Operator algebras can be used to study arbitrary sets of operators with little algebraic relation
simultaneously. From this point of view, operator algebras can be regarded as a generalization of
spectral theory of a single operator. In general, operator algebras are
non-commutative rings. An operator algebra is typically required to be
closed in a specified operator
topology inside the whole algebra of continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties are
axiomatized and algebras with certain topological structure become the subject of the research. Though algebras of operators are studied in various contexts (for example, algebras of
pseudo-differential operators acting on spaces of
distributions), the term
operator algebra is usually used in reference to algebras of
bounded operators on a
Banach space or, even more specifically in reference to algebras of operators on a
separable Hilbert space, endowed with the
operator norm topology. In the case of operators on a Hilbert space, the
Hermitian adjoint map on operators gives a natural
involution, which provides an additional algebraic structure that can be imposed on the algebra. In this context, the best studied examples are
self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include
C*-algebras,
von Neumann algebras, and
AW*-algebras. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebras can be identified to a certain closed
subalgebra of the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.
Commutative self-adjoint operator algebras can be regarded as the algebra of
complex-valued continuous functions on a
locally compact space, or that of
measurable functions on a
standard measurable space. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of the
base space on which the functions are defined. This point of view is elaborated as the philosophy of
noncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras. Examples of operator algebras that are not self-adjoint include: •
nest algebras, • many
commutative subspace lattice algebras, • many
limit algebras. ==See also==