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In mathematics, specifically infunctional analysis, aC^∗-algebra (pronounced "C-star") is aBanach algebra together with aninvolution satisfying the properties of theadjoint. A particular case is that of acomplexalgebraA ofcontinuous linear operators on acomplexHilbert space with two additional properties:

• A is a topologicallyclosed set in thenorm topology of operators.
• A is closed under the operation of takingadjoints of operators.

Another important class of non-Hilbert C*-algebras includes the algebra${\displaystyle C_0(X)}$ of complex-valued continuous functions onX that vanish at infinity, whereX is alocally compactHausdorff space.

C*-algebras were first considered primarily for their use inquantum mechanics tomodel algebras of physicalobservables. This line of research began withWerner Heisenberg'smatrix mechanics and in a more mathematically developed form withPascual Jordan around 1933. Subsequently,John von Neumann attempted to establish a general framework for these algebras, which culminated in a series of papers onrings of operators. These papers considered a special class of C*-algebras that are now known asvon Neumann algebras.

Around 1943, the work ofIsrael Gelfand andMark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space.

C*-algebras are now an important tool in the theory ofunitary representations oflocally compact groups, and are also used in algebraic formulations of quantum mechanics. Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simplenuclear C*-algebras.

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

A C*-algebra,A, is aBanach algebra over the field ofcomplex numbers, together with amap$x\mapsto x^{\ast }$ for$x\in A$ with the following properties:

• It is aninvolution, for everyx inA:

${\displaystyle x^{\ast \ast }=(x^{\ast }{)}^{\ast }=x}$

• For allx,y inA:

${\displaystyle (x+y{)}^{\ast }=x^{\ast }+y^{\ast }}$

${\displaystyle (xy{)}^{\ast }=y^{\ast }{x}^{\ast }}$

• For every complex number${\displaystyle \lambda \in \mathbb{C}}$ and everyx inA:

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

• For allx inA:

${\displaystyle \Vert x{x}^{\ast }\Vert =\Vert x\Vert \Vert x^{\ast }\Vert .}$

Remark. The first four identities say thatA is a*-algebra. The last identity is called theC* identity and is equivalent to:

${\displaystyle \Vert x{x}^{\ast }\Vert =\Vert x{\Vert }^2,}$

which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see thehistory section below.

The C*-identity is a very strong requirement. For instance, together with thespectral radius formula, it implies that the C*-norm is uniquely determined by the algebraic structure:

${\displaystyle \Vert x{\Vert }^2=\Vert x^{\ast }x\Vert =sup\{|\lambda |:x^{\ast }x-\lambda 1\text{ is not invertible}\}.}$

Abounded linear map,π :A →B, between C*-algebrasA andB is called a*-homomorphism if

• Forx andy inA

${\displaystyle \pi (xy)=\pi (x)\pi (y)}$

• Forx inA

${\displaystyle \pi (x^{\ast })=\pi (x{)}^{\ast }}$

In the case of C*-algebras, any *-homomorphismπ between C*-algebras iscontractive, i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C*-algebras isisometric. These are consequences of the C*-identity.

A bijective *-homomorphismπ is called aC*-isomorphism, in which caseA andB are said to beisomorphic.

The term B*-algebra was introduced byC. E. Rickart in 1946 to describeBanach *-algebras that satisfy the condition:

• ${\displaystyle \Vert x{x}^{\ast }\Vert =\Vert x{\Vert }^2}$ for allx in the given B*-algebra. (B*-condition)

This condition automatically implies that the *-involution is isometric, that is,${\displaystyle \Vert x\Vert =\Vert x^{\ast }\Vert }$. Hence,${\displaystyle \Vert x{x}^{\ast }\Vert =\Vert x\Vert \Vert x^{\ast }\Vert }$, and therefore, a B*-algebra is also a C*-algebra. Conversely, the C*-condition implies the B*-condition. This is nontrivial, and can be proved without using the condition${\displaystyle \Vert x\Vert =\Vert x^{\ast }\Vert }$.^[1] For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C*-algebra'.

The term C*-algebra was introduced byI. E. Segal in 1947 to describe norm-closed subalgebras ofB(H), namely, the space of bounded operators on some Hilbert spaceH. 'C' stood for 'closed'.^[2][3] In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space".^[4]

C*-algebras have a large number of properties that are technically convenient. Some of these properties can be established by using thecontinuous functional calculus or by reduction to commutative C*-algebras. In the latter case, we can use the fact that the structure of these is completely determined by theGelfand isomorphism.

