C*-algebra

A Banach algebra over the field of complex numbers, with an involution , , such that the norm and the involution are connected by the relation for any element . -algebras were introduced in 1943 [1] under the name of totally regular rings; they are also known under the name of -algebras. The most important examples of -algebras are:

1) The algebra of continuous complex-valued functions on a locally compact Hausdorff space which tend towards zero at infinity (i.e. continuous functions on such that, for any , the set of points which satisfy the condition is compact in ); has the uniform norm

The involution in is defined as transition to the complex-conjugate function: . Any commutative -algebra is isometrically and symmetrically isomorphic (i.e. is isomorphic as a Banach algebra with involution) to the -algebra , where is the space of maximal ideals of endowed with the Gel'fand topology [1], [2], [3].

2) The algebra of all bounded linear operators on a Hilbert space , considered with respect to the ordinary linear operations and operator multiplication. The involution in is defined as transition to the adjoint operator, and the norm is defined as the ordinary operator norm.

A subset is said to be self-adjoint if , where . Any closed self-adjoint subalgebra of a -algebra is a -algebra with respect to the linear operations, multiplication, involution, and norm taken from ; is said to be a -subalgebra of . Any -algebra is isometrically and symmetrically isomorphic to a -subalgebra of some -algebra of the form . Any closed two-sided ideal in a -algebra is self-adjoint (thus is a -subalgebra of ), and the quotient algebra , endowed with the natural linear operations, multiplication, involution, and quotient space norm, is a -algebra. The set of completely-continuous linear operators on a Hilbert space is a closed two-sided ideal in . If is a -algebra and is the algebra with involution obtained from by addition of a unit element, there exists a unique norm on which converts into a -algebra and which extends the norm on . Moreover, the operations of bounded direct sum and tensor product [3], [4] have been defined for -algebras.

As in all symmetric Banach algebras with involution, in a -algebra it is possible to define the following subsets: the real linear space of Hermitian elements; the set of normal elements; the multiplicative group of unitary elements (if contains a unit element); and the set of positive elements. The set is a closed cone in , , , and the cone converts into a real ordered vector space. If contains a unit element 1, then 1 is an interior point of the cone . A linear functional on is called positive if for all ; such a functional is continuous. If , where is a -subalgebra of , the spectrum of in coincides with the spectrum of in . The spectrum of a Hermitian element is real, the spectrum of a unitary element lies on the unit circle, and the spectrum of a positive element is non-negative. A functional calculus for the normal elements of a -algebra has been constructed. Any -algebra has an approximate unit, located in the unit ball of and formed by positive elements of . If are closed two-sided ideals in , then is a closed two-sided ideal in and . If is a closed two-sided ideal in and is a closed two-sided ideal in , then is a closed two-sided ideal in . Any closed two-sided ideal is the intersection of the primitive two-sided ideals in which it is contained; any closed left ideal in is the intersection of the maximal regular left ideals in which it is contained.

Any *-isomorphism of a -algebra is isometric. Any *-isomorphism of a Banach algebra with involution into a -algebra is continuous, and for all . In particular, all representations of a Banach algebra with involution (i.e. all *-homomorphism of into a -algebra of the form ) are continuous. The theory of representations of -algebras forms a significant part of the theory of -algebras, and the applications of the theory of -algebras are related to the theory of representations of -algebras. The properties of representations of -algebras make it possible to construct for each -algebra a topological space , called the spectrum of the -algebra , and to endow this space with a Mackey–Borel structure. In the general case, the spectrum of a -algebra does not satisfy any separation axiom, but is a locally compact Baire space.

A -algebra is said to be a CCR-algebra (respectively, a GCR-algebra) if the relation (respectively, ) is satisfied for any non-null irreducible representation of the -algebra in a Hilbert space .

A -algebra is said to be an NGCR-algebra if does not contain non-zero closed two-sided -ideals (i.e. ideals which are -algebras). Any -algebra contains a maximal two-sided -ideal , and the quotient algebra is an -algebra. Any -algebra contains an increasing family of closed two-sided ideals , indexed by ordinals , , such that , , is a -algebra for all , and for limit ordinals . The spectrum of a -algebra contains an open, everywhere-dense, separable, locally compact subset.

A -algebra is said to be a -algebra of type I if, for any representation of the -algebra in a Hilbert space , the von Neumann algebra generated by the family in is a type I von Neumann algebra. For a -algebra, the following conditions are equivalent: a) is a -algebra of type I; b) is a -algebra; and c) any quotient representation of the -algebra is a multiple of the irreducible representation. If satisfies these conditions, then: 1) two irreducible representations of the -algebra are equivalent if and only if their kernels are identical; and 2) the spectrum of the -algebra is a -space. If is a separable -algebra, each of the conditions 1) and 2) is equivalent to the conditions a)–c). In particular, each separable -algebra with a unique (up to equivalence) irreducible representation, is isomorphic to the -algebra for some Hilbert space .

