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A Lie algebra over an algebraically closed field is called exceptional if it is constructed from one of the root systems E_6, E_7, E_8, F_4, and G_2 by the Chevalley ...

The Killing form is an inner product on a finite dimensional Lie algebra g defined by B(X,Y)=Tr(ad(X)ad(Y)) (1) in the adjoint representation, where ad(X) is the adjoint ...

A solvable Lie group is a Lie group G which is connected and whose Lie algebra g is a solvable Lie algebra. That is, the Lie algebra commutator series ...

An approximate unit for a C^*-algebra A is an increasing net {u_lambda}_(lambda in Lambda) of positive elements in the closed unit ball of A such that a=lim_(lambda)au_lambda ...

Let A be a commutative complex Banach algebra. The space of all characters on A is called the maximal ideal space (or character space) of A. This space equipped with the ...

Let A be a unital C^*-algebra. An element u in A is called unitary if u^*u=uu^*=1. For example, for each self-adjoint element a in A, the element ...

There are two completely different definitions of Cayley numbers. The first and most commonly encountered type of Cayley number is the eight elements in a Cayley algebra, ...

A bounded left approximate identity for a normed algebra A is a bounded net {e_alpha}_(alpha in I) with the property lim_(alpha)e_alphaa=a for a in A. Bounded right and ...

A representation of a C^*-algebra A is a pair (H,phi) where H is a Hilbert space and phi:A->B(H) is a *-homomorphism. (H,phi) is said to be faithful if phi is injective. For ...

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a ...