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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
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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
fo
r 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
-algeb
ra
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, "![]() |
[4] | S. Sakai, "![]() ![]() |
[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, "![]() |
[a3] | L.G. Brown, R.G. Douglas, P.A. Filmore, "Extensions of ![]() ![]() |
[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, "![]() ![]() |
[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, "To
pological invariants of elliptic operators I. ![]() |
[a9] | M. Takesaki, "Theory of operator algebras" , 1 , Springer (1979) MR0548728 Zbl 0436.46043 |
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