Algebras

## Algebras

Last updates: 1 April 2010

## Algebras

An algebra is a vector space $A$ with an associative multiplication $A×A$ which satisfies the distributive laws, ie, such that $A$ is a ring. A Banach algebra is a Banach space $A$ with a multiplication such that $A$ is an algebra and $∥ a 1 a 2 ∥≤∥ a 1 ∥∥ a 2 ∥,for all a 1 , a 2 ∈A.$

A $*$-algebra is a Banach algebra with an involution $*:A\to A$ such that

An element $a$ in a $*$-algebra is hermitian, or self-adjoint, if $a*=a.$ A $C*$-algebra is a $*$-algebra $A$ such that $∥ a*a∥- ∥a∥ 2 ,for alla∈A.$

An idempotented algebra is an algebra $A$ with a set of idempotents $ℰ$ such that

1. For each pair ${e}_{1},{e}_{2}\in ℰ$ there is an ${e}_{0}\in ℰ$ such that ${e}_{0}{e}_{1}={e}_{1}{e}_{0}={e}_{1}$ and ${e}_{0}{e}_{2}={e}_{2}{e}_{0}={e}_{2},$ and
2. For each $a\in A$ there is an $e\in ℰ$ such that $ae=ea=a.$ A von Neumann algebra is an algebra $A$ of operators on a Hilbert space $H$ such that
1. $A$ is closed under taking adjoints,
2. $A$ coincides with its bicommutant.

Examples

1. The algebra $B\left(H\right)$ of bounded linear operators on a Hilbert space $H$ with the operator norm ???? and involution given by adjoint ??? is a Banach algebra.
2. Let $G$ be a locally compact Hausdorff topological group $G$ and let $\mu$ be a Haar measure on $G.$ The vector space $L 2 Gμ = f:G→ℂ| ∥f∥ 2 <∞$ is a Hilbert space under the operations defined ????.
3. Let $V$ be a vector space. Then $\mathrm{End}\left(V\right)$ is an algebra.