DISCRETIZED CCR ALGEBRAS

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

To the memory of John Bunce

Abstract. We discuss how the canonical commutation relations must be modified in order to make appropriate numerical models of quantum systems. The C^{∗}-algebras associated with the discretized CCRs are the non-commutative spheres of Bratteli, Elliott, Evans and Kishimoto.

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. C^{∗}-algebras, commutation relations, quantum mechanics. This research was supported in part by NSF grant DMS89-12362

Typeset by AMS-TEX

1. Introduction.

We consider the problem of discretizing the Hamiltonian of a one-dimensional quantum system in a form that is appropriate for carrying out numerical studies. Specifically, we start with a formal Schrödinger operator

H = \frac{1}{2}P^{2} + v(Q)

acting on the Hilbert space L^{2}(ℝ), where P and Q are the canonical operators

P = −i\frac{d}{dx}
Q = multiplication by x,

and v is a real-valued continuous function of a real variable. The problem of discretizing H is that of finding an approximation to H which satisfies two requirements: (a) the basic principles of numerical analysis are satisfied, and (b) the uncertainty principle is preserved.

In [1, §§ 1–2], we argued that in order to satisfy these two conditions one must first replace P, Q with the pair

P_{τ} = \frac{1}{τ} sin(τP)
Q_{τ} = \frac{1}{τ} sin(τQ).

Here, τ is a fixed positive real number, the numerical step size. The discretized Hamiltonian is then defined as the following bounded self-adjoint operator on L^{2}(ℝ):

H_{τ} = \frac{1}{2}P_{τ}^{2} + v(Q_{τ}).

Obviously, H_{τ} belongs to the unital C^{∗}-algebra C^{∗}(P_{τ} , Q_{τ}) generated by P_{τ} and Q_{τ}. We show that when τ^{2}/π is irrational (e.g., when τ is a rational number), C^{∗}(P_{τ} , Q_{τ}) is isomorphic to the non-commutative sphere B_{τ^{2}} of Bratteli, Evans, Elliott and Kishimoto [5][6]; hence it is a simple C^{∗}-algebra with a unique trace. We also describe the way in which the canonical commutation relations must be “discretized” in order to accommodate pairs of operators (P_{τ} , Q_{τ}) of this type. Together, these observations serve to make a more philosophical point, namely noncommutative spheres will arise in any serious attempt to model quantum systems on a computer.

In the “linear” case where v has the form v(x) = cx^{2}/2, c being a positive constant, the operator H_{τ} turns out to be unitarily equivalent to an operator of the form λM + μI, where λ and μ are real constants and M is the almost Mathieu Hamiltonian

M = U + U^{∗} + c(V + V^{∗}),

associated with a pair of unitary operators U, V satisfying

V U = e^{i4τ^{2}}UV.

An extensive amount of work has been done to compute the spectra of such operators. Here, we mention only [2], [3], [4], [9], [16] and refer the reader to the monograph [8] for further references.

Finally, I would like to thank Larry Schweitzer for pointing out the references [11] and [13] (as well as the relevance of his own work [17]) in connection with the spectral invariance property of the Banach ∗-algebra l^{1}(ℤ ⊕ ℤ, ω).

2. Discretized CCR algebras.

Let θ be a real number such that θ/π is irrational, and let ω be the bicharacter of the discrete abelian group G = ℤ ⊕ ℤ defined by

2.1.  ω((m, n), (p, q)) = e^{i(np−mq)θ/2}

A uniformly bounded family {D_{x} : x ∈ G} of self-adjoint operators on a Hilbert space H is said to satisfy the discretized canonical commutation relations if

2.2  D_{x}D_{y} = ω(x, y)D_{x+y} + ω(y, x)D_{x−y},  x, y ∈ G.

Remarks. Notice that (2.2) is a generalization of the elementary trigonometric identity

2 cos A cos B = cos(A + B) + cos(A − B),

in which phase shifts have been added by way of the cocycle ω. Indeed, for any pair of real numbers α, β, the function D : G → ℝ defined by

D(m, n) = 2 cos(αm + βn)

satisfies (2.2) for the trivial cocycle ω = 1. It is related to formula (2.2) of [5], except that our operators are self-adjoint and the phase factor is associated with a nondegenerate bicharacter ω.

