C *-algebras are studied in the mathematical branch of functional analysis. It is an abstraction of bounded linear operators on a Hilbert space, therefore they play in the mathematical description of quantum mechanics play a role. C *-algebras are special Banach algebras, where a close relationship between algebraic and topological properties exists; the category of locally compact spaces turns out to be equivalent to the category of commutative C *-algebras, therefore the theory of C *-algebras is also considered as non-commutative topology.

Definition and properties

A C * - algebra over the body or is a Banach algebra with an involution with the following properties


From the C * - property follows that the involution is isometric, which makes together with the first three properties, a C * - algebra to a Banach *-algebra ( = involutive Banach algebra ).

One speaks of a commutative C * - algebra if the multiplication is commutative. Most authors mean by a C * - algebra is always a Banach algebra and write - accurate real C * - algebra if- Banach algebras are allowed.

Standard examples; the sets of Gelfand - Neumark and Gelfand - Neumark - Segal

The best known example of a C *-algebra is the algebra of bounded linear operators on a Hilbert space, and more generally, any closed in the norm topology of self-adjoint subalgebra of. Conversely, has, by the theorem of Gelfand - Neumark - Segal each C * - algebra of this form, so it is a isomorphic to a norm closed self-adjoint subalgebra.

The complex-valued, continuous and vanishing at infinity functions on a locally compact Hausdorff space form a commutative C * - algebra with respect to the supremum norm and the complex conjugation as involution. The Gelfand - Neumark states that every commutative C * - algebra is isomorphic to such an algebra of functions.

Other properties of C * - algebras

Homomorphisms between C *-algebras

Are and C *-algebras, then that means a picture * - homomorphism if it is linear, multiplicative and compatible with the involution.

Every * - homomorphism is contracting, that is, it holds for arbitrary, and hence in particular continuous.

Injective * - homomorphisms are automatically isometric, that is, it holds for arbitrary.

Finite dimensional C *-algebras

The algebra of complex matrices, which can be identified with the linear operators to form a C * - algebra with the operator norm. One can show that every finite-dimensional C * - algebra is isomorphic to a direct sum of such Matrixalgebren.

Construction of new C *-algebras from given

  • Direct sums, direct products, inductive Limites of C * - algebras are again C *-algebras with a suitable standard definition.
  • There is a minimum and a maximum possibility of tensor products of C * - algebras to complete to C *-algebras, see " Spatial tensor " and " Maximum tensor ".
  • In the article on dual C *-algebras restricted products are defined.
  • A closed two-sided ideal is automatically closed under the involution and the quotient algebra is the quotient norm again a C * - algebra.
  • For a C * -dynamical system, more C *-algebras can be constructed, the cross product and the reduced cross product.

Identity elements

C *-algebras must have no identity element. But you can always adjoin a unit element, or use as a replacement for a missing one element a limited approximation of the one that exists in every C * - algebra.

Hilbert space representations

Is a Hilbert space, it is called a * - homomorphism a Hilbert space representation, or simply displaying. The theory of Hilbert space representations is an important tool for further analysis of C * - algebras.

Examples and special cases of C * - algebras

  • The compact operators on a Hilbert space form a C * - subalgebra of. is a closed two- sided ideal in. The quotient algebra is called Calkin algebra.
  • Von Neumann algebras are strongly closed * - subalgebras of. Since the strong operator topology is weaker than the norm topology, the von Neumann algebras are complete in the norm topology and hence in particular C * - algebras.
  • Group C *-algebras ( = enveloping C * - algebra of the group algebra)
  • Dual C *-algebras (= C *-algebras of compact operators)
  • Liminal C *-algebras ( = CCR - algebras )
  • Postliminale C *-algebras ( = GCR - algebras or type IC *-algebras ), for example, the Toeplitz algebra
  • UHF - algebras, such as the CAR- algebra
  • AF-C *-algebras,
  • Bunce - Deddens algebras
  • Nuclear C *-algebras,
  • Irrational rotation algebra,
  • Cuntz algebra,
  • Cuntz - Krieger algebra,

Historical Remarks

A B *-algebra is by Gelfand and Neumark ( 1943) an involutive Banach algebra A ( with unit 1 ) with the two properties

A C * - algebra has been defined as a standard and completed with respect to the involution closed subalgebra of the algebra of operators on a Hilbert space. Gelfand and Neumark could then show that every B *-algebra is a C * - algebra. The already suspected them of redundancy of the second condition could be shown only in the 1950s by M. Fukamiya and I. Kaplansky. The term B *-algebra as an abstractly defined (ie, not represented on a Hilbert space ) algebra has become expendable by the Gelfand - Neumark, which is why the term B *-algebra only can still be found in older literature.

The C * - condition for all could be further weakened in the 1960s for all, which can be derived from the set of Vidav - Palmer, which in turn characterizes the C *-algebras among all Banach algebras. This weakening of the C * - condition, however, plays no special role in the theory of C * - algebras.