Let be a -algebra, and let be a set of elements such that the function is finite and continuous on the spectrum of . If the linear envelope of is everywhere dense in , then is said to be a -algebra with continuous trace. The spectrum of such a -algebra is separable and, under certain additional conditions, a -algebra with a continuous trace may be represented as the algebra of vector functions on its spectrum [3].

Let be a -algebra, let be the set of positive linear functionals on with norm and let be the set of non-zero boundary points of the convex set . Then will be the set of pure states of . Let be a -subalgebra of . If is a -algebra and if separates the points of the set , i.e. for any , , there exists an such that , then (the Stone–Weierstrass theorem). If is any -algebra and separates the points of the set , then .

The second dual space of a -algebra is obviously provided with a multiplication converting into a -algebra isomorphic to some von Neumann algebra; this algebra is named the von Neumann algebra enveloping the -algebra [3], [4].

The theory of -algebras has numerous applications in the theory of representations of groups and symmetric algebras [3], the theory of dynamical systems [4], statistical physics and quantum field theory [5], and also in the theory of operators on a Hilbert space [6].

References

[1]	I.M. Gel'fand, M.A. [M.A. Naimark] Neumark, "On the imbedding of normed rings in the rings of operators in Hilbert space" Mat. Sb. , 12 (54) : 2 (1943) pp. 197–213 MR9426
[2]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian) MR1292007 MR0355601 MR0355602 MR0205093 MR0110956 MR0090786 MR0026763 Zbl 0218.46042 Zbl 0137.31703 Zbl 0089.10102 Zbl 0073.08902
[3]	J. Dixmier, " algebras" , North-Holland (1977) (Translated from French) MR0498740 MR0458185 Zbl 0372.46058 Zbl 0346.17010 Zbl 0339.17007
[4]	S. Sakai, "-algebras and -algebras" , Springer (1971) MR0442701 MR0399878 MR0318902 MR0293415 MR0293414
[5]	D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974) MR0289084 Zbl 0997.82506 Zbl 1016.82500 Zbl 0177.57301
[6]	R.G. Douglas, "Banach algebra techniques in operator theory" , Acad. Press (1972) MR0361893 Zbl 0247.47001

Comments

If over is an algebra with involution, i.e. if there is an operation satisfying , , , the Hermitian, normal and positive elements are defined as follows. The element is a Hermitian element if ; it is a normal element if and it is a positive element if for some . An element is a unitary element if . An algebra with involution is also sometimes called a symmetric algebra (or symmetric ring), cf., e.g., [2]. However, this usage conflicts with the concept of a symmetric algebra as a special kind of Frobenius algebra, cf. Frobenius algebra.

Recent discoveries have revealed connections with, and applications to, algebraic topology. If is a compact metrizable space, a group, , can be formed from -extensions of the compact operators by ,

In [a3], is shown to be a homotopy invariant functor of which may be identified with the topological -homology group, . In [a1] M.F. Atiyah attempted to make a description of -homology, , in terms of elliptic operators [a5], p. 58. In [a7], [a8] G.G. Kasparov developed a solution to this problem. Kasparov and others have used the equivariant version of Kasparov -theory to prove the strong Novikov conjecture on higher signatures in many cases (see [a2], pp. 309-314).

In addition, deep and novel connections between -theory and operator algebras (cf. Operator ring) were recently discovered by A. Connes [a4]. Finally, V.F.R. Jones [a6] has exploited operator algebras to provide invariants of topological knots (cf. Knot theory).

Further details on recent developments may be found in [a2], [a5].

References

[a1]	M.F. Atiyah, "Global theory of elliptic operators" , Proc. Internat. Conf. Funct. Anal. Related Topics , Univ. Tokyo Press (1970) MR0266247 Zbl 0193.43601
[a2]	B. Blackadar, "-theory for operator algebras" , Springer (1986) MR0859867 Zbl 0597.46072
[a3]	L.G. Brown, R.G. Douglas, P.A. Filmore, "Extensions of -algebras and -homology" Ann. of Math. (2) , 105 (1977) pp. 265–324
[a4]	A. Connes, "Non-commutative differential geometry" Publ. Math. IHES , 62 (1986) pp. 257–360 Zbl 0657.55006 Zbl 0592.46056 Zbl 0564.58002
[a5]	R.G. Douglas, "-algebra extensions and -homology" , Princeton Univ. Press (1980) MR0571362
[a6]	V.F.R. Jones, "A polynomial invariant for knots via von Neumann algebras" Bull. Amer. Math. Soc. , 12 (1985) pp. 103–111 MR0766964 Zbl 0564.57006
[a7]	G.G. Kasparov, "The generalized index of elliptic operators" Funct. Anal. and Its Appl. , 7 (1973) pp. 238–240 Funkt. Anal. i Prilozhen. , 7 (1973) pp. 82–83 MR445561 Zbl 0305.58017
[a8]	G.G. Kasparov, "Topological invariants of elliptic operators I. -homology" Math. USSR-Izv. , 9 (1975) pp. 751–792 Izv. Akad. Nauk SSSR , 4 (1975) pp. 796–838 MR488027
[a9]	M. Takesaki, "Theory of operator algebras" , 1 , Springer (1979) MR0548728 Zbl 0436.46043