Wikipedia / Banach algebra
Wikipedia / Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric
This ensures that the multiplication operation is continuous with respect to the metric topology.
A Banach algebra is called unital if it has an identity element and commutative if its multiplication is commutative (whether it is unital or not) can be embedded isometrically so as to form a closed ideal. Often one assumes a priori and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.
The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.
Banach algebras can also be defined over fields of p-adic numberss. This is part of p-adic analysis.
Examples
, the space of (complex-valued) continuous functions, defined on a locally compact Hausdorff space, that vanish at infinity is compact. The complex conjugation being an involution is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra by definition.
The set of real (or complex) numbers is a Banach algebra with norm given by the absolute value.
matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremum norm) is a unital Banach algebra.
The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.
The algebra of all continuous linear (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators
is a locally compact Hausdorff topological group is its Haar measure becomes a Banach algebra under the convolution
Uniform algebra (which must be a compact Hausdorff space).
C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space.
Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution of measures.
The algebra of the quaternions is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.
An affinoid algebra is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks in rigid analytic geometry.
Properties
Several elementary functions that are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric series remains valid in general unital Banach algebras. The binomial theorem also holds for two commuting elements of a Banach algebra.
The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a topological group under multiplication.
cannot be a commutator have the same spectrum
The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:
Every real Banach algebra that is a division algebra is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as the Gelfand–Mazur theorem.)
Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.
Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
Permanently singular elements in Banach algebras are topological divisors of zero
Spectral theory
Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum, consists of all those complex scalars and thus is compact is non-empty and satisfies the spectral radius
the holomorphic functional calculus holomorphic Furthermore, the spectral mapping theorem holds:
coincides with the usual one in operator theory
of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.
(the complex case of the Gelfand–Mazur theorem).
Ideals and characters
be a unital commutative belongs to some maximal ideal is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem is called the structure space or character space.
A character (that is, the topology induced by the weak-* topology is a compact Hausdorff space.
is the Gelfand representation
As an algebra, a unital commutative Banach algebra is semisimple (that is, its Jacobson radical
Banach *-algebras
is a Banach algebra over the field of complex numbers that has the following properties:
(so the map is an involution).
denotes the complex conjugate
that is also a *-algebra.
In most natural examples, one also has that the involution is isometric Some authors include this isometric property in the definition of a Banach *-algebra.
is a C*-algebra.
See also
Notes
Citations
References