Self-adjoint elements are those of the form${\displaystyle x=x^{\ast }}$. The set of elements of a C*-algebraA of the form${\displaystyle x^{\ast }x}$ forms a closedconvex cone. This cone is identical to the elements of the form${\displaystyle x{x}^{\ast }}$. Elements of this cone are callednon-negative (or sometimespositive, even though this terminology conflicts with its use for elements of${\displaystyle \mathbb{R}}$)

The set of self-adjoint elements of a C*-algebraA naturally has the structure of apartially orderedvector space; the ordering is usually denoted${\displaystyle \ge }$. In this ordering, a self-adjoint element${\displaystyle x\in A}$ satisfies${\displaystyle x\ge 0}$ if and only if thespectrum of${\displaystyle x}$ is non-negative, if and only if${\displaystyle x=s^{\ast }s}$ for some${\displaystyle s\in A}$. Two self-adjoint elements${\displaystyle x}$ and${\displaystyle y}$ ofA satisfy${\displaystyle x\ge y}$ if${\displaystyle x-y\ge 0}$.

This partially ordered subspace allows the definition of apositive linear functional on a C*-algebra, which in turn is used to define thestates of a C*-algebra, which in turn can be used to construct thespectrum of a C*-algebra using theGNS construction.

Any C*-algebraA has anapproximate identity. In fact, there is a directed family {e_λ}_λ∈I of self-adjoint elements ofA such that

${\displaystyle x{e}_{\lambda }\to x}$

${\displaystyle 0\le e_{\lambda }\le e_{\mu }\le 1\text{ whenever }\lambda \le \mu .}$

In caseA is separable,A has a sequential approximate identity. More generally,A will have a sequential approximate identity if and only ifA contains astrictly positive element, i.e. a positive elementh such thathAh is dense inA.

Using approximate identities, one can show that the algebraicquotient of a C*-algebra by a closed proper two-sidedideal, with the natural norm, is a C*-algebra.

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

The algebra M(n,C) ofn ×nmatrices overC becomes a C*-algebra if we consider matrices as operators on the Euclidean space,Cⁿ, and use theoperator norm ||·|| on matrices. The involution is given by theconjugate transpose. More generally, one can consider finitedirect sums of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras aresemisimple, from which fact one can deduce the following theorem ofArtin–Wedderburn type:

Theorem. A finite-dimensional C*-algebra,A, iscanonically isomorphic to a finite direct sum

${\displaystyle A=\underset{e\in minA}{⨁}Ae}$

where minA is the set of minimal nonzero self-adjoint central projections ofA.

Each C*-algebra,Ae, is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(e),C). The finite family indexed on minA given by {dim(e)}_e is called thedimension vector ofA. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language ofK-theory, this vector is thepositive cone of theK₀ group ofA.

A†-algebra (or, more explicitly, a†-closed algebra) is the name occasionally used inphysics^[5] for a finite-dimensional C*-algebra. Thedagger, †, is used in the name because physicists typically use the symbol to denote aHermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) †-algebras feature prominently inquantum mechanics, and especiallyquantum information science.

An immediate generalization of finite dimensional C*-algebras are theapproximately finite dimensional C*-algebras.

The prototypical example of a C*-algebra is the algebraB(H) of bounded (equivalently continuous)linear operators defined on a complexHilbert spaceH; herex* denotes theadjoint operator of the operatorx :H →H. In fact, every C*-algebra,A, is *-isomorphic to a norm-closed adjoint closed subalgebra ofB(H) for a suitable Hilbert space,H; this is the content of theGelfand–Naimark theorem.

LetH be aseparable infinite-dimensional Hilbert space. The algebraK(H) ofcompact operators onH is anorm closed subalgebra ofB(H). 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. IfA is a C*-subalgebra ofK(H), then there exists Hilbert spaces {H_i}_i∈I such that

${\displaystyle A\cong \underset{i\in I}{⨁}K(H_i),}$

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

ThoughK(H) does not have an identity element, a sequentialapproximate identity forK(H) can be developed. To be specific,H is isomorphic to the space of square summable sequencesl²; we may assume thatH =l². For each natural numbern letH_n be the subspace of sequences ofl² which vanish for indicesk ≥n and lete_n be the orthogonal projection ontoH_n. The sequence {e_n}_n is an approximate identity forK(H).

K(H) is a two-sided closed ideal ofB(H). For separable Hilbert spaces, it is the unique ideal. Thequotient ofB(H) byK(H) is theCalkin algebra.

