1c. Free spheres

[math] \newcommand{\mathds}{\mathbb}[/math]

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With the above formalism is hand, we can go ahead, and construct two free quadruplets [math](S,T,U,K)[/math], in analogy with those corresponding to the classical real and complex geometries. Let us begin with the spheres. Following , , we have:

Definition

We have free real and complex spheres, defined via

[[math]] C(S^{N-1}_{\mathbb R,+})=C^*\left(x_1,\ldots,x_N\Big|x_i=x_i^*,\sum_ix_i^2=1\right) [[/math]]

[[math]] C(S^{N-1}_{\mathbb C,+})=C^*\left(x_1,\ldots,x_N\Big|\sum_ix_ix_i^*=\sum_ix_i^*x_i=1\right) [[/math]]

where the symbol [math]C^*[/math] stands for universal enveloping [math]C^*[/math]-algebra.

All this deserves some explanations. Given an integer [math]N\in\mathbb N[/math], consider the free complex algebra on [math]2N[/math] variables, denoted [math]x_1,\ldots,x_N[/math] and [math]x_1^*,\ldots,x_N^*[/math]:

[[math]] A=\Big \lt x_1,\ldots,x_N,x_1^*,\ldots,x_N^*\Big \gt [[/math]]

This algebra has an involution [math]*:A\to A[/math], given by [math]x_i\leftrightarrow x_i^*[/math]. Now let us consider the following [math]*[/math]-algebra quotients of our [math]*[/math]-algebra [math]A[/math]:

[[math]] \begin{eqnarray*} A_R&=&A\Big/\Big \lt x_i=x_i^*,\sum_ix_i^2=1\Big \gt \\ A_C&=&A\Big/\Big \lt \sum_ix_ix_i^*=\sum_ix_i^*x_i=1\Big \gt \end{eqnarray*} [[/math]]

Since the first relations imply the second ones, we have quotient maps as follows:

[[math]] A\to A_C\to A_R [[/math]]

Our claim now is both [math]A_C,A_R[/math] admit enveloping [math]C^*[/math]-algebras, in the sense that the biggest [math]C^*[/math]-norms on these [math]*[/math]-algebras are bounded. We only have to check this for the bigger algebra [math]A_C[/math]. But here, our claim follows from the following estimate:

[[math]] ||x_i||^2 =||x_ix_i^*|| \leq\Big|\Big|\sum_ix_ix_i^*\Big|\Big| =1 [[/math]]

Summarizing, our claim is proved, so we can define [math]C(S^{N-1}_{\mathbb R,+}),C(S^{N-1}_{\mathbb C,+})[/math] as being the enveloping [math]C^*[/math]-algebras of [math]A_R,A_C[/math], and so Definition 1.16 makes sense.

In order to formulate now some results, let us introduce as well:

Definition

Given a compact quantum space [math]X[/math], its classical version is the usual compact space [math]X_{class}\subset X[/math] obtained by dividing [math]C(X)[/math] by its commutator ideal:

[[math]] C(X_{class})=C(X)/I\quad,\quad I= \lt [a,b] \gt [[/math]]

In this situation, we also say that [math]X[/math] appears as a “liberation” of [math]X[/math].

In other words, the space [math]X_{class}[/math] appears as the Gelfand spectrum of the commutative [math]C^*[/math]-algebra [math]C(X)/I[/math]. Observe in particular that [math]X_{class}[/math] is indeed a classical space. As a first result now, regarding the above free spheres, we have:

Theorem

We have embeddings of compact quantum spaces, as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ S^{N-1}_{\mathbb R,+}\ar[r]&S^{N-1}_{\mathbb C,+}\\ S^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_\mathbb C\ar[u] } [[/math]]

and the spaces on top appear as liberations of the spaces on the bottom.

