In this monograph, we consider the connection between graphs and
Hilbert space operators. In particular, we are interested in the
algebraic structures, called graph groupoids, embedded in operator
algebras. In Part 1, we consider the connection from graphs to
partial isometries. Every element in graph groupoids assigns an
operator, which is either a partial isometry or a projection, under
suitable representations. The von Neumann algebras induced by the
dynamical systems of graph groupoids are characterized. In Part 2,
we observe the connection from partial isometries to graphs. We
show that a finite family of partial isometries on a fixed Hilbert
space H creates the corresponding graph, and the graph groupoid of
it is an embedded groupoid inside B(H). Moreover, the C*-subalgebra
generated by the family is *-isomorphic to the groupoid algebra
generated by the graph groupoid of the corresponding graph. As
application, we consider the C*-subalagebras generated by a single
operator.
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