Banach algebra

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.

If a Banach algebra has unit ${\displaystyle \mathbf{1},}$ then ${\displaystyle \mathbf{1}}$ cannot be a commutator; that is, ${\displaystyle xy-yx\ne \mathbf{1}}$ for any ${\displaystyle x,y\in A.}$ This is because ${\displaystyle xy}$ and ${\displaystyle yx}$ have the same spectrum except possibly ${\displaystyle 0.}$

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, that is, considering extensions ${\displaystyle B}$ of Banach algebras ${\displaystyle A}$ some elements that are singular in the given algebra ${\displaystyle A}$ have a multiplicative inverse element in a Banach algebra extension ${\displaystyle B.}$ Topological divisors of zero in ${\displaystyle A}$ are permanently singular in any Banach extension ${\displaystyle B}$ of ${\displaystyle A.}$

Spectral theory

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element ${\displaystyle x\in A,}$ denoted by ${\displaystyle \sigma (x)}$, consists of all those complex scalars ${\displaystyle \lambda }$ such that ${\displaystyle x-\lambda \mathbf{1}}$ is not invertible in ${\displaystyle A.}$ The spectrum of any element ${\displaystyle x}$ is a closed subset of the closed disc in ${\displaystyle \mathbb{C}}$ with radius ${\displaystyle \Vert x\Vert }$ and center ${\displaystyle 0,}$ and thus is compact. Moreover, the spectrum ${\displaystyle \sigma (x)}$ of an element ${\displaystyle x}$ is non-empty and satisfies the spectral radius formula: $${\displaystyle sup\{|\lambda |:\lambda \in \sigma (x)\}=\underset{n\to \mathrm{\infty }}{lim}\Vert x^n{\Vert }^{1/n}.}$$

Given ${\displaystyle x\in A,}$ the holomorphic functional calculus allows to define ${\displaystyle f(x)\in A}$ for any function ${\displaystyle f}$ holomorphic in a neighborhood of ${\displaystyle \sigma (x).}$ Furthermore, the spectral mapping theorem holds: $${\displaystyle \sigma (f(x))=f(\sigma (x)).}$$

When the Banach algebra ${\displaystyle A}$ is the algebra ${\displaystyle L(X)}$ of bounded linear operators on a complex Banach space ${\displaystyle X}$ (for example, the algebra of square matrices), the notion of the spectrum in ${\displaystyle A}$ coincides with the usual one in operator theory. For ${\displaystyle f\in C(X)}$ (with a compact Hausdorff space ${\displaystyle X}$), one sees that: $${\displaystyle \sigma (f)=\{f(t):t\in X\}.}$$

The norm of a normal element ${\displaystyle x}$ of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.

Let ${\displaystyle A}$ be a complex unital Banach algebra in which every non-zero element ${\displaystyle x}$ is invertible (a division algebra). For every ${\displaystyle a\in A,}$ there is ${\displaystyle \lambda \in \mathbb{C}}$ such that ${\displaystyle a-\lambda \mathbf{1}}$ is not invertible (because the spectrum of ${\displaystyle a}$ is not empty) hence ${\displaystyle a=\lambda \mathbf{1}:}$ this algebra ${\displaystyle A}$ is naturally isomorphic to ${\displaystyle \mathbb{C}}$ (the complex case of the Gelfand–Mazur theorem).

Ideals and characters

Let ${\displaystyle A}$ be a unital commutative Banach algebra over ${\displaystyle \mathbb{C}.}$ Since ${\displaystyle A}$ is then a commutative ring with unit, every non-invertible element of ${\displaystyle A}$ belongs to some maximal ideal of ${\displaystyle A.}$ Since a maximal ideal ${\displaystyle \mathfrak{m}}$ in ${\displaystyle A}$ is closed, ${\displaystyle A/\mathfrak{m}}$ 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 ${\displaystyle A}$ and the set ${\displaystyle \mathrm{\Delta }(A)}$ of all nonzero homomorphisms from ${\displaystyle A}$ to ${\displaystyle \mathbb{C}.}$ The set ${\displaystyle \mathrm{\Delta }(A)}$ is called the structure space or character space of ${\displaystyle A}$.

A character ${\displaystyle \chi \in \mathrm{\Delta }(A)}$ is a linear functional on ${\displaystyle A}$ that is at the same time multiplicative, ${\displaystyle \chi (ab)=\chi (a)\chi (b),}$ and satisfies ${\displaystyle \chi (\mathbf{1})=1.}$ Every character is automatically continuous from ${\displaystyle A}$ to ${\displaystyle \mathbb{C},}$ 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 ${\displaystyle A}$ (that is, the topology induced by the weak-* topology of ${\displaystyle A^{\ast }}$), the character space, ${\displaystyle \mathrm{\Delta }(A),}$ is a compact Hausdorff space.

For any ${\displaystyle x\in A,}$ $${\displaystyle \sigma (x)=\sigma (\hat{x})}$$ where ${\displaystyle \hat{x}}$ is the Gelfand representation of ${\displaystyle x}$ defined as follows: ${\displaystyle \hat{x}}$ is the continuous function from ${\displaystyle \mathrm{\Delta }(A)}$ to ${\displaystyle \mathbb{C}}$ given by ${\displaystyle \hat{x}(\chi )=\chi (x).}$ The spectrum of ${\displaystyle \hat{x},}$ in the formula above, is the spectrum as element of the algebra ${\displaystyle C(\mathrm{\Delta }(A))}$ of complex continuous functions on the compact space ${\displaystyle \mathrm{\Delta }(A).}$ Explicitly, $${\displaystyle \sigma (\hat{x})=\{\chi (x):\chi \in \mathrm{\Delta }(A)\}.}$$

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 ${\displaystyle A}$ is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between ${\displaystyle A}$ and ${\displaystyle C(\mathrm{\Delta }(A)).}$

Banach *-algebras

A Banach *-algebra ${\displaystyle A}$ is a Banach algebra over the field of complex numbers, together with a map ${\displaystyle {}^{\ast }:A\to A}$ that has the following properties:

1. ${\displaystyle {\left(x^{\ast }\right)}^{\ast }=x}$ for all ${\displaystyle x\in A}$ (so the map is an involution).
2. ${\displaystyle (x+y{)}^{\ast }=x^{\ast }+y^{\ast }}$ for all ${\displaystyle x,y\in A.}$
3. ${\displaystyle (\lambda x{)}^{\ast }=\overline{\lambda }{x}^{\ast }}$ for every ${\displaystyle \lambda \in \mathbb{C}}$ and every ${\displaystyle x\in A;}$ here, ${\displaystyle \overline{\lambda }}$ denotes the complex conjugate of ${\displaystyle \lambda .}$
4. ${\displaystyle (xy{)}^{\ast }=y^{\ast }{x}^{\ast }}$ for all ${\displaystyle x,y\in A.}$

In other words, a Banach *-algebra is a Banach algebra over ${\displaystyle \mathbb{C}}$ that is also a *-algebra.

In most natural examples, one also has that the involution is isometric, that is, $${\displaystyle \Vert x^{\ast }\Vert =\Vert x\Vert \text{ for all }x\in A.}$$ Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying ${\displaystyle \Vert x^{\ast }x\Vert =\Vert x^{\ast }\Vert \Vert x\Vert }$ is a C*-algebra.