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In mathematics, specifically in functional analysis, a **C ^{∗}-algebra**(pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra

- Abstract characterization
- Some history: B*-algebras and C*-algebras
- Structure of C*-algebras
- Self-adjoint elements
- Quotients and approximate identities
- Examples
- Finite-dimensional C*-algebras
- C*-algebras of operators
- C*-algebras of compact operators
- Commutative C*-algebras
- C*-enveloping algebra
- Von Neumann algebras
- Type for C*-algebras
- C*-algebras and quantum field theory
- See also
- Notes
- References

*A*is a topologically closed set in the norm topology of operators.*A*is closed under the operation of taking adjoints of operators.

Another important class of non-Hilbert C*-algebras includes the algebra of complex-valued continuous functions on *X* that vanish at infinity, where *X* is a locally compact Hausdorff space.

C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. 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.

Around 1943, the work of Israel Gelfand and Mark 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 of unitary representations of locally 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 simple nuclear 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 a Banach algebra over the field of complex numbers, together with a map for with the following properties:

- It is an involution, for every
*x*in*A*:

- For all
*x*,*y*in*A*:

- For every complex number λ in
**C**and every*x*in*A*:

- For all
*x*in*A*:

**Remark.** The first three identities say that *A* is a *-algebra. The last identity is called the **C* identity** and is equivalent to:

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

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

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

- For
*x*and*y*in*A*

- For
*x*in*A*

In the case of C*-algebras, any *-homomorphism *π* between C*-algebras is contractive, i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C*-algebras is isometric. 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**.

The term B*-algebra was introduced by C. E. Rickart in 1946 to describe Banach *-algebras that satisfy the condition:

- for all
*x*in the given B*-algebra. (B*-condition)

This condition automatically implies that the *-involution is isometric, that is, . Hence, , 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 .^{ [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 by I. E. Segal in 1947 to describe norm-closed subalgebras of *B*(*H*), namely, the space of bounded operators on some Hilbert space *H*. '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 the continuous 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 the Gelfand isomorphism.

Self-adjoint elements are those of the form *x*=*x**. The set of elements of a C*-algebra *A* of the form *x*x* forms a closed convex cone. This cone is identical to the elements of the form *xx**. 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 this ordering, a self-adjoint element *x* of *A* satisfies *x* ≥ 0 if and only if the spectrum of *x* is non-negative,^{[ clarification needed ]} if and only if *x* = *s*s* for some *s*. Two self-adjoint elements *x* and *y* of *A* satisfy *x* ≥ *y* if *x*−*y* ≥ 0.

This partially ordered subspace allows the definition of a positive linear functional on a C*-algebra, which in turn is used to define the states of a C*-algebra, which in turn can be used to construct the spectrum of a C*-algebra using the GNS construction.

Any C*-algebra *A* has an approximate identity. In fact, there is a directed family {*e*_{λ}}_{λ∈I} of self-adjoint elements of *A* such that

- In case
*A*is separable,*A*has a sequential approximate identity. More generally,*A*will have a sequential approximate identity if and only if*A*contains a**strictly positive element**, i.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.

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

The algebra M(*n*, **C**) of *n* × *n* matrices over **C** becomes a C*-algebra if we consider matrices as operators on the Euclidean space, **C**^{n}, and use the operator norm ||·|| on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct 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 are semisimple, from which fact one can deduce the following theorem of Artin–Wedderburn type:

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

where min

Ais 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 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. In the language of K-theory, this vector is the positive cone of the *K*_{0} group of *A*.

A **†-algebra** (or, more explicitly, a *†-closed algebra*) is the name occasionally used in physics ^{ [5] } for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a Hermitian 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 in quantum mechanics, and especially quantum information science.

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

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 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.

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*). 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.IfAis a C*-subalgebra ofK(H), then there exists Hilbert spaces {H}_{i}_{i∈I}such that

where the (C*-)direct sum consists of elements (

T) of the Cartesian product Π_{i}K(H) with ||_{i}T|| → 0._{i}

Though *K*(*H*) does not have an identity element, a sequential approximate identity for *K*(*H*) can be developed. To be specific, *H* is isomorphic to the space of square summable sequences *l*^{2}; we may assume that *H* = *l*^{2}. For each natural number *n* let *H _{n}* be the subspace of sequences of

*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.

