In analogy to the doubling of vector spaces introduced for the [L.sub.[infinity]] realization of Theorem 1 we will show that every nonassociative algebra has a realization as an [A.sub.[infinity]] algebra.

Let (V,*) be a nonassociative algebra and [V.sup.*] a vector space isomorphic to V with the isomorphism denoted by V [contains as member of] a [??] [a.sup.*] [member of] [V.sup.*].

This completes the proof that any nonassociative algebra can be embedded into an [A.sub.[infinity]] algebra.

By defining a new binary operation a-b = (1/2)(ab + ba) on an associative algebra over a field whose characteristic is not equal to 2, we obtain another important nonassociative algebra known as Jordan algebra.

Moreover, we introduce a new class of a nonassociative algebra called left almost algebra.

The Role of

Nonassociative Algebra in Projective Geometry

Maltsev used the name "Moufang-Lie algebras" for these

nonassociative algebras while Sagle [8] introduced the term "Malcev algebras." Equivalent defining identities of Maltsev algebras are pointed out in [8].

Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor 1994; ActaAppl.

Nonlinear Elliptic Equations and Nonassociative Algebras

Nadirashvili, Tkachev, and Vladut present students, academics, and mathematicians with a collection of applications of noncommutative and nonassociative algebras, used to construct unusual solutions to nonlinear elliptic partial differential equations of the second order.

An Introduction to

Nonassociative Algebras. Dover Publications Inc., New York, 1995.

The original motivation to introduce the class of

nonassociative algebras known as Jordan algebras came from quantum mechanics (see [?]).