, it follows that X and Y are homeomorphic. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Topological equivalence redirects here see also Topological equivalence (dynamical systems. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs. Noncommutative topology in Mathematics is a term applied to the strictly C*-algebraic part of the Noncommutative geometry program Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the

C*-algebras of compact operators

Let H be a separable infinite-dimensional Hilbert space. The algebra K(H) of compact operators on H is a norm closed subalgebra of B(H). In Functional analysis, Compact operators on Hilbert spaces are a direct extension of matrices in the Hilbert spaces they are precisely the closure of Finite In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of It is also closed under involution; hence it is a C*-algebra.

Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras.

Theorem. If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {H_i}_{i ∈ I} such that A is isomorphic to the following direct sum

$\bigoplus_{i \in I } K(H_i),$

where the (C*-)direct sum consists of elements (T_i) of the Cartesian product Π K(H_i) with ||T_i|| → 0.

Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be easily displayed. In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda To be specific, H is isomorphic to the space of square summable sequences l²; we may assume that

$H = \ell^2. \quad$

For each natural number n let H_n be the subspace of sequences of l² which vanish for indices

$k \geq n$

and let

$e_n \quad$

be the orthogonal projection onto H_n. The sequence {e_n}_n is an approximate identity for K(H).

K(H) is a two-sided closed ideal of B(H). For separable Hilbert spaces, it is the unique ideal. The quotient of B(H) by K(H) is the Calkin algebra. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the In Functional analysis, the Calkin algebra is the quotient of B ( H) the ring of Bounded linear operators on a separable

C*-enveloping algebra

Given a B*-algebra A with an approximate identity, there is a unique (up to C*-isomorphism) C*-algebra E(A) and *-morphism π from A into E(A) which is universal, that is every other B*-morphism π': A → B factors uniquely through π. In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism E(A) is called the C*-enveloping algebra of the B*-algebra A.

Of particular importance is the C*-algebra of a locally compact group G. In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space. This is defined as the enveloping C*-algebra of the group algebra of G. In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach The C*-algebra of G provides context for general harmonic analysis of G in the case G is non-abelian. Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra. See spectrum of a C*-algebra. The spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of Unitary equivalence classes of irreducible

von Neumann algebras

von Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*-algebra. In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the They are required to be closed in the weak operator topology, which is weaker than the norm topology. In Functional analysis, the weak operator topology, often abbreviated WOT is the weakest Topology on the set of Bounded operators on a Hilbert space Their study is a specialized area of functional analysis.

Properties of C*-algebras

C*-algebras have a large number of properties which are technically convenient. These properties can be established by use the continuous functional calculus or by reduction to commutative C*-algebras. In Mathematics, the continuous functional calculus of Operator theory and C*-algebra theory allows applications of continuous functions to normal elements In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism. In Mathematics, the Gelfand representation in Functional analysis (named after I

The set of elements of a C*-algebra A of the form x*x forms a closed convex cone. In Linear algebra, a convex cone is a Subset of a Vector space that is closed under Linear combinations with positive coefficients This cone is identical to the elements of the form x x*. Elements of this cone are called non-negative (or sometimes positive, even though this terminology conflicts with its use for elements of R. )

The set of self-adjoint elements of a C*-algebra A naturally has the structure of a partially ordered vector space; the ordering is usually denoted ≥. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In this ordering, a self-adjoint element x of A satisfies x ≥ 0 if and only if the spectrum of x is non-negative. In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices Two self-adjoint elements x and y of A satisfy x ≥ y if x - y ≥ 0.

Any C*-algebra A has an approximate identity. In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda In fact, there is a directed family {e_λ}_{λ ∈ I} of self-adjoint elements of A such that

$x e_\lambda \rightarrow x$

$0 \leq e_\lambda \leq e_\mu \leq 1\quad \mbox{ whenever } \lambda \leq \mu.$

In case A is separable, A has a sequential approximate identity. In Mathematics and in Physics, separability may refer to properties of Separable spaces in Topology. More generally, A will have a sequential approximate identity if and only if A contains a strictly positive element, i. In Operator algebras a hereditary C*-subalgebra of a C*-algebra A is a particular type of C*-subalgebra whose structure is closely related to that of e. a positive element h such that hAh is dense in A.

Using approximate identities, one can show that the algebraic quotient of a C*-algebra by a closed proper two-sided ideal, with the natural norm, is a C*-algebra. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the iDEAL is an Internet payment method in The Netherlands, based on online banking

Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.

Type for C*-algebras

A C*-algebra A is of type I if and only if for all non-degenerate representations π of A the von Neumann algebra π(A)′′ (that is, the bicommutant of π(A)) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i. e. representations π for which π(A)′′ is a factor.

A locally compact group is said to be of type I if and only if its group C*-algebra is type I. In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach

However, if a C*-algebra has non-type I representations, then by results of James Glimm it also has representations of type II and type III. James Gilbert Glimm is an American Mathematical physicist, and Professor at the State University of New York at Stony Brook. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.

C*-algebras and quantum field theory

In quantum field theory, one typically describes a physical system with a C*-algebra A with unit element; the self-adjoint elements of A (elements x with x* = x) are thought of as the observables, the measurable quantities, of the system. In quantum field theory (QFT the forces between particles are mediated by other particles A state of the system is defined as a positive functional on A (a C-linear map φ : A → C with φ(u u*) > 0 for all u∈A) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).

See Local quantum physics. The Haag-Kastler Axiomatic framework for Quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local

References

W. Arveson, An Invitation to C*-Algebra, Springer-Verlag, 1976. ISBN 0387901760. An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article

A. Connes, Non-commutative geometry, ISBN 0-12-185860-X. Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.

J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969. ISBN 0720407621. This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.

G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972. ISBN 0471239003. Mathematically rigorous reference which provides extensive physics background.

A. I. Shtern (2001), “C* algebra”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

S. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Sakai, C*-algebras and W*-algebras , Springer (1971) ISBN 3-540-63633-1

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