12a. Matrix models

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

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

You can model everything with random matrices, the saying in analysis goes. We have already seen an instance of this phenomenon in chapter 9, when talking about half-liberation. To be more precise, for certain manifolds [math]X\subset S^{N-1}_{\mathbb C,*}[/math], we have constructed embeddings of algebras of the following type, with [math]Y[/math] being a certain classical manifold, and [math]T_1,\ldots,T_N\in M_2(C(Y))[/math] being certain suitable antidiagonal [math]2\times 2[/math] matrices:

[[math]] \pi:C(X)\subset M_2(C(Y))\quad,\quad x_i\to T_i [[/math]]

These models, which are quite powerful, were used afterwards in order to establish several non-trivial results on the original half-classical manifolds [math]X\subset S^{N-1}_{\mathbb C,*}[/math]. Indeed, some knowledge and patience helping, any computation inside the target algebra [math]M_2(C(Y))[/math] can only be fun and doable, and produce results about [math]X\subset S^{N-1}_{\mathbb C,*}[/math] itself.

We discuss here, following , modeling questions for general manifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math], by using the same idea, suitably modified and generalized, as to cover most of the manifolds that we are interested in. Let us start with a broad definition, as follows:

Definition

A model for a real algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is a morphism of [math]C^*[/math]-algebras of the following type,

[[math]] \pi:C(X)\to B [[/math]]

with [math]B[/math] being a [math]C^*[/math]-algebra, called target of the model. We say that the model is faithful if [math]\pi[/math] is faithful, in the usual sense.

Obviously, this is something too broad, because we can simply take [math]B=C(X)[/math], and we have in this way our faithful model, which is of course something unuseful:

[[math]] id:C(X)\to C(X) [[/math]]

Thus, we must suitably restrict the class of target algebras [math]B[/math] that we use, to algebras that we “know well”. However, this is something quite tricky, because if we want our model to be faithful, we cannot use simple algebras like the algebras [math]M_2(C(Y))[/math] used in the half-classical setting. In short, we are running into some difficulties here, of functional analytic nature, and a systematic discussion of all this is needed.

As a first objective, let us try to understand if an arbitrary manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] can be modelled by using familiar variables such as usual matrices, or operators. The answer here is yes, when using operators on a separable Hilbert space, with this coming from the GNS representation theorem, that we know from chapter 1, which is as follows:

Theorem

Any [math]C^*[/math]-algebra [math]A[/math] appears as closed [math]*[/math]-algebra of operators on a Hilbert space, [math]A\subset B(H)[/math], in the following way:

• In the commutative case, where [math]A=C(X)[/math], we can set [math]H=L^2(X)[/math], with respect to some probability measure on [math]X[/math], and use the embedding [math]g\to(g\to fg)[/math].
• In general, we can set [math]H=L^2(A)[/math], with respect to some faithful positive trace [math]tr:A\to\mathbb C[/math], and then use a similar embedding, [math]a\to(b\to ab)[/math].

Show Proof

This is something that we already know, from chapter 1, coming from basic measure theory and functional analysis, the idea being as follows:

(1) In the commutative case, where [math]A=C(X)[/math] by the Gelfand theorem, we can pick a probability measure on [math]X[/math], and then we have an embedding as follows:

[[math]] C(X)\subset B(L^2(X))\quad,\quad f\to(g\to fg) [[/math]]

(2) In general, assuming that a linear form [math]\varphi:A\to\mathbb C[/math] has suitable positivity properties, we can define a scalar product on [math]A[/math], by the following formula:

[[math]] \lt a,b \gt =\varphi(ab^*) [[/math]]

By completing we obtain a Hilbert space [math]H[/math], and we have a representation as follows, called GNS representation of our algebra with respect to the linear form [math]\varphi[/math]:

[[math]] A\to B(H)\quad,\quad a\to(b\to ab) [[/math]]

Moreover, when [math]\varphi:A\to\mathbb C[/math] has suitable faithfulness properties, making it analogous to the integration functionals [math]\int_X:A\to\mathbb C[/math] from the commutative case, with respect to faithful probability measures on [math]X[/math], this representation is faithful, as desired.