The purpose of this section is to associate a C^{∗}-algebra with the relations (2.2), and to point out some of its basic properties. Let {D_{x} : x ∈ G} satisfy (2.2). It is clear that the norm closed linear span

D = \overline{span}\{D_{x} : x ∈ G\}

is a separable C^{∗}-algebra. Thus by passing from H to the subspace [DH] if necessary, we can assume that D is nondegenerate.

Proposition 2.3.

(i) D_{0} = 2I.

(ii) D_{−x} = D_{x}.

(iii) ∥D_{x}∥ ≤ 2, for every x ∈ G.

proof. Setting y = 0 in (2.2) we obtain D_{x}D_{0} = 2D_{x} for all x ∈ G, from which (i) is evident. Setting x = 0 in (2.2) now leads to 2D_{y} = D_{0}D_{y} = D_{y} + D_{−y}, hence (ii). For (iii), let

M = sup_{x∈G} ∥D_{x}∥.

By hypothesis, M < ∞. Moreover, setting y = x in (2.2) gives

D_{x}^{2} = D_{2x} + D_{0} = D_{2x} + 2I

and thus M^{2} ≤ M + 2. This inequality implies that −1 ≤ M ≤ 2, hence (iii) □

We now construct a Banach ∗-algebra whose representations are associated with operator realizations of (2.2). Let l^{1}(G, ω) denote the Banach space of all absolutely summable complex functions on G, endowed with the multiplication and involution

f ∗ g(x) = ∑_{y} ω(y, x)f(y)g(x − y)
f^{∗}(x) = \overline{f(−x)}.

It is easily checked that the linear subspace

D_{θ} = {f ∈ l^{1}(G, ω) : f(−x) = f(x), x ∈ G}

is in fact a ∗-subalgebra of l^{1}(G, ω). Of course, the adjoint operation in D_{θ} simplifies to f^{∗}(x) = \overline{f(x)}. Moreover, D_{θ} is linearly spanned by the elements

d_{x} = δ_{x} + δ_{−x},

δ_{x} denoting the unit function supported at x, and one has

d_{x}d_{y} = ω(x, y)d_{x+y} + ω(y, x)d_{x−y}
∥d_{x}∥ = 2
d_{x} = d_{−x} = d_{x}^{∗}.

Proposition 2.4. Let {D_{x} : x ∈ G} be a uniformly bounded family of self-adjoint operators on a Hilbert space H satisfying (2.2). Then there is a unique representation π : D_{θ} → B(H) such that

π(d_{x}) = D_{x}, x ∈ G.

proof. By proposition (2.3), we know that ∥D_{x}∥ ≤ 2; hence

π(f) = \frac{1}{2} ∑_{x∈G} f(x)D_{x}

defines a contractive self-adjoint linear mapping of D_{θ} into B(H). Moreover, using (2.2) we have

π(f)π(g) = \frac{1}{4} ∑_{x,y} f(x)g(y)(ω(x, y)D_{x+y} + ω(y, x)D_{x−y})

= \frac{1}{4} ∑_{z,x} f(x)g(z − x)ω(x, z)D_{z} + \frac{1}{4} ∑_{z,x} f(x)g(x − z)ω(−z, x)D_{z}.

Using the fact that g(x − z) = g(z − x) and ω(−z, x) = ω(x, z), the right side becomes

\frac{1}{2} ∑_{z}(∑_{x} f(x)g(z − x)ω(x, z))D_{z} = π(f ∗ g),

as required.

Finally, taking f = δ_{x} + δ_{−x} = d_{x} and using (ii) of (2.3), we find that

π(d_{x}) = \frac{1}{2}(f(x)D_{x} + f(−x)D_{x}) = D_{x}

as required □

Remarks. It follows that the enveloping C^{∗}-algebra C^{∗}(D_{θ}) is the universal C^{∗}- algebra generated by the commutation relations (2.2).

Let α be an automorphism of the discrete abelian group ℤ ⊕ ℤ. Then α is given by a 2 × 2 integer matrix

( \begin{matrix}a & b \\ c & d \end{matrix} )

by way of α(m, n) = (am + bn, cm + dn), where a, b, c, d ∈ ℤ satisfy the condition

det α = ad − bc = ±1.

It follows that

ω(αx, αy) = \{ \begin{matrix}ω(x, y), & if det α = +1 \\ ω(y, x), & if det α = −1. \end{matrix}

Hence the group SL(2, ℤ) of determinant 1 automorphisms acts naturally on D_{θ} (resp. C^{∗}(D_{θ})) as a group of ∗-automorphisms. Any α ∈ aut(ℤ ⊕ ℤ) satisfying det α = −1 gives rise to a ∗-anti-automorphism of D_{θ} (resp. C^{∗}(D_{θ})).

