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A nonassociative algebra named after physicist Pascual Jordan which satisfies xy=yx (1) and (xx)(xy)=x((xx)y)). (2) The latter is equivalent to the so-called Jordan identity ...

Given a commutative ring R, an R-algebra H is a Hopf algebra if it has additional structure given by R-algebra homomorphisms Delta:H->H tensor _RH (1) (comultiplication) and ...

A Banach algebra A is called contractible if H^1(A,X)=Z^1(A,X)/B^1(A,X)=0 for all Banach A-bimodules X (Helemskii 1989, 1997). A C^*-algebra is contractible if and only if it ...

Every finite-dimensional Lie algebra of characteristic p=0 has a faithful finite-dimensional representation.

An operator Gamma=sum_(i=1)^me_i^Ru^(iR) on a representation R of a Lie algebra.

Let A be a unital C^*-algebra, then an element u in A is called co-isometry if uu^*=1.

Every finite-dimensional Lie algebra of characteristic p!=0 has a faithful finite-dimensional representation.

The identity (xy)x^2=x(yx^2) satisfied by elements x and y in a Jordan algebra.

A simple root of a Lie algebra is a positive root that is not the sum of two positive roots.

Let A be a C^*-algebra, then a state is a positive linear functional on A of norm 1.