Let *X* be a locally compact Hausdorff space. The space of complex-valued continuous functions on *X* that *vanish at infinity* (defined in the article on local compactness) form a commutative C*-algebra under pointwise multiplication and addition. The involution is pointwise conjugation. has a multiplicative unit element if and only if is compact. As does any C*-algebra, has an approximate identity. In the case of this is immediate: consider the directed set of compact subsets of , and for each compact let be a function of compact support which is identically 1 on . Such functions exist by the Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions is an approximate identity.

The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra , where is the space of characters equipped with the weak* topology. Furthermore, if is isomorphic to as C*-algebras, it follows that and are homeomorphic. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs.

Given a Banach *-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*) that is universal, that is, every other continuous *-morphism π ' : *A* → *B* factors uniquely through π. The algebra **E**(*A*) is called the **C*-enveloping algebra** of the Banach *-algebra *A*.

Of particular importance is the C*-algebra of a locally compact group *G*. This is defined as the enveloping C*-algebra of the group algebra of *G*. The C*-algebra of *G* provides context for general harmonic analysis of *G* in the case *G* 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. See spectrum 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 the weak operator topology, which is weaker than the norm topology.

The Sherman–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*-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.

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. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.

In quantum mechanics, 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. 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*).

This C*-algebra approach is used in the Haag-Kastler axiomatization of local quantum field theory, where every open set of Minkowski 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

- ↑ Doran & Belfi 1986 , pp. 5–6, Google Books.
- ↑ Doran & Belfi 1986 , p. 6, Google Books.
- ↑ Segal 1947
- ↑ Segal 1947 , p. 75
- ↑ John A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction."
*Quantum Information Processing*. Volume 2, Number 5, pp. 381–419. Oct 2003.

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, especially functional analysis, a **Banach algebra**, named after Stefan Banach, is an associative algebra *A* over the real or complex numbers that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy

In abstract algebra, the **direct sum** is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

In mathematics, particularly linear algebra and functional analysis, a **spectral theorem** is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In mathematics, a **trace-class** operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.

In mathematics, a **von Neumann algebra** or **W*-algebra** is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.

In functional analysis, a discipline within mathematics, given a C*-algebra *A*, the **Gelfand–Naimark–Segal construction** establishes a correspondence between cyclic *-representations of *A* and certain linear functionals on *A*. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In mathematics, **spectral theory** is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

In functional analysis, a branch of mathematics, a **compact operator** is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of . Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces.

In mathematics, **operator theory** is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

In functional analysis, a branch of mathematics, the **Borel functional calculus** is a *functional calculus*, which has particularly broad scope. Thus for instance if *T* is an operator, applying the squaring function *s* → *s*^{2} to *T* yields the operator *T*^{2}. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential

In mathematics, the **spectrum of a C*-algebra** or **dual of a C*-algebra***A*, denoted *Â*, is the set of unitary equivalence classes of irreducible *-representations of *A*. A *-representation π of *A* on a Hilbert space *H* is **irreducible** if, and only if, there is no closed subspace *K* different from *H* and {0} which is invariant under all operators π(*x*) with *x* ∈ *A*. We implicitly assume that irreducible representation means *non-null* irreducible representation, thus excluding trivial representations on one-dimensional spaces. As explained below, the spectrum *Â* is also naturally a topological space; this is similar to the notion of the spectrum of a ring.

In mathematics, the **essential spectrum** of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".

In functional analysis, the concept of a **compact operator on Hilbert space** is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

**Hilbert C*-modules** are mathematical objects that generalise the notion of a Hilbert space, in that they endow a linear space with an "inner product" that takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras. In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.

In mathematics, a **commutation theorem** explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacques Dixmier as part of the theory of **Hilbert algebras**. It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.

The mathematical concept of a **Hilbert space**, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.

In mathematics, **Jordan operator algebras** are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called **Jordan Banach algebras**. The theory has been extensively developed only for the subclass of **JB algebras**. The axioms for these algebras were devised by Alfsen, Schultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called **JC algebras**. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called **JB* algebras** or **Jordan C* algebras**. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of **JBW algebras**. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called **JW algebras**. The JBW algebras with trivial center, so-called **JBW factors**, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the **spin factors**, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.

This is a glossary for the terminology in a mathematical field of functional analysis.

- 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 basic functional analysis. - Connes, Alain,
*Non-commutative geometry*, 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 .

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