Finally, notice that there is a natural ∗-homomorphism which carries D_{θ} into the irrational rotation C^{∗}-algebra A_{θ}. Indeed, D_{θ} is obviously contained in the larger Banach ∗-algebra l^{1}(ℤ ⊕ ℤ, ω) obtained by simply dropping the requirement that f(−x) = f(x). It is clear that l^{1}(ℤ ⊕ ℤ, ω) is the universal Banach ∗-algebra generated by unitary operators {W_{x} : x ∈ ℤ ⊕ ℤ} satisfying

W_{x}W_{y} = ω(x, y)W_{x+y}, x, y ∈ ℤ ⊕ ℤ.

Because of the formula (2.1) giving ω in terms of θ, the unitary elements U, V defined by U = W_{(1,0)}, V = W_{(0,1)} satisfy V U = e^{iθ}UV, and of course they generate l^{1}(ℤ ⊕ ℤ, ω) as a Banach ∗-algebra. It follows that the enveloping C^{∗}-algebra of l^{1}(ℤ ⊕ ℤ, ω) is A_{θ}. Thus we obtain a morphism of D_{θ} into A_{θ} by simply restricting the completion map

γ : l^{1}(ℤ ⊕ ℤ, ω) → A_{θ}

to D_{θ}. By the universal property of enveloping C^{∗}-algebras there is correspondingly a unique morphism of C^{∗}-algebras

γ_{B} : C^{∗}(D_{θ}) → A_{θ}.

In the next section it will be shown that γ_{B} is injective and we will identify its range.

3. Spectral Invariance and Extensions of States.

Let A be a Banach ∗-algebra with unit, and let

A^{+} = \overline{\{a_{1}^{∗}a_{1} + a_{2}^{∗}a_{2} + ⋯ + a_{n}^{∗}a_{n} : a_{k} ∈ A, n ≥ 1\}}

denote the closed positive cone in A. For simplicity, we assume throughout this section that the completion map γ of A into its enveloping C^{∗}-algebra is injective.

Let B be a unital self-adjoint Banach subalgebra of A. We are interested in determining whether or not the C^{∗}-algebra obtained by closing γ(B) in the norm of C^{∗}(A) is the enveloping C^{∗}-algebra of B. More precisely, we seek conditions under which the ∗-homomorphism γ_{B} : C^{∗}(B) → C^{∗}(A) defined by the commutative diagram

(3.1)  \begin{matrix}B & ⟶^{incl} & A \\ ↓ & & ↓ \\ C^{∗}(B) & ⟶^{γ_{B}} & C^{∗}(A) \end{matrix}

should be injective. Elementary considerations show that the following three conditions are equivalent:

(1) γ_{B} is injective.

(2) Every positive linear functional on B can be extended to a positive linear functional on A.

(3) A^{+} ∩ B ⊆ B^{+}. Note, for example, that the implication (3) ⟹ (2) is the extension theorem of M. G. Krein [15, p. 227], whereas (2) ⟹ (3) follows from a standard separation theorem. It is not hard to find examples showing that these conditions are not always satisfied (see Appendix).

A is said to have the spectral invariance property if for every element a ∈ A which is invertible in C^{∗}(A), we have a^{−1} ∈ A. This is equivalent to the assertion that the spectrum of any element of A is the same whether it is computed in A or in C^{∗}(A), or that A is closed under the holomorphic functional calculus of C^{∗}(A) (see [10, p. 52] for further significant consequences of spectral invariance in more general Frèchet algebras).

A familiar Tauberian theorem of Wiener asserts that if a continuous function on the unit circle never vanishes and has an absolutely convergent Fourier series

f(e^{iθ}) = ∑_{n=−∞}^{+∞} a_{n}e^{inθ},

∑ |a_{n}| < ∞, then 1/f has an absolutely convergent Fourier series. Of course, this is precisely the assertion that the group algebra l^{1}(ℤ) has the spectral invariance property. While this theorem has a simple proof using the Gelfand theory, it is certainly not a triviality.

The significance of spectral invariance for our purposes derives from the following.

Proposition 3.2. Let A be a unital Banach ∗-algebra which admits spectral invariance. Then for every self-adjoint unital Banach subalgebra B of A, the natural ∗-homomorphism

θ_{B} : C^{∗}(B) → C^{∗}(A)

is injective.

proof. We will verify property (3) above by showing that A^{+} ∩ B ⊆ B^{+}. We may clearly assume that A ⊆ C^{∗}(A), as a self-adjoint subalgebra which is a Banach algebra relative to a larger norm than that of C^{∗}(A).

