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A
[[Banach algebra|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
{{Cite|GeNe}} 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
{{Cite|GeNe}},
{{Cite|Na}},
{{Cite|Di}}.

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
{{Cite|Di}},
{{Cite|Sa}} 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|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|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|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 $ $
{{Cite|Di}}.

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
{{Cite|Di}},
{{Cite|Sa}}.

The theory of $ $-algebras has numerous applications in the theory of representations of groups and symmetric algebras
{{Cite|Di}}, the theory of dynamical systems
{{Cite|Sa}}, statistical physics and quantum field theory
{{Cite|Ru}}, and also in the theory of operators on a Hilbert space
{{Cite|Do}}.

====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.,
{{Cite|Na}}. However, this usage conflicts with the concept of a symmetric algebra as a special kind of Frobenius algebra, cf.
[[Frobenius algebra|Frobenius algebra]].

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

$$ $$
In
{{Cite|BrDoFi}}, $ $ is shown to be a homotopy invariant functor of $ $ which may be identified with the topological $ $-homology group, $ $. In
{{Cite|At}} M.F. Atiyah attempted to make a description of $ $-homology, $ $, in terms of elliptic operators
{{Cite|Do2}}, p. 58. In
{{Cite|Ka}},
{{Cite|Ka2}} 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
{{Cite|Bl}}, pp. 309-314).

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

Further details on recent developments may be found in
{{Cite|Bl}},
{{Cite|Do2}}.



====References====
{|
|-
|valign="top"|{{Ref|At}}||valign="top"| M.F. Atiyah, "Global theory of elliptic operators", ''Proc. Internat. Conf. Funct. Anal. Related Topics'', Univ. Tokyo Press (1970) {{MR|0266247}} {{ZBL|0193.43601}}
|-
|valign="top"|{{Ref|Bl}}||valign="top"| B. Blackadar, "$ $-theory for operator algebras", Springer (1986) {{MR|0859867}} {{ZBL|0597.46072}}
|-
|valign="top"|{{Ref|BrDoFi}}||valign="top"| L.G. Brown, R.G. Douglas, P.A. Filmore, "Extensions of $ $-algebras and $ $-homology" ''Ann. of Math. (2)'', '''105''' (1977) pp. 265–324
|-
|valign="top"|{{Ref|Co}}||valign="top"| A. Connes, "Non-commutative differential geometry" ''Publ. Math. IHES'', '''62''' (1986) pp. 257–360 {{MR|}} {{ZBL|0657.55006}} {{ZBL|0592.46056}} {{ZBL|0564.58002}}
|-
|valign="top"|{{Ref|Di}}||valign="top"| J. Dixmier, "$ $ algebras", North-Holland (1977) (Translated from French) {{MR|0498740}} {{MR|0458185}} {{ZBL|0372.46058}} {{ZBL|0346.17010}} {{ZBL|0339.17007}}
|-
|valign="top"|{{Ref|Do}}||valign="top"| R.G. Douglas, "Banach algebra techniques in operator theory", Acad. Press (1972) {{MR|0361893}} {{ZBL|0247.47001}}
|-
|valign="top"|{{Ref|Do2}}||valign="top"| R.G. Douglas, "$ $-algebra extensions and $ $-homology", Princeton Univ. Press (1980) {{MR|0571362}} {{ZBL|}}
|-
|valign="top"|{{Ref|GeNe}}||valign="top"| 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 {{MR|9426}} {{ZBL|}}
|-
|valign="top"|{{Ref|Jo}}||valign="top"| V.F.R. Jones, "A polynomial invariant for knots via von Neumann algebras" ''Bull. Amer. Math. Soc.'', '''12''' (1985) pp. 103–111 {{MR|0766964}} {{ZBL|0564.57006}}
|-
|valign="top"|{{Ref|Ka}}||valign="top"| 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 {{MR|445561}} {{ZBL|0305.58017}}
|-
|valign="top"|{{Ref|Ka2}}||valign="top"| G.G. Kasparov, "Topological invariants of elliptic operators I. $ $-homology" ''Math. USSR-Izv.'', '''9''' (1975) pp. 751–792 ''Izv. Akad. Nauk SSSR'', '''4''' (1975) pp. 796–838 {{MR|488027}} {{ZBL|}}
|-
|valign="top"|{{Ref|Na}}||valign="top"| M.A. Naimark, "Normed rings", Reidel (1984) (Translated from Russian) {{MR|1292007}} {{MR|0355601}} {{MR|0355602}} {{MR|0205093}} {{MR|0110956}} {{MR|0090786}} {{MR|0026763}} {{ZBL|0218.46042}} {{ZBL|0137.31703}} {{ZBL|0089.10102}} {{ZBL|0073.08902}}
|-
|valign="top"|{{Ref|Ru}}||valign="top"| D. Ruelle, "Statistical mechanics: rigorous results", Benjamin (1974) {{MR|0289084}} {{ZBL|0997.82506}} {{ZBL|1016.82500}} {{ZBL|0177.57301}}
|-
|valign="top"|{{Ref|Sa}}||valign="top"| S. Sakai, "$ $-algebras and $ $-algebras", Springer (1971) {{MR|0442701}} {{MR|0399878}} {{MR|0318902}} {{MR|0293415}} {{MR|0293414}} {{ZBL|}}
|-
|valign="top"|{{Ref|Ta}}||valign="top"| M. Takesaki, "Theory of operator algebras", '''1''', Springer (1979) {{MR|0548728}} {{ZBL|0436.46043}}
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