1d. Axiomatization fix
Let us get back now to the bad functoriality properties of the Gelfand correspondence, coming from the fact that certain compact quantum spaces, such as the duals [math]\widehat{\Gamma}[/math] of the discrete groups [math]\Gamma[/math], can be represented by several [math]C^*[/math]-algebras, instead of one. We can fix these issues by using the GNS theorem, as follows:
The category of compact quantum measured spaces [math](X,\mu)[/math] is the category of the [math]C^*[/math]-algebras with faithful traces [math](A,tr)[/math], with the arrows reversed. In the case where we have a [math]C^*[/math]-algebra [math]A[/math] with a non-faithful trace [math]tr[/math], we can still talk about the corresponding space [math](X,\mu)[/math], by performing the GNS construction.
Observe that this definition fixes the functoriality problem with Gelfand duality, at least for the group algebras. Indeed, in the context of the comments following Definition 1.25, consider an arbitrary intermediate [math]C^*[/math]-algebra, as follows:
If we perform the GNS construction with respect to the canonical trace, we obtain the reduced algebra [math]C^*_{red}(\Gamma)[/math]. Thus, all these algebras [math]A[/math] correspond to a unique compact quantum measured space in the above sense, which is the abstract group dual [math]\widehat{\Gamma}[/math]. Let us record a statement about this finding, as follows:
The category of group duals [math]\widehat{\Gamma}[/math] is a well-defined subcategory of the category of compact quantum measured spaces, with each [math]\widehat{\Gamma}[/math] corresponding to the full group algebra [math]C^*(\Gamma)[/math], or the reduced group algebra [math]C^*_{red}(\Gamma)[/math], or any algebra in between.
This is more of an empty statement, coming from the above discussion.
With this in hand, it is tempting to go even further, namely forgetting about the [math]C^*[/math]-algebras, and trying to axiomatize instead the operator algebras of type [math]L^\infty(X)[/math]. Such an axiomatization is possible, and the resulting class of operator algebras consists of a certain special type of [math]C^*[/math]-algebras, called “finite von Neumann algebras”.
However, and here comes our point, doing so would be bad, and would lead to a weak theory, because many spaces such as the compact groups, or the compact homogeneous spaces, do not come with a measure by definition, but rather by theorem.
In short, our “fix” is not a very good fix, and if we want a really strong theory, we must invent something else. In order to do so, our idea will be that of restricting the attention to certain special classes of quantum algebraic manifolds, as follows:
A real algebraic submanifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is a closed quantum subspace defined, at the level of the corresponding [math]C^*[/math]-algebra, by a formula of type
Observe that any family [math]f_i\in\mathbb C \lt x_1,\ldots,x_N \gt [/math] produces such a manifold [math]X[/math], simply by defining an algebra [math]C(X)[/math] as above. Observe also that the use of [math]S^{N-1}_{\mathbb C,+}[/math] is essential in all this, because the quadratic condition [math]\sum_ix_ix_i^*=\sum_ix_i^*x_i=1[/math] gives by positivity [math]||x_i||\leq1[/math] for any [math]i[/math], and so guarantees the fact that the universal [math]C^*[/math]-norm is bounded.
We have already met such manifolds, in the context of the free spheres, free tori, and more generally in Proposition 1.22 above. Here is a list of examples:
The following are algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math]:
- The spheres [math]S^{N-1}_\mathbb R\subset S^{N-1}_\mathbb C,S^{N-1}_{\mathbb R,+}\subset S^{N-1}_{\mathbb C,+}[/math].
- Any compact Lie group, [math]G\subset U_n[/math], when [math]N=n^2[/math].
- The duals [math]\widehat{\Gamma}[/math] of finitely generated groups, [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math].
These facts are all well-known, the proof being as follows:
(1) This is true by definition of our various spheres.
(2) Given a closed subgroup [math]G\subset U_n[/math], we have indeed an embedding [math]G\subset S^{N-1}_\mathbb C[/math], with [math]N=n^2[/math], given in double indices by:
We can further compose this embedding with the standard embedding [math]S^{N-1}_\mathbb C\subset S^{N-1}_{\mathbb C,+}[/math], and we obtain an embedding as desired. As for the fact that we obtain indeed a real algebraic manifold, this is well-known, coming either from Lie theory or from Tannakian duality. We will be back to this later on, in a more general context.
(3) This follows from the fact that the variables [math]x_i=\frac{g_i}{\sqrt{N}}[/math] satisfy the quadratic relations [math]\sum_ix_ix_i^*=\sum_ix_i^*x_i=1[/math], with the algebricity claim of the manifold being clear.
At the level of the general theory, we have the following version of the Gelfand theorem, which is something very useful, and that we will use many times in what follows:
When [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is an algebraic manifold, given by
This is something that already met, in the context of the free spheres. In general, the proof is similar, by using the Gelfand theorem. Indeed, if we denote by [math]X_{class}'[/math] the manifold constructed in the statement, then we have a quotient map of [math]C^*[/math]-algebras as follows, mapping standard coordinates to standard coordinates:
Conversely now, from [math]X\subset S^{N-1}_{\mathbb C,+}[/math] we obtain [math]X_{class}\subset S^{N-1}_\mathbb C[/math], and since the relations defining [math]X_{class}'[/math] are satisfied by [math]X_{class}[/math], we obtain an inclusion of subspaces [math]X_{class}\subset X_{class}'[/math]. Thus, at the level of algebras of continuous functions, we have a quotient map of [math]C^*[/math]-algebras as follows, mapping standard coordinates to standard coordinates:
Thus, we have constructed a pair of inverse morphisms, and we are done.
With these results in hand, we are now ready for formulating our second “fix” for the functoriality issues of the Gelfand correspondence, as follows:
The category of the real algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is the category of the universal [math]C^*[/math]-algebras of type
In other words, what we are doing here is that of proposing a definition for the morphisms between the compact quantum spaces, in the particular case where these compact quantum spaces are algebraic submanifolds of the free complex sphere [math]S^{N-1}_{\mathbb C,+}[/math]. And the point is that this “fix” perfectly works for the group duals, as follows:
The category of finitely generated groups [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], with the morphisms being the group morphisms mapping generators to generators, embeds contravariantly via [math]\Gamma\to\widehat{\Gamma}[/math] into the category of real algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math].
We know from Proposition 1.30 above that, given a finitely generated group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], we have an embedding of algebraic manifolds [math]\widehat{\Gamma}\subset S^{N-1}_{\mathbb C,+}[/math], given by [math]x_i=\frac{g_i}{\sqrt{N}}[/math]. Now since a morphism [math]C[\Gamma]\to C[\Lambda][/math] mapping coordinates to coordinates means a morphism of groups [math]\Gamma\to\Lambda[/math] mapping generators to generators, our notion of isomorphism is indeed the correct one, as claimed.
We will see later on that Theorem 1.33 has various extensions to the quantum groups and quantum homogeneous spaces that we will be interested in, which are all algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math]. We will also see that all these manifolds have Haar integration functionals, which are traces, and so that for these manifolds, our functoriality fix from Definition 1.32 coincides with the “von Neumann” fix from Definition 1.27.
So, this will be our formalism, and operator algebra knowledge required. We should mention that our approach heavily relies on Woronowicz's philosophy in [1]. Also, part of the above has been folklore for a long time, with the details worked out in [2].
General references
Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].