1 Definition

C*-algebrasMathworldPlanetmathPlanetmath are a type of involutive Banach algebras which arise in the study of operators on Hilbert spacesMathworldPlanetmath, Lie group representations, locally compact topological spaces, knots, noncommutative , among other topics in mathematics and theoretical physics . Their study was initiated in the 1930’s with the purpose of axiomatizing quantum mechanics, and still today, C*-algebras play a decisive role in formulations of quantum statistical mechanics and quantum .

The defining property of these algebras is that the norm and the involution are related in a very special way.

Definition 1 - A C*-algebra A is a Banach *-algebra such that a*a=a2 for all aA.

The equality in Definition 1 is sometimes called the C* axiom. It turns out that one can weaken this condition and still specify the same of algebras.

Definition 2 - A C*-algebra A is a Banach algebraMathworldPlanetmath with an antilinear involution * such that a2a*a for all aA.

Definition 3 - A C*-algebra A is a Banach algebra with an antilinear involution * such that a*a=a*a

2 C* Norm

C*-algebras are a very peculiar type of topological algebrasMathworldPlanetmath. The C* axiom, deceptively , imposes severe on the the algebraic and topological structure of a C*-algebra.

A most striking consequence of the C* axiom is that the norm is solely determined by the algebraic structurePlanetmathPlanetmath of the algebra. More specifically,


where Rσ(x) denotes the spectral radius of the element x𝒜. For C* algebras with an identity elementMathworldPlanetmath e we can specify even further: the norm of an element a𝒜 is determined by

a2=sup{|λ|:λanda*a-λeis not invertible}

This also implies that the norm in a C*-algebra is unique, in the sense that there is no other norm in the algebra that satisfies that C* axiom, i.e. that turns the algebra into a C*-algebra. This is a stark contrast to the case of general normed algebras, where one may find many norms which are with the algebraic structure.

Moreover, the C* norm occupies a unique amongst the possible norms for an involutive algebra. Suppose that 𝒜 is a C* algebra with norm C*. If B is any other norm for which 𝒜 is a Banach *-algebra, then we must have


Hence we see that the C* norm enjoys an extremal property — it is the smallest possible norm for which 𝒜 is a Banach *-algebra.

There are many other surprising consequences of the C* axiom, like: *-homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between C*-algebras are automatically continuousMathworldPlanetmathPlanetmath and every C*-algebra is semi-simple, which again are not true for general involutive algebras.

3 Elements of a C*-algebra

Like in involutory rings, there are some special elements in C*-algebras that deserve some attention. We recall some definitions here:

Let 𝒜 be a C*-algebra with identity element e. An element a𝒜 is said to be

  • self-adjoint if a*=a

  • unitaryPlanetmathPlanetmath if a*a=aa*=e

  • positivePlanetmathPlanetmath if a=b*b for some element b𝒜

It is many times useful to have some interpretationMathworldPlanetmathPlanetmath for this elements. One of this interpretations comes from complex analysis: we regard the elements of a C*-algebra as functions with values in and the involution as complex conjugation.

In this frame, self-adjoint elements correspond to real functions, unitary elements correspond to functions whose values lie in the unit circle in and positive elementsMathworldPlanetmathPlanetmath correspond to positive functions (functions with values in 𝟘+).

It is easily seen that self-adjoint elements are closed underPlanetmathPlanetmath addition, multiplication and multiplication by real numbers. It can be proven the same for positive elements (with multiplication by positive numbers).

There are some decompositions of elements in a C*-algebra analogous to some decompositions in complex analysis. For instance, every element a in a C*-algebra has a unique decomposition of the form


where x,y are self-adjoint. This is similar to the decomposition of a complex valued function in its real and imaginary parts.

Moreover, every self-adjoint element a is of the form


where x,y are positive elements. This is similar to the decomposition of real valued functions in its positive and negative parts.

There are many other aspects of the theory of C*-algebras for which this kind of interpretation proves to be very insightful.

For example, C*-algebras happen to have a natural partial ordering. One can define an ordering by declaring that x>y when x-y is positive. Given this ordering, one can then speak of such things as monotonic functionsMathworldPlanetmath, monotonic sequences, and positive linear functionalsMathworldPlanetmath on the algebra. These notions, in turn, prove to be extremely useful in the study of C*-algebras.

