  We develop a version of quantum mechanics that can handle nonassociative
algebras of observables and which reduces to standard quantum theory in the
traditional associative setting. Our algebraic approach is naturally
probabilistic and is based on using the universal enveloping algebra of a
general nonassociative algebra to introduce a generalized notion of associative
composition product. We formulate properties of states together with notions of
trace, and use them to develop GNS constructions. We describe Heisenberg and
Schr\"odinger pictures of completely positive dynamics, and we illustrate our
formalism on the explicit examples of finite-dimensional matrix Jordan algebras
as well as the octonion algebra.
