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C∗-algebras (pronounced "C-star") are an important area of research in functional analysis, a branch of mathematics. A C *-algebra is a complex algebra ''A'' of continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. C *-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C *-algebras which are now known as von Neumann algebras. Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C *-algebras making no reference to operators on a Hilbert space. C *-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics. Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple nuclear C *-algebras. == Abstract characterization == We begin with the abstract characterization of C *-algebras given in the 1943 paper by Gelfand and Naimark. A C *-algebra, ''A'', is a Banach algebra over the field of complex numbers, together with a map * : ''A'' → ''A''. One writes ''x *'' for the image of an element ''x'' of ''A''. The map * has the following properties: * It is an involution, for every ''x'' in ''A'' :: * For all ''x'', ''y'' in ''A'': :: :: * For every complex number λ in C and every ''x'' in ''A'': :: * For all ''x'' in ''A'': :: Remark. The first three identities say that ''A'' is a *-algebra. The last identity is called the C * identity and is equivalent to: which is sometimes called the B *-identity. For history behind the names C *- and B *-algebras, see the history section below. The C *-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies that the C *-norm is uniquely determined by the algebraic structure: :: A bounded linear map, π : ''A'' → ''B'', between C *-algebras ''A'' and ''B'' is called a *-homomorphism if * For ''x'' and ''y'' in ''A'' :: * For ''x'' in ''A'' :: In the case of C *-algebras, any *-homomorphism π between C *-algebras is non-expansive, i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C *-algebras is isometric. These are consequences of the C *-identity. A bijective *-homomorphism π is called a C *-isomorphism, in which case ''A'' and ''B'' are said to be isomorphic. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「C*-algebra」の詳細全文を読む スポンサード リンク
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