 Solution

A state on a C-star algebra is a positive linear functional of norm one. A cyclic representation of a C-star algebra is a representation of a C-star algebra $\pi :A\to B(H)$ with a vector $\xi$ whose orbit $\pi (A)\xi$ has dense linear span in $H$. On $c$, pointwise multiplication and coordinatewise complex conjugation preserve convergence; the constant sequence one is the unit. The supremum norm is submultiplicative and obeys $\Vert {\alpha }^{\ast }\alpha {\Vert }_{\infty }=\Vert \alpha {\Vert }_{\infty }^2$. Completeness is given, so $c$ is a commutative unital C-star algebra.

If $F=\varphi (\beta )$ is a state, testing its positivity on the coordinate projections gives ${\beta }_{j+1}\ge 0$. To recover the independent coefficient at infinity, let $q_N$ have its first $N$ coordinates zero and all later ones one. This is a positive element of $c$, and $F(q_N)={\beta }_1+{\sum }_{j>N}{\beta }_{j+1}\ge 0$. Passing to the limit gives ${\beta }_1\ge 0$. By the given isometry, $\Vert F\Vert =\Vert \beta {\Vert }_1={\sum }_j{\beta }_j$, which is one. Conversely these nonnegative unit-sum coefficients make $F({\alpha }^{\ast }\alpha )$ a sum of nonnegative weighted squares, and the isometry gives norm one. Thus the stated coefficient conditions are necessary and sufficient.

For the specified state, all other coefficients are zero because they are nonnegative and the first three already sum to one. Its GNS null ideal isEvaluation at infinity and the first two coordinates identifies $c/N_x$ with ${\mathbb{C}}^3$, with inner product $\frac{1}{2}{u}_{\infty }\overline{v_{\infty }}+\frac{1}{4}{u}_1\overline{v_1}+\frac{1}{4}{u}_2\overline{v_2}$. Left multiplication induces the diagonal representation; the class of the unit is $(1,1,1)$. In the equivalent standard inner product the resulting GNS triple isIts coefficient is exactly $\frac{1}{2}{\alpha }_{\infty }+\frac{1}{4}{\alpha }_1+\frac{1}{4}{\alpha }_2=F(\alpha )$. The three evaluations can be assigned arbitrary independent values by a convergent sequence, and all three entries of ${\xi }_x$ are nonzero. Therefore its orbit spans ${\mathbb{C}}^3$, verifying cyclicity and the required state identity directly.