14a. Functional analysis
We have seen so far that the theory of the compact quantum Lie groups, [math]G\subset U_N^+[/math], can be developed with some inspiration from the theory of compact Lie groups, [math]G\subset U_N[/math]. In this chapter we discuss an alternative approach to this, by looking at the finitely generated discrete quantum groups [math]\Gamma=\widehat{G}[/math] which are dual to our objects. Thus, the idea will be that of developing the theory of the finitely generated discrete quantum groups, [math]\widehat{U_N^+}\to\Gamma[/math], with inspiration from the theory of finitely generated discrete groups, [math]F_N\to\Gamma[/math].
Normally the theory is already there, as developed in the previous chapters, which equally concern the compact quantum group [math]G[/math] and its discrete dual [math]\Gamma=\widehat{G}[/math]. However, from the discrete group viewpoint, what has been worked out so far looks more like specialized mathematics, and there are still a lot of basic things, to be developed. In short, what we will be doing here will be a complement to the material from the previous chapters, obtained by using a different, and somehow opposite, philosophy.
Let us begin with a reminder regarding the cocommutative Woronowicz algebras, which will be our main objects in this chapter, coming before the commutative ones, that we are so used to have in the [math]\#1[/math] spot. As explained in chapter 3 above, we have:
For a Woronowicz algebra [math]A[/math], the following are equivalent:
- [math]A[/math] is cocommutative, [math]\Sigma\Delta=\Delta[/math].
- The irreducible corepresentations of [math]A[/math] are all [math]1[/math]-dimensional.
- [math]A=C^*(\Gamma)[/math], for some group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], up to equivalence.
This follows from the Peter-Weyl theory, as follows:
[math](1)\implies(2)[/math] The assumption [math]\Sigma\Delta=\Delta[/math] tells us that the inclusion [math]\mathcal A_{central}\subset\mathcal A[/math] is an isomorphism, and by using Peter-Weyl theory we conclude that any irreducible corepresentation of [math]A[/math] must be equal to its character, and so must be 1-dimensional.
[math](2)\implies(3)[/math] This follows once again from Peter-Weyl, because if we denote by [math]\Gamma[/math] the group formed by the 1-dimensional corepresentations, then we have [math]\mathcal A=\mathbb C[\Gamma][/math], and so [math]A=C^*(\Gamma)[/math] up to the standard equivalence relation for Woronowicz algebras.
[math](3)\implies(1)[/math] This is something trivial, that we already know from chapter 2.
The above result is not the end of the story, because one can still ask what are the cocommutative Woronowicz algebras, without reference to the equivalence relation. More generally, we are led in this way into the question, that we have usually avoided so far, as being not part of the “compact” philosophy, of computing the equivalence class of a given Woronowicz algebra [math]A[/math]. We first have here the following construction:
Given a Woronowicz algebra [math](A,u)[/math], the enveloping [math]C^*[/math]-algebra [math]A_{full}[/math] of the algebra of “smooth functions” [math]\mathcal A= \lt u_{ij} \gt [/math] has morphisms
There are several assertions here, the idea being as follows:
(1) Consider indeed the algebra [math]A_{full}[/math], obtained by completing the [math]*[/math]-algebra [math]\mathcal A\subset A[/math] with respect to its maximal [math]C^*[/math]-norm. We have then a quotient map, as follows:
By universality of [math]A_{full}[/math], the comultiplication, counit and antipode of [math]A[/math] lift into morphisms [math]\Delta,\varepsilon,S[/math] as in the statement, and the Woronowicz algebra axioms are satisfied.
(2) The fact that we have an equivalence [math]A_{full}\sim A[/math] is clear from definitions, because at the level of [math]*[/math]-algebras of coefficients, the above quotient map [math]\pi[/math] is an isomorphism.
(3) Finally, in the cocommutative case, where [math]A\sim C^*(\Gamma)[/math], the coefficient algebra is [math]\mathcal A=\mathbb C[\Gamma][/math], and the corresponding enveloping [math]C^*[/math]-algebra is [math]A_{full}=C^*(\Gamma)[/math].
Summarizing, in connection with our equivalence class question, we already have an advance, with the construction of a biggest object in each equivalence class:
We can of course stop our study here, by formulating the following statement, which apparently terminates any further discussion about equivalence classes:
Let us call a Woronowicz algebra “full” when the following canonical quotient map is an isomorphism:
The first assertion is clear from Theorem 14.2, which tells us that we have [math]A\sim A_{full}[/math], and the second assertion holds as well, for exactly the same reason.
