Suppose you have a finite dimensional real Hilbert space $V$ and you form the tensor algebra
$$T(V) = \mathbb{R} \oplus V \oplus (V\otimes V) \oplus (V\otimes V \otimes V) \oplus \cdots$$
where the multiplication is given by the tensor product. Now imagine you have a finite set of vectors $S$ which you use to form an ideal $I(S)$. From this, we can get the quotient algebra $Q = T(V)/I(S)$ (see  [this Wikipedia article](http://en.wikipedia.org/wiki/Quotient_algebra)).


I am interested in comparing elements of $Q$, ideally using an inner product, if it is possible to define one, or else using a distance, if it is possible to define a norm.


The approach I've taken so far is, given an element $x$ of $T(V)$ is to try to project it onto the orthogonal complement of $I(S)$ (call this $P(x)$) in the assumption that there will be a unique element of the equivalence class of $x$ that lives in this orthogonal complement, and that we can then use the inner product of $T(V)$. However I think this assumption is false. Consider the case where $S =$ {$a\otimes a - a$} for a one dimensional vector space $V$ with basis vector $a$. Then $I(S)$ is the vector space with basis $a - a\otimes a$, $a\otimes a - a\otimes a\otimes a$, $a\otimes a\otimes a - a\otimes a\otimes a\otimes a$, $\ldots$.


Intuitively, I would expect $Q$ in this case to be a two dimensional vector space with basis $1, [a]$. Is this correct? Projecting onto the orthogonal complement seems to give zero however.


Is there a standard way to define a norm or an inner product on a quotient algebra such as this? How can it be computed?



Norm on quotient algebra of a tensor algebra

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> Intuitively, I would expect Q in this case to be a two dimensional vector space with basis 1,[a]. Is this correct? Projecting onto the orthogonal complement seems to give zero however.
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> 


The first part is correct. This may be easiest to see by considering the isomorphism, in the present case where $V$ is one dimensional, of $T(V)$ with $\mathbb{R}[x]$. Then $Q$ is $\mathbb{R}[x]/(x^2-x)$, which is 2-dimensional by the division algorithm.


The orthogonal complement of $I(S)$ in this case is not zero, but it is one dimensional and your concern is valid. Thinking again in $\mathbb{R}[x]$ for convenience, the condition that $\sum\_k a\_k x^k$ is orthogonal to $x^{m+2} - x^{m+1}$ for all $m\geq0$ says that $a\_{m+2}=a\_{m+1}$ for all $m\geq0$; thus, $a\_k=0$ for $k\geq1$. Note that the conclusion is the same if we first complete to a Hilbert space by taking power series with square summable coefficients, so that unfortunately Andrew Stacey's desperate hope appears to be shattered. I have been assuming in this paragraph that $a$ (or $x$) has norm 1; it makes no difference for the conclusion in the non-completed case, but it simplifies the relation. (In other cases when $\dim(V)>1$ the Hilbert space quotient can be far too large for the vector space isomorphism to be possible, having dimension $2^{\aleph\_0}$, so completion is probably not the way to go.)


This shows that there is no hope in general of using the standard inner product to define a norm on $T(V)/I(S)$. However, the problem in the above case arose because the ideal was not homogeneous; this forces relations on coefficients of different degree for elements of the orthogonal complement, thereby forcing too many coefficients to be zero. For this construction to work I recommend considering the case where $S$ is a set of homogeneous elements of $T(V)$ (i.e., each element of $S$ lies in one of the tensor powers $V^{\otimes k}$). I don't know what else to tell you in the nonhomogeneous case.


### Gratuitous Add-on


The Hilbert space completion of the free algebra $T(V)$ is the full (or free) Fock space $F(V)$. The latter is not an algebra, but a Banach algebra completion of $T(V)$ can be obtained by first representing $T(V)$ on $F(V)$ by choosing an orthonormal basis $v\_1,\ldots,v\_n$ for $V$ and sending $v\_j\in T(V)$ to the "creation operator" $S\_j$ on $F(V)$ defined by $S\_j(w)=v\_j\otimes w$, and then closing the image in your favorite topology. These algebras were first studied by G. Popescu circa 1991 (but over $\mathbb{C}$ rather than $\mathbb{R}$ and with either norm or ultraweak closure), and a similar construction was earlier used by D. Evans to study the Cuntz algebra $\mathcal{O}\_n$ in the 1980 paper "On $\mathcal{O}\_n$".