■

Now back to our questions, the above result tells us that we have:

Theorem

Given an algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math], coming via

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

we have a morphism of [math]C^*[/math]-algebras as follows,

[[math]] \pi:C(X)\to B(H)\quad,\quad x_i\to T_i [[/math]]

whenever the operators [math]T_i\in B(H)[/math] satisfy the following relations:

[[math]] \sum_iT_iT_i^*=\sum_iT_i^*T_i=1\quad,\quad f_\alpha(T_1,\ldots,T_N)=0 [[/math]]

Moreover, we can always find a Hilbert space [math]H[/math] and operators [math]\{T_i\}[/math] such that [math]\pi[/math] is faithful.

Show Proof

Here the first assertion is more of an empty statement, explaining the definition of the algebra [math]C(X)[/math], via generators and relations, and the second assertion is something non-trivial, coming as a consequence of the GNS theorem.

■

In practice now, all this is a bit too general, and not very useful. We need a good family of target algebras [math]B[/math], that we understand well. And here, we can use:

Definition

A random matrix [math]C^*[/math]-algebra is an algebra of type

[[math]] B=M_K(C(T)) [[/math]]

with [math]T[/math] being a compact space, and [math]K\in\mathbb N[/math] being an integer.

The terminology here comes from the fact that, in practice, the space [math]T[/math] usually comes with a probability measure on it, which makes the elements of [math]B[/math] “random matrices”. Observe that we can write our random matrix algebra as follows:

[[math]] B=M_K(\mathbb C)\otimes C(T) [[/math]]

Thus, the random matrix algebras appear by definition as tensor products of the simplest types of [math]C^*[/math]-algebras that we know, namely the full matrix algebras, [math]M_K(\mathbb C)[/math] with [math]K\in\mathbb N[/math], and the commutative algebras, [math]C(T)[/math], with [math]T[/math] being a compact space. Getting back now to our modelling questions for manifolds, we can formulate:

Definition

A matrix model for a noncommutative algebraic manifold

[[math]] X\subset S^{N-1}_{\mathbb C,+} [[/math]]

is a morphism of [math]C^*[/math]-algebras of the following type,

[[math]] \pi:C(X)\to M_K(C(T)) [[/math]]

with [math]T[/math] being a compact space, and [math]K\in\mathbb N[/math] being an integer.

As a first observation, when [math]X[/math] happens to be classical, we can take [math]K=1[/math] and [math]T=X[/math], and we have a faithful model for our manifold, namely:

[[math]] id:C(X)\to M_1(C(X)) [[/math]]

In general, we cannot use [math]K=1[/math], and the smallest value [math]K\in\mathbb N[/math] doing the job, if any, will correspond somehow to the “degree of noncommutativity” of our manifold.

Before getting into this, we would like to clarify a few abstract issues. As mentioned above, the algebras of type [math]B=M_K(C(T))[/math] are called random matrix [math]C^*[/math]-algebras. The reason for this is the fact that most of the interesting compact spaces [math]T[/math] come by definition with a natural probability measure of them. Thus, [math]B[/math] is a subalgebra of the bigger algebra [math]B''=M_K(L^\infty(T))[/math], usually known as a “random matrix algebra”.

This perspective is quite interesting for us, because most of our examples of manifolds [math]X\subset X^{N-1}_{\mathbb C,+}[/math] appear as homogeneous spaces, and so are measured spaces too. Thus, we can further ask for our models [math]C(X)\to M_K(C(T))[/math] to extend into models of the following type, which can be of help in connection with integration problems:

[[math]] L^\infty(X)\to M_K(L^\infty(T)) [[/math]]

In short, time now to talk about [math]L^\infty[/math]-functions, in the noncommutative setting.

General references

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