Noncommutative Torus

Definitions#

In terms of commutative relations#

An \(n\)-dimensional noncommutative torus is an associative algebra with involution, having unitary generators \(u_1,\ldots,u_n\), obeying the relations

\[u_k u_j = \exp(2\pi i \theta_{kj}) u_j u_k\]

where \(\theta\) is an antisymmetric matrix. The same name is used for different completions of this algebra, but we shall consider it as a \(C^\ast\)-algebra \(A_\theta\), the universal \(C^\ast\)-algebra generated by \(n\) unitary operators satisfying the above relation.

In terms of twisted group \(C^\ast\)-algebras#

First let's recall what is a twisted group \(C^\ast\)-algebra. Let \(G\) be a locally compact group and let \(c\) be a 2-cocylce on \(G\) with values in the circle group \(\mathbb{T}\):

\[c(r,st)c(s,t)=c(r,s)c(rs,t)\]

Let the definition of convolution of functions on \(G\) be modified by a “twist”:

\[(f\ast_c g)(t) = \int f(s) g(s^{-1}t) c(s,s^{-1}t) dt\]

and undergo the same process as forming the group \(C^\ast\)-algebra, the resulting algebra is a twisted group \(C^\ast\)-algebra.

When we take \(G=\mathbb{Z}^n\) and let \(c\) be \(c_\theta\), in the form of

\[c_\theta(u,v) = \exp(2\pi i \langle \theta u,v\rangle)\]

where \(\theta\) is an operator on \(\mathbb{R}^n\) that is skew-adjoint for its usual inner product. This results in a noncommutative torus. Why so can be seen in the next definition in terms of deformation quantization of a torus.

Group \(C^\ast\)-algebras#

We shall recall the definition of group \(C^\ast\)-algebra and indicate what it is for. \(L^1(G)\) (actually \(L^1(G,\mu)\) where we have suppressed the left Haar measure, but since given a different left Haar measure \(\mu' = \alpha \mu\) we have by \(f\mapsto \alpha f\) an isomorphism between \(L^1(G\),\mu) and \(L^1(G,\mu')\) we don't care) is a Banach algebra under this multiplication; \(f^\ast\), the involution, is defined by \(f^\ast(t) = \Delta_G(t^{-1})\bar{f}(t^{-1})\) making \(L^1(G)\) into a Banach \(\ast\)-algebra.

Suppose that \(\pi\) is a strongly continuous unitary representation of \(G\) on a Hilbert space \(\mathcal{H}\), which is continuous for the strong operator topology. For \(f\in L^1(G)\) the operator

\[\pi(f) = \int_G f(t)\pi(t) d\mu(t)\]

is well defined and bounded, and \(f\mapsto \pi(f)\) is a nondegenerate \(\ast\)-homomorphism from the Banach \(\ast\)-algebra \(L^1(G)\) to \(\mathcal{L}(\mathcal{H})\), this representation of \(L^(G)\) is called the integrated form of \(\pi\).

A theorem:

Every non-degenerate representation of \(L^1(G)\) on a Hilbert space arises from a strongly continuous unitary representation of \(G\); thus, there's a 1-to-1 correspondence between the representation theories of \(G\) and \(L^1(G)\).

Now the (full) group \(C^\ast\)-algebra \(C^\ast(G)\) is the universal enveloping \(C^\ast\)-algebra of the Banach \(\ast\)-algebra \(L^1(G)\), that is, if \(f\in L^1(G)\), define \(||f|| = ||f||_{C^\ast(G)}\) by

\[||f|| = \mathrm{sup}\{||\pi(f)||:\pi\text{ a representation of }L^1(G)\}\]

and let \(C^\ast(G)\) be the completion of \((L^1(G),||\cdot||)\). Actually \(||f|| = \mathrm{sup} \{||\pi(f)||\}\) where \(\pi\) runs over all nondegenerate \(\ast\)-homomorphisms \(L^1(G)\to \mathcal{L}(\mathcal{E})\) for all Hilbert modules \(\mathcal{E}\).

In terms of deformation quantization#

Let \(\mathbb{T}^n\) be an \(n\)-torus. Let \(C^\infty(\mathbb{T}^n)\) be the algebra of smooth complex-valued functions on \(\mathbb{T}^n\). Let \(\theta\) be a real antisymmetric \(n\times n\) matrix. On \(C^\ast(\mathbb{T}^n)\), \(\theta\) defines a Poisson bracket:

\[\{f,g\} = \sum_{jk} \theta_{jk} (\partial f/\partial x_j)(\partial g/\partial x_k)\]

for \(f,g\in C^\infty(\mathbb{T}^n)\). Fourier transform \(C^\infty(\mathbb{T}^n)\) to \(\mathcal{S}(\mathbb{Z}^n)\), the space of complex-valued Schwartz functions on \(\mathbb{Z}^n\), then the Fourier transform carries the pointwise multiplication on \(C^\infty(\mathbb{T}^n)\) to convolution on \(\mathcal{S}(\mathbb{Z}^n)\), and the Poisson bracket is carried to

\[\{\phi,\psi\}(p) = -4\pi^2 \sum_q \phi(q) \psi(p-q)\gamma(q,p-q)\]

for \(\phi,\psi \in \mathcal{S}(\mathbb{Z}^n)\) and \(p,q\in\mathbb{Z}^n\), together with

\[\gamma(p,q) = \sum_{j,k} p_j q_k.\]

Now let \(\sigma_\hbar\) be a 2-cocycle on \(\mathbb{Z}^n\), defined by

\[\sigma_\hbar(p,q) = \exp(-\pi i \gamma(p,q)),\]

and put

\[(\phi\ast_\hbar\psi)(p) = \sum_q\phi(q)\psi(p-q)\sigma_\hbar(q,p-q),\]

for each \(\hbar\in\R\) and define the involution on \(\mathcal{S}(\mathbb{Z}^n)\) to be that coming from complex conjugation on \(C^\ast(\mathbb{T}^n)\) so that \(\phi^\ast(p) = \bar{\phi}(-p)\).

Now for each \(\hbar\) define the norm \(||\cdot||_\hbar\) on \(\mathcal{S}(\Z^n)\) to be the operator norm for the action of \(\mathcal{S}(\Z^n)\) on \(\ell^2(\Z^n)\) given by the formula as used above to define the product \(\ast_\hbar\). Let \(C_\hbar\) be \(C^\infty(\mathbb{T}^n)\) equipped with product, involution, and norm obtained by pulling back through the inverse Fourier transform the product \(\ast_\hbar\), involution, and norm \(||\cdot||_\hbar\). Then the completions of the \(C_\hbar\)'s form a continuous field of \(C^\ast\)-algebras, and that for \(f,g\in C^\infty(\mathbb{T}^n)\) one has

\[||(f\ast_\hbar g - g\ast_\hbar f)/i\hbar - \{f,g\}||_\hbar \to 0\]

as \(\hbar\to 0\). This means that the \(C_\hbar\)'s form a strict deformation quantization of \(C^\ast(\mathbb{T}^n)\) in the direction of the Poisson bracket defined with \(\theta\), since \(C_0\) is just \(C^\ast(\mathbb{T}^n)\) with its usual pointwise multiplication and supremum norm.

Let the \(C^\ast\) completion of the algebra for \(\hbar = 1\) be denoted by \(A_\theta\), this is a noncommutative \(n\)-torus.