Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers and $d\in\mathbb{N}$. Let $W_{2d}$, with concatenation $(u,v)\mapsto uv$ and adjoint words $w\mapsto w^{*}$, be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words. Let $\mathbb{C}$ be the field of complex numbers with conjugation $z\mapsto\overline{z}$ and modulus $|z|$. Sums over a nonempty finite index set are those of Sum over a Finite Index Set, and a sum over the empty index set is read as $0$.

1. (Word polynomials)¶ A word polynomial is a map $X:W_{2d}\to\mathbb{C}$ for which there is a nonempty finite set $F\subseteq W_{2d}$, a support set of $X$, with $X(w)=0$ for every $w\in W_{2d}\setminus F$. For $w\in W_{2d}$, $e_{w}$ is the word polynomial with value $1$ at $w$ and $0$ elsewhere, with support set $\{w\}$. Sums $X+Y$ and multiples $zX$, $z\in\mathbb{C}$, are pointwise; they are word polynomials with support sets $F\cup G$ and $F$, where $G$ is a support set of $Y$ and $F\cup G$ is finite by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §finite-union.

2. (Product)¶ For word polynomials $X,Y$ with support sets $F,G$ and $w\in W_{2d}$,

$$(XY)(w)=\sum_{(u,v)\in P_{w}}X(u)\,Y(v),\qquad P_{w}=\{(u,v)\in F\times G:\ uv=w\}.$$

Here $P_{w}$ is finite by claim 3 of Basic Properties of Finite Sets, as a subset of $F\times G$. The value does not depend on the support sets: for two choices $(F,G)$ and $(F',G')$, both sums equal the sum over the corresponding set of pairs in $(F\cup F')\times(G\cup G')$, because the additional pairs $(u,v)$ satisfy $X(u)Y(v)=0$; when that larger set is nonempty this is the second part of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, and when a smaller set is empty while the larger one is not, the first part of that clause shows that the sum over the larger set is $0$. $XY$ is a word polynomial with support set $\{uv:(u,v)\in F\times G\}$, which is finite by claim 4 of Basic Properties of Finite Sets, being the image of the finite set $F\times G$ (finite by claim 1 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets).

3. (Adjoint)¶ The adjoint of a word polynomial $X$ with support set $F$ is the map $X^{*}(w)=\overline{X(w^{*})}$; it is a word polynomial with support set $\{u^{*}:u\in F\}$, finite by claim 4 of Basic Properties of Finite Sets, because $(w^{*})^{*}=w$ for every $w\in W_{2d}$: inverse letters satisfy $(l^{-1})^{-1}=l$ by Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters, and reversal is an involution by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal. $X$ is self-adjoint if $X^{*}=X$.

4. ($\ell^{1}$ norm)¶ For a word polynomial $X$ with support set $F$, $\lVert X\rVert_{1}=\sum_{w\in F}|X(w)|$, a nonnegative real number by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative, independent of $F$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing applied to $F$, $G$ and $F\cup G$ for two support sets $F,G$.