Show Proof

The first assertion, regarding the inclusions, comes from the fact that at the level of the associated [math]C^*[/math]-algebras, we have surjective maps, as follows:

[[math]] \xymatrix@R=15mm@C=15mm{ C(S^{N-1}_{\mathbb R,+})\ar[d]&C(S^{N-1}_{\mathbb C,+})\ar[d]\ar[l]\\ C(S^{N-1}_\mathbb R)&C(S^{N-1}_\mathbb C)\ar[l] } [[/math]]

For the second assertion, we must establish the following isomorphisms, where the symbol [math]C^*_{comm}[/math] stands for “universal commutative [math]C^*[/math]-algebra generated by”:

[[math]] C(S^{N-1}_\mathbb R)=C^*_{comm}\left(x_1,\ldots,x_N\Big|x_i=x_i^*,\sum_ix_i^2=1\right) [[/math]]

[[math]] C(S^{N-1}_\mathbb C)=C^*_{comm}\left(x_1,\ldots,x_N\Big|\sum_ix_ix_i^*=\sum_ix_i^*x_i=1\right) [[/math]]

As a first observation, it is enough to establish the second isomorphism, because the first one will follow from it, simply by dividing by the relations [math]x_i=x_i^*[/math]. So, consider the second universal commutative [math]C^*[/math]-algebra [math]A[/math] constructed above. Since the standard coordinates on [math]S^{N-1}_\mathbb C[/math] satisfy the defining relations for [math]A[/math], we have a quotient map of as follows, mapping standard coordinates to standard coordinates:

[[math]] A\to C(S^{N-1}_\mathbb C) [[/math]]

Conversely, let us write [math]A=C(S)[/math], by using the Gelfand theorem. The variables [math]x_1,\ldots,x_N[/math] become in this way true coordinates, providing us with an embedding [math]S\subset\mathbb C^N[/math]. Also, the quadratic relations become [math]\sum_i|x_i|^2=1[/math], so we have [math]S\subset S^{N-1}_\mathbb C[/math]. Thus, we have a quotient map [math]C(S^{N-1}_\mathbb C)\to A[/math], as desired, and this gives all the results.

■

Summarizing, we are done with the spheres. Before getting into tori, let us talk about algebraic manifolds. By using the free spheres constructed above, we can formulate:

Definition

A real algebraic manifold [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

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_i(x_1,\ldots,x_N)=0\Big \gt [[/math]]

for certain family of noncommutative polynomials, as follows:

[[math]] f_i\in\mathbb C \lt x_1,\ldots,x_N \gt [[/math]]

We denote by [math]\mathcal C(X)[/math] the [math]*[/math]-subalgebra of [math]C(X)[/math] generated by the coordinates [math]x_1,\ldots,x_N[/math].

As a basic example here, we have the free real sphere [math]S^{N-1}_{\mathbb R,+}[/math]. The classical spheres [math]S^{N-1}_\mathbb C,S^{N-1}_\mathbb R[/math], and their real submanifolds, are covered as well by this formalism. At the level of the general theory, we have the following version of the Gelfand theorem:

Theorem

If [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is an algebraic manifold, as above, we have

[[math]] X_{class}=\left\{x\in S^{N-1}_\mathbb C\Big|f_i(x_1,\ldots,x_N)=0\right\} [[/math]]

and [math]X[/math] appears as a liberation of [math]X_{class}[/math].

Show Proof

This is something that we 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:

[[math]] C(X_{class})\to C(X_{class}') [[/math]]

Conversely now, from [math]X\subset S^{N-1}_{\mathbb C,+}[/math] we obtain [math]X_{class}\subset S^{N-1}_\mathbb C[/math]. Now since the relations defining [math]X_{class}'[/math] are satisfied by [math]X_{class}[/math], we obtain an inclusion [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:

[[math]] C(X_{class}')\to C(X_{class}) [[/math]]

Thus, we have constructed a pair of inverse morphisms, and we are done.

■

Finally, once again at the level of the general theory, we have:

Definition

We agree to identify two real algebraic submanifolds [math]X,Y\subset S^{N-1}_{\mathbb C,+}[/math] when we have a [math]*[/math]-algebra isomorphism between [math]*[/math]-algebras of coordinates

[[math]] f:\mathcal C(Y)\to\mathcal C(X) [[/math]]

mapping standard coordinates to standard coordinates.

We will see later the reasons for making this convention, coming from amenability.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].