A C*-algebra isometry is an element 
of a unital C*-algebra satisfying
.
Then 
is a projection onto its range in any
unital Hilbert space representation. The element
is unitary precisely when 
as well. Thus an isometry need not be surjective in a
representation. A unital algebra is stably
finite when every isometry in every matrix algebra
over it is unitary.
C^*-Algebra Isometry
See also
C*-Algebra, C*-Algebra Projection, Stably Finite C*-Algebra, Unitary ElementExplore with Wolfram|Alpha
References
Blackadar, B. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Berlin, Germany: Springer, 2006. https://doi.org/10.1007/3-540-28517-2.Cite this as:
Weisstein, Eric W. "C^*-Algebra Isometry." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/C-Star-AlgebraIsometry.html