There are many cases of these algebras which may obviously be combined into natural classes, but the consideration of this portion of the subject will be reserved to subsequent researches. Benjamin Peirce
Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements. Omar Khayyám
In all other algebras both relations must be combined, and the algebra must conform to the character of the relations. Benjamin Peirce
Abstract characterization We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gelfand and Naimark. Source: Internet
A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. Source: Internet
A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. Source: Internet