Banach Algebra - Properties

Properties

Several elementary functions which 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 homeomorphism) so that it forms a topological group under multiplication.

If a Banach algebra has unit 1, then 1 cannot be a commutator; i.e.,   for any x, y ∈ A.

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 which is a division algebra is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra which 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, i.e., considering extensions B of Banach algebras A some elements that are singular in the given algebra A have a multiplicative inverse element in a Banach algebra extension B. Topological divisors of zero in A are permanently singular in all Banach extension B of A.