Choose x ∈ A^{+}∩B; without loss of generality we may assume that the B-norm of x is less than 1. Since x belongs to the positive cone of C^{∗}(A) its spectrum in C^{∗}(A) is nonnegative. By spectral invariance we have σ_{A}(x) ⊆ [0, 1). Moreover, since σ_{A}(x) cannot separate the complex plane, we see from the spectral permanence theorem that σ_{B}(x) = σ_{A}(x) ⊆ [0, 1). Hence for sufficiently small ϵ we have σ_{B}(x + ϵ1) ⊆ (ϵ, 1). Thus we may apply the power series

\sqrt{t} = ∑_{n=0}^{∞} a_{n}(1 − t)^{n}, |1 − t| < 1

to the element x + ϵ1 to obtain a square root in B, i.e., a self-adjoint element h ∈ B satisfying x + ϵ1 = h^{2}. This shows that x + ϵ1 ∈ B^{+}, and we obtain the desired conclusion by allowing ϵ to tend to zero □

We now apply this to show that the enveloping C^{∗}-algebra C^{∗}(D_{θ}) is isomorphic to the non-commutative sphere B_{θ} of [6]. If we realize the irrational rotation C^{∗}-algebra A_{θ} as the C^{∗}-algebra generated by a pair of unitary operators U, V satisfying V U = e^{iθ}UV, then there is a unique automorphism σ of A_{θ} satisfying σ(U) = U^{−1}, σ(V) = V^{−1}. In case θ/π is irrational, B_{θ} is defined to be the fixed subalgebra

B_{θ} = {a ∈ A_{θ} : σ(a) = a}.

Let {W_{x} : x ∈ ℤ ⊕ ℤ} be the family of unitary operators in A_{θ} defined by

W_{(m,n)} = e^{imnθ/2}U^{m}V^{n}, m, n ∈ ℤ.

One verifies easily that

W_{x}W_{y} = ω(x, y)W_{x+y},

where ω is the bicharacter on ℤ ⊕ ℤ defined in (2.1), and moreover the action of σ is given by

σ(W_{x}) = W_{−x}, x ∈ ℤ ⊕ ℤ.

Since A_{θ} is spanned by {W_{x} : x ∈ ℤ ⊕ ℤ}, we conclude that B_{θ} is spanned by {W_{x} + W_{−x} : x ∈ ℤ ⊕ ℤ}.

Corollary. Suppose θ is not a rational multiple of π, and let α : D_{θ} → A_{θ} be the morphism defined by

α(d_{x}) = W_{x} + W_{−x}, x ∈ ℤ × ℤ.

Then the natural extension α̃ : C^{∗}(D_{θ}) → A_{θ} gives an isomorphism of C^{∗}-algebras

C^{∗}(D_{θ}) ≅ B_{θ}.

proof. Let G = ℤ ⊕ ℤ and let ω : G × G → 𝕋 be the bicharacter of (2.1). Consider the Banach ∗-algebra l^{1}(G, ω), where multiplication and involution are defined respectively by

f ∗ g(x) = ∑_{y} ω(y, x)f(y)g(y − x)
f^{∗}(x) = \overline{f(−x)}.

Notice first that C^{∗}(G, ω) is naturally identified with the irrational rotation C^{∗}- algebra A_{θ} = C^{∗}(U, V), where U and V are unitary operators satisfying the above relation V U = e^{iθ}UV. Indeed, letting {W_{x} : x ∈ G} be the operators of A_{θ} defined in the preceding remarks, it is clear that we can define a morphism of l^{1}(G, ω) into A_{θ} by

γ(δ_{x}) = W_{x}, x ∈ G,

δ_{x} denoting the unit function at x. The range of γ is dense in A_{θ}, and the natural extension of γ to C^{∗}(l^{1}(G, ω)) is injective because of the familiar universal property of such pairs U, V.

D_{θ} is indentified (via an isometric isomorphism of Banach ∗-algebras) with a subalgebra of l^{1}(G, ω),

D_{θ} = {f ∈ l^{1}(G, ω) : σ_{0}(f) = f},

where σ_{0} is the ∗-automorphism of l^{1}(G, ω) given by

σ_{0}(f)(x) = f(−x), x ∈ G.

