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 $∥ a_{1} a_{2} ∥ \leq ∥ a_{1} ∥ ∥ a_{2} ∥, for all a_{1}, a_{2} \in A .$

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 $∥ a * a ∥ - {∥ a ∥}^{2}, for all a \in A .$

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

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
For each a∈A there is an e∈ℰ such that ae=ea=a. A von Neumann algebra is an algebra A of operators on a Hilbert space H such that

Examples

The algebra $B (H)$ 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 $μ$ be a Haar measure on $G .$ The vector space $L^{2} (G, μ) = \{f : G \to ℂ | {∥ f ∥}_{2} < \infty\}$ is a Hilbert space under the operations defined ????.
Let $V$ be a vector space. Then $End (V)$ 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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