III.12 C*-Algebras

A BANACH SPACE [III.62] is both a VECTOR SPACE [I.3 §2.3] and a METRIC SPACE [III.56], and the study of Banach spaces is therefore a mixture of linear algebra and analysis. However, one can arrive at more sophisticated mixtures of algebra and analysis if one looks at Banach spaces that have more algebraic structure. In particular, while one can add two elements of a Banach space together, one cannot in general multiply them. However, sometimes one can: a vector space with a multiplicative structure is called an algebra, and if the vector space is also a Banach space, and if the multiplication has the property that ||xy|| ≤ ||x|| ||y|| for any two elements x and y, then it is called a Banach algebra. (This name does not really reflect historical reality, since the basic theory of Banach algebras was not worked out by Banach. A more appropriate name might have been Gelfand algebras.)

A C*- algebra is a Banach algebra with an involution, which means a function that associates with each element x another element x* in such a way that x* * = x, ||x*|| = ||x||, (x+y)* = x*+y*, and (xy)* = y*x* for any elements x and y; this involution is required to satisfy the C*-identity ||xx*|| = ||x||₂ A basic example of a C*- algebra is the algebra B(H) of all continuous linear maps T defined on a HILBERT SPACE [III.37] H. The norm of T is defined to be the smallest constant M such that ||Tx|| ≤ M||x|| for every x ∈ H, and the involution takes T to its adjoint. This is a map T* that has the property that (x, Ty) = (T*x, y) for every x and y in H. (It can be shown that there is exactly one map with this property.) If H is finite dimensional, then T can be thought of as an n ×n matrix for some n, and T * is then the complex conjugate of the transpose of T.

A fundamental theorem of Gelfand and Naimark states that every C*- algebra can be represented as a subalgebra of B(H) for some Hilbert space H. For more information, see OPERATOR ALGEBRAS [IV.15 §3].