## Algebras

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

and

Department of Mathematics

University of Wisconsin, Madison

Madison, WI 53706 USA

ram@math.wisc.edu

Last updates: 1 April 2010

## Algebras

An **algebra** is a vector space $A$ with an associative multiplication $A\times A$ 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 $$\parallel {a}_{1}{a}_{2}\parallel \le \parallel {a}_{1}\parallel \parallel {a}_{2}\parallel ,\phantom{\rule{2em}{0ex}}\text{for all}\phantom{\rule{2em}{0ex}}{a}_{1},{a}_{2}\in A.$$

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 $$\parallel a*a\parallel -{\parallel a\parallel}^{2},\phantom{\rule{2em}{0ex}}\text{for all}\phantom{\rule{2em}{0ex}}a\in A.$$

An **idempotented** algebra is an algebra $A$ with a set of idempotents $\mathcal{E}$ such that

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

Examples

- 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.
- Let $G$ be a locally compact Hausdorff topological group $G$ and let $\mu $ be a Haar measure on $G.$ The vector space $${L}^{2}\left(G,\mu \right)=\left\{f:G\to \u2102\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{\parallel f\parallel}_{2}<\infty \right\}$$ is a Hilbert space under the operations defined ????.
- Let $V$ be a vector space. Then $\mathrm{End}\left(V\right)$ is an algebra.

## References **[PLACEHOLDER]**

[BG]
A. Braverman and
D. Gaitsgory,
* Crystals via the affine Grassmanian*,
Duke Math. J.
**107** no. 3, (2001), 561-575;
arXiv:math/9909077v2,
MR1828302 (2002e:20083)

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