LetX be alocally compact Hausdorff space. The space${\displaystyle C_0(X)}$ of complex-valued continuous functions onX thatvanish at infinity (defined in the article onlocal compactness) forms a commutative C*-algebra${\displaystyle C_0(X)}$ under pointwise multiplication and addition. The involution is pointwise conjugation.${\displaystyle C_0(X)}$ has a multiplicative unit element if and only if${\displaystyle X}$ is compact. As does any C*-algebra,${\displaystyle C_0(X)}$ has anapproximate identity. In the case of${\displaystyle C_0(X)}$ this is immediate: consider the directed set of compact subsets of${\displaystyle X}$, and for each compact${\displaystyle K}$ let${\displaystyle f_K}$ be a function of compact support which is identically 1 on${\displaystyle K}$. Such functions exist by theTietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions${\displaystyle \{f_K\}}$ is an approximate identity.

TheGelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra${\displaystyle C_0(X)}$, where${\displaystyle X}$ is the space ofcharacters equipped with theweak* topology. Furthermore, if${\displaystyle C_0(X)}$ isisomorphic to${\displaystyle C_0(Y)}$ as C*-algebras, it follows that${\displaystyle X}$ and${\displaystyle Y}$ arehomeomorphic. This characterization is one of the motivations for thenoncommutative topology andnoncommutative geometry programs.

Given a Banach *-algebraA with anapproximate identity, there is a unique (up to C*-isomorphism) C*-algebraE(A) and *-morphism π fromA intoE(A) that isuniversal, that is, every other continuous *-morphismπ ' :A →B factors uniquely through π. The algebraE(A) is called theC*-enveloping algebra of the Banach *-algebraA.

Of particular importance is the C*-algebra of alocally compact groupG. This is defined as the enveloping C*-algebra of thegroup algebra ofG. The C*-algebra ofG provides context for generalharmonic analysis ofG in the caseG is non-abelian. In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra. Seespectrum of a C*-algebra.

Von Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in theweak operator topology, which is weaker than the norm topology.

TheSherman–Takeda theorem implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.

A C*-algebraA is of type I if and only if for all non-degenerate representations π ofA 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 itsgroup C*-algebra is type I.

However, if a C*-algebra has non-type I representations, then by results ofJames Glimm it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.

Inquantum mechanics, one typically describes a physical system with a C*-algebraA with unit element; the self-adjoint elements ofA (elementsx withx* =x) are thought of as theobservables, the measurable quantities, of the system. Astate of the system is defined as a positive functional onA (aC-linear map φ :A →C with φ(u*u) ≥ 0 for allu ∈A) such that φ(1) = 1. The expected value of the observablex, if the system is in state φ, is then φ(x).

This C*-algebra approach is used in the Haag–Kastler axiomatization oflocal quantum field theory, where every open set ofMinkowski spacetime is associated with a C*-algebra.

• Banach algebra
• Banach *-algebra
• *-algebra
• Hilbert C*-module
• Operator K-theory
• Operator system, a unital subspace of a C*-algebra that is *-closed.
• Gelfand–Naimark–Segal construction
• Jordan operator algebra

• Arveson, W. (1976),An Invitation to C*-Algebra, Springer-Verlag,ISBN 0-387-90176-0. An excellent introduction to the subject, accessible for those with a knowledge of basicfunctional analysis.
• Connes, Alain (1994),Non-commutative geometry, Gulf Professional,ISBN 0-12-185860-X. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
• Dixmier, Jacques (1969),Les C*-algèbres et leurs représentations, Gauthier-Villars,ISBN 0-7204-0762-1. 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.
• Doran, Robert S.; Belfi, Victor A. (1986),Characterizations of C*-algebras: The Gelfand-Naimark Theorems, CRC Press,ISBN 978-0-8247-7569-8.
• Emch, G. (1972),Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience,ISBN 0-471-23900-3. Mathematically rigorous reference which provides extensive physics background.
• A.I. Shtern (2001) [1994],"C*-algebra",Encyclopedia of Mathematics,EMS Press
• Sakai, S. (1971),C*-algebras and W*-algebras, Springer,ISBN 3-540-63633-1.
• Segal, Irving (1947), "Irreducible representations of operator algebras",Bulletin of the American Mathematical Society,53 (2):73–88,doi:10.1090/S0002-9904-1947-08742-5.

• C*-algebras
• Functional analysis

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