4 Examples

Having discussed these algebras in general terms, it is high time that we illustrate the definition with some examples.

Example 1

As our first class of examples, we consider algebras of functions. Let X be a compactPlanetmathPlanetmath Hausdorff topological space and let C(X) be the algebra of continuous functions from X to . For the involution operation, we take pointwise complex conjugation and for the norm we take the norm of uniform convergence:


It is a routine matter to check that the norm and involution satisfy the appropriate algebraic requirements. Completeness under this norm follows from the fact that the uniform limit of continuous functions on a locally compact Hausdorff topological space is continuous.

More generally, instead of a compact space, we can take a locally compact Hausdorff spacePlanetmathPlanetmath X and consider the algebra C0(X) of continuous functions X that vanish at infinity, endowed with the same norm and involution. These are important examples of C*-algebras.

Example 2

As our second class of examples, we consider operator algebras. Let H be a complex Hilbert space with inner product , and let B(H) be the algebra of bounded operatorsMathworldPlanetmathPlanetmath on H. For the involution, we take the adjoint operation and as a norm we take the usual operator norm:


Again, it is straightforward to verify that the norm and involution satisfy the appropriate algebraic requirements, as is done in an attachment to this entry. Completeness under the norm follows from a well-known theorem of functional analysisMathworldPlanetmath.

5 Commutative and noncommutative C*-algebras

The algebras C0(X) in Example 1 above are more than just an example. In fact, all commutativePlanetmathPlanetmathPlanetmathPlanetmath C*-algebras are *-isomorphic to C0(X) for some locally compact Hausdorff space X. Moreover, X is compact if and only if the C*-algebra has an identity element. This is the content of the Gelfand-Naimark theoremMathworldPlanetmathPlanetmath.

Furthermore, there is a correspondence between properties of the topological spaceMathworldPlanetmath and properties of the C*-algebra. For example: a compactification of the space corresponds to a unitizationMathworldPlanetmath of the C*-algebra; the space is connectedPlanetmathPlanetmath if and only if the C*-algebra has no non-trivial projections, among many other interesting correspondences.

For this reason, the theory of (noncommutative) C*-algebras is many times called noncommutative topologyMathworldPlanetmathPlanetmathPlanetmath (click on the link for more information).

The second example is also more than just an example of C*-algebras. In fact, by the Gelfand-Naimark representation theorem, all C*-algebras are *-isomorphic to a norm closed *-subalgebraMathworldPlanetmathPlanetmath of B(H), for some Hilbert space H.

Note, however, that this does not provide a “classification” of C*-algebras since we do not know in general what are the closed *-subalgebras of B(H). This is merely a (very-important) structural theorem. The classification problem for C*-algebras is still open.

6 Additional Examples

Example 3

Compact operatorsMathworldPlanetmath in a Hilbert space H form a closed ideal of B(H). Moreover, this ideal is also closed for the involution of operators. Hence, the algebra of compact operators, K(H), is a C*-algebra.

Example 4

Let (X,𝔅,μ) be a measure space. The space L(X) (http://planetmath.org/LpSpace) is an algebra under pointwise operations. We can define an involution again by complex conjugation and we consider the essential supremum norm . It can be readily verified that, under these operations and norm, L(X) is a C*-algebra.

The algebras L(X) are also particularly important since they are examples of von Neumann algebrasMathworldPlanetmathPlanetmathPlanetmath, which are a specific kind of C*-algebras.

Title C*-algebra
Canonical name Calgebra
Date of creation 2013-03-22 12:57:55
Last modified on 2013-03-22 12:57:55
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 34
Author asteroid (17536)
Entry type Definition
Classification msc 46L05
Classification msc 46L87
Synonym C*-algebra
Synonym C* algebra
Related topic GroupCAlgebra
Related topic VonNeumannAlgebra
Related topic NoncommutativeGeometry
Related topic GroupoidCConvolutionAlgebra
Related topic GroupoidCDynamicalSystem
Related topic CAlgebra3
Related topic NuclearCAlgebra
Related topic HomomorphismsOfCAlgebrasAreContinuous
Related topic ContinuousLinearMapping
Related topic OperatorNorm
Related topic C_cG
Related topic UniformContinuityOverLocallyCompa
Defines C* axiom