Evolution algebras are non-associative algebras. In this work we provide an extension of this class of algebras, in the context of Hilbert spaces, capable to deal with infinite-dimensional spaces. We illustrate the applicability of our approach by discussing a connection with discrete-time Markov chains with countable state space.

Evolution algebras are a special class of non-associative algebras exhibiting connections with different fields of Mathematics. Hilbert evolution algebras generalize the concept through a framework of Hilbert spaces. This allows to deal with a wide class of infinitedimensional spaces. In this work we study Hilbert evolution algebras associated to a graph. Inspired in definitions of evolution algebras we define the Hilbert evolution algebra associated to a given graph and the Hilbert evolution algebra associated to the symmetric random walk on a graph. For a given graph, we provide conditions under which these structures are or are not isomorphic. Our definitions and results extend to graphs with infinitely many vertices a similar theory developed for evolution algebras associated to finite graphs.

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