As a first observation, restricting the attention to the full Woronowicz algebras is more or less what we have being doing so far in this book, with all the algebras that we introduced and studied being full by definition. However, there are several good reasons for not leaving things like this, and for further getting into the subject, one problem for instance coming from the fact that for the non-amenable groups [math]\Gamma[/math], we have:
To be more precise, on the right we have the group von Neumann algebra [math]L(\Gamma)[/math], appearing by definition as the weak closure of [math]\mathbb C[\Gamma][/math], in the left regular representation. It is known that the above non-inclusion happens indeed in the non-amenable case, and in terms of the quantum group [math]G=\widehat{\Gamma}[/math], we are led to the following bizarre conclusion:
In other words, we have noncommutative continuous functions which are not measurable. This is something that we must clarify. Welcome to functional analysis.
Before anything, we must warn the reader that a lot of modesty and faith is needed, in order to deal with such questions. We are basically doing quantum mechanics here, where the moving objects don't have clear positions or speeds, and where the precise laws of motion are not known, and where any piece of extra data costs a few billion dollars. Thus, the fact that we have [math]C(G)\not\subset L^\infty(G)[/math] is just one problem, among many other.
With this discussion made, let us go back now to Theorem 14.2. As a next step in our study, we can attempt to construct a smallest object [math]A_{red}[/math] in each equivalence class. The situation here is more tricky, and we have the following statement:
Given a Woronowicz algebra [math](A,u)[/math], its quotient [math]A\to A_{red}[/math] by the null ideal of the Haar integration [math]tr:A\to\mathbb C[/math] has morphisms as follows,
We have several assertions here, the idea being as follows:
(1) Consider indeed the algebra [math]A_{red}[/math], obtained by dividing [math]A[/math] by the null ideal of the Haar integration [math]tr:A\to\mathbb C[/math]. We have then a quotient map, as follows:
Also, by GNS construction, we have an embedding as follows:
By using these morphisms [math]\pi,i[/math], we can see that the comultiplication, counit and antipode of the [math]*[/math]-algebra [math]\mathcal A[/math] lift into morphisms [math]\Delta,\varepsilon,S[/math] as in the statement, or, equivalently, that the comultiplication, counit and antipode of the [math]C^*[/math]-algebra [math]A[/math] factorize into morphisms [math]\Delta,\varepsilon,S[/math] as in the statement. Thus, we have our morphisms, as claimed.
(2) In the case where the morphisms [math]\Delta,\varepsilon,S[/math] that we just constructed lift, as indicated in the statement, the Woronowicz algebra axioms are clearly satisfied, and so the algebra [math]A_{red}[/math], together with the matrix [math]u=(u_{ij})[/math], is a Woronowicz algebra, in our sense.
(3) The fact that we have an equivalence [math]A_{red}\sim A[/math] is clear from definitions, because at the level of [math]*[/math]-algebras of coefficients, the above quotient map [math]\pi[/math] is an isomorphism.
(4) Finally, in the cocommutative case, where [math]A\sim C^*(\Gamma)[/math], the above embedding [math]i[/math] is the left regular representation, and so we have [math]A_{red}=C^*_{red}(\Gamma)[/math], as claimed.
With the above result in hand, which is complementary to Theorem 14.2, we can now answer some of our philosophical questions, the idea being as follows:
- In the group dual case we have [math]C^*_{red}(\Gamma)\subset L(\Gamma)[/math], as subalgebras of [math]B(l^2(\Gamma))[/math], and so in terms of the compact quantum group [math]G=\widehat{\Gamma}[/math], the conclusion is that we have [math]C(G)\subset L^\infty(G)[/math], as we should, with the convention [math]C(G)=C^*_{red}(\Gamma)[/math].
- In view of this, it is tempting to modify our Woronowicz algebra axioms, with [math]\Delta,\varepsilon,S[/math] being redefined as in the first part of Theorem 14.4, as to include the reduced group algebras [math]C^*_{red}(\Gamma)[/math], and more generally, all the algebras [math]A_{red}[/math].
- With such a modification done, we could call then a Woronowicz algebra “reduced” when the quotient map [math]A\to A_{red}[/math] is an isomorphism. This would lead to a nice situation like in Proposition 14.3, with 1 object per equivalence class.
- However, we will not do this, simply because the bulk of the present book, which is behind us, is full of interesting examples of Woronowicz algebras constructed with generators and relations, which are full by definition.
In short, nevermind for the philosophy, we will keep our axioms which are nice, simple and powerful, keeping however in mind the fact that the full picture is as follows:
Given a Woronowicz algebra [math]A[/math], we have morphisms
This is something rather philosophical, coming by putting together the results that we have, namely Theorem 14.2 and Theorem 14.4.
General references
Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].