This ensures that the multiplication operation is
continuous.
A Banach algebra is called unital if it has an
identity element for the multiplication whose norm is and commutative if its multiplication is
commutative.
Any Banach algebra (whether it has an
identity element or not) can be embedded
isometrically into a unital Banach algebra so as to form a
closedideal of . Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering 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.
The set of real (or complex) numbers is a Banach algebra with norm given by the
absolute value.
The set of all real or complex -by-matrices becomes a
unital Banach algebra if we equip it with a sub-multiplicative
matrix norm.
Take the Banach space (or ) with norm and define multiplication componentwise:
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
continuouslinear operators on a Banach space (with functional composition as multiplication and the
operator norm as norm) is a unital Banach algebra. The set of all
compact operators on is a Banach algebra and closed ideal. It is without identity if [1]
Uniform algebra: A Banach algebra that is a subalgebra of the complex algebra with the supremum norm and that contains the constants and separates the points of (which must be a compact Hausdorff space).
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.
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.[3]
If a Banach algebra has unit then cannot be a
commutator; that is, for any This is because and have the same
spectrum except possibly
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.[4]
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, that is, considering extensions of Banach algebras some elements that are singular in the given algebra have a multiplicative inverse element in a Banach algebra extension Topological divisors of zero in are permanently singular in any Banach extension of
Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element denoted by , consists of all those complex
scalars such that is not invertible in The spectrum of any element is a closed subset of the closed disc in with radius and center and thus is
compact. Moreover, the spectrum of an element is
non-empty and satisfies the
spectral radius formula:
When the Banach algebra is the algebra of bounded linear operators on a complex Banach space (for example, the algebra of square matrices), the notion of the spectrum in coincides with the usual one in
operator theory. For (with a compact Hausdorff space ), one sees that:
The norm of a normal element of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.
Let be a complex unital Banach algebra in which every non-zero element is invertible (a division algebra). For every there is such that
is not invertible (because the spectrum of is not empty) hence this algebra is naturally isomorphic to (the complex case of the Gelfand–Mazur theorem).
Ideals and characters
Let be a unital commutative Banach algebra over Since is then a commutative ring with unit, every non-invertible element of belongs to some
maximal ideal of Since a maximal ideal in is closed, is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of and the set of all nonzero homomorphisms from to The set is called the "
structure space" or "character space" of and its members "characters".
A character is a linear functional on that is at the same time multiplicative, and satisfies Every character is automatically continuous from to since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on (that is, the topology induced by the weak-* topology of ), the character space, is a Hausdorff compact space.
For any
where is the
Gelfand representation of defined as follows: is the continuous function from to given by The spectrum of in the formula above, is the spectrum as element of the algebra of complex continuous functions on the compact space Explicitly,
As an algebra, a unital commutative Banach algebra is
semisimple (that is, its
Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between and [a]
Banach *-algebras
A Banach *-algebra is a Banach algebra over the field of
complex numbers, together with a map that has the following properties:
^Proof: Since every element of a commutative C*-algebra is normal, the Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense by the
Stone–Weierstrass theorem.
Mosak, R. D. (1975). Banach algebras. Chicago Lectures in Mathematics. University of Chicago Press).
ISBN0-226-54203-3.
Takesaki, M. (1979). Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences. Vol. 124 (1st ed.). Berlin Heidelberg: Springer-Verlag.
ISBN978-3-540-42248-8.
ISSN0938-0396.