Definition:Cyclic Representation of C*-Algebra

Definition

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.

Let $\tuple {\pi, \HH}$ be a representation of $\struct {A, \ast, \norm {\, \cdot \,} }$.

We call $\tuple {\pi, \HH}$ cyclic if and only if there exists $e \in \HH$ such that:

$\set {\map \pi a e : a \in A}$ is everywhere dense in $\HH$.

We call $e$ a cyclic vector.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {VIII}.5.6$