Banach algebra

A Banach algebra is an associative algebra with a complete submultiplicative norm, $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$.

Every complex unital Banach algebra in which every nonzero element is invertible is isometrically isomorphic to $\mathbb{C}$.

An algebra norm is a norm satisfying $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$. Its completion is a Banach algebra in which the original algebra embeds densely.

The unitization adjoins an identity to a possibly nonunital algebra. For $C_0(X)$ it is naturally $C(X^{+})$ on the one-point compactification.

If $a$ belongs to a unital Banach algebra and $\Vert a\Vert <1$, thenAbsolute convergence and multiplication of partial sums prove the identity.

The spectrum of $a$ in a unital complex algebra $A$ isFor a nonunital algebra it is defined in the unitization of an algebra.

The resolvent is $R(\lambda ,a)=(\lambda 1-a{)}^{-1}$ on the complement of the spectrum of an element. In a Banach algebra it is analytic there.

If a closed unital subalgebra $A\subseteq B$ contains $x$, then ${\sigma }_A(x)$ is obtained from ${\sigma }_B(x)$ by adjoining some bounded connected components of its complement. Membership of $(x-\lambda 1{)}^{-1}$ in $A$ is constant on every connected component of the resolvent set, and it always holds on the unbounded component.

For $x$ in a unital Banach algebra and $f$ holomorphic near $\sigma (x)$, the holomorphic functional calculus defineswhere $\Gamma$ winds once around the spectrum. It is a continuous unital algebra homomorphism and satisfies the spectral mapping theorem $\sigma (f(x))=f(\sigma (x))$.

The full spectrum of an element is its spectrum together with every bounded component of its complement. Its complement is the unbounded resolvent component, and the maximum modulus principle controls a holomorphic function on each filled hole by its values on the spectral boundary.

A character on a complex algebra is a nonzero multiplicative linear functional $\phi :A\to \mathbb{C}$.

The character space ${\Phi }_A$ is the set of all characters on $A$.

The Gelfand topology on ${\Phi }_A$ is the weak-star topology inherited from $A^{\ast }$, equivalently the coarsest topology making every map $\phi \mapsto \phi (a)$ continuous.

For an algebra of functions on a space, evaluation at $x$ is the character ${\delta }_x(f)=f(x)$.

For a commutative Banach algebra $A$, the Gelfand transform sends $a\in A$ to the continuous function $\hat{a}(\phi )=\phi (a)$ on its character space of an algebra. It is a contractive unital algebra homomorphism into $C({\Phi }_A)$.

The closed unital subalgebra generated by $x$ is the norm closure of the polynomials in $x$. Its character space is homeomorphic to $\sigma (x)$ through $\phi \mapsto \phi (x)$, and the complement of that spectrum is connected.

A C-star algebra is a complex Banach algebra with an involution satisfying $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.

An element $a$ of a C-star algebra is positive when $a=b^{\ast }b$ for some $b$, equivalently when $a=a^{\ast }$ and $\sigma (a)\subseteq [0,\infty )$. Continuous functional calculus gives it a unique positive square root $a^{1/2}$.

An operator $U$ on a Hilbert space is a partial isometry when it is isometric on $(\ker U{)}^{\perp }$. Equivalently, $U^{\ast }U$ is the orthogonal projection onto this initial space.

Every bounded operator on a Hilbert space has a polar decomposition $T=U|T|$, where $|T|=(T^{\ast }T{)}^{1/2}$ and $U$ is a partial isometry with $\ker U=\ker T$. On $\mathrm{im}|T|$, it is defined by $U(|T|x)=Tx$.