It is clear that the restriction of γ to D_{θ} carries d_{x} = δ_{x} + δ_{−x} to W_{x} + W_{−x}; hence by the preceding remarks γ(D_{θ}) is a dense ∗-subalgebra of B_{θ}. Thus γ extends naturally to a surjective ∗-homomorphism of C^{∗}(l^{1}(G, ω)) onto B_{θ}, and it remains only to show that the latter morphism is injective.

Now it is known that l^{1}(G, ω) admits spectral invariance (see [13, Satz 5] for example, or apply Theorem 1.1.3 of [17] together with the results of [11] on the symmetry of the group algebra of the rank 3 discrete Heisenberg group); hence Proposition 3.2 implies that γ ↾_{D_{θ}} extends uniquely to a ∗-isomorphism of C^{∗}(D_{θ}) onto \overline{γ(D_{θ})} = D_{θ} □

4. Representations.

In this section we make some general comments about the representation theory of the discretized CCRs (2.2). We assume throughout that θ is a real number such that θ/π is irrational.

Remark 4.1: Finite representations.

The unique trace on the irrational rotation algebra A_{θ} gives rise to a representation of A_{θ} which generates the hyperfinite II_{1} factor R. The closure of B_{θ} in this representation is a sub von Neumann algebra of R. Since B_{θ} has a unique tracial state [5], it follows that the closure of B_{θ} is a subfactor of R, and hence is also isomorphic to R. Moreover, since B_{θ} is also simple [5], any finite representation of B_{θ} is quasi-equivalent to this one.

It is not hard to show that the subfactor of R generated by B_{θ} in the above representation has Jones index 2. Since any two subfactors of R of index 2 are known to be isomorphic [12], we have here a very stable invariant for the embedding of the discretized CCR algebra in the irrational rotation algebra A_{θ}.

In particular, by the corollary of 3.2 we may conclude from these remarks that there is a representation of the discretized CCRs (2.2) which generates R as a von Neumann algebra; moreover any finite representation of the discretized CCRs is quasi-equivalent to this one.

Now let τ be a positive real number such that τ^{2}/π is irrational, and let P_{τ} and Q_{τ} be the discretized canonical operators on L^{2}(ℝ) associated with the step size τ as in section 1. We want to make explicit the relation that exists between the pair (P_{τ} , Q_{τ}) and the C^{∗}-algebra C^{∗}(B_{τ^{2}}) discussed in section 2.

Theorem 4.2. There is a unique representation π of D_{τ^{2}} on L^{2}(ℝ) satisfying

π(d_{(1,0)}) = 2τQ_{τ} ,
π(d_{(0,1)}) = 2τP_{τ} .

π(D_{τ^{2}}) and {P_{τ} , Q_{τ} } generate the same unital C^{∗}-algebra. Thus, the three C^{∗}- algebras

C^{∗}(D_{τ^{2}}), B_{τ^{2}}, C^{∗}(P_{τ} , Q_{τ})

are mutually isomorphic.

proof. Let U, V be the one-parameter groups

U_{t}f(x) = e^{itx}f(x),
V_{t}f(x) = f(x + t)  f ∈ L^{2}(ℝ).

As in section 1 we have

4.3  \begin{aligned}Q_{τ} & = \frac{1}{2iτ} (U_{τ} − U_{−τ}) = \frac{1}{τ} sin(τQ), \\ P_{τ} & = \frac{1}{2iτ} (V_{τ} − V_{−τ}) = \frac{1}{τ} sin(τP). \end{aligned}

We claim first that the sines in (4.3) can be replaced by cosines in the sense that the pair (P_{τ} , Q_{τ}) is unitarily equivalent to the pair (P̃_{τ} , Q̃_{τ}) given by

4.4  \begin{aligned}Q̃_{τ} & = \frac{1}{2τ} (U_{τ} + U_{−τ}) \\ P̃_{τ} & = \frac{1}{2τ} (V_{τ} + V_{−τ}). \end{aligned}

To see this, put λ = π/2τ and let R denote the reflection on L^{2}(ℝ) given by Rf(x) = f(−x). Consider the unitary operator

W = RU_{−λ}V_{λ}.

Using the commutation relations V_{t}U_{s} = e^{ist}U_{s}V_{t} together with RU_{s}R^{∗} = U_{−s} and RV_{t}R^{∗} = V_{−t}, one finds that

W U_{s}W^{∗} = e^{iλs}U_{−s}
W V_{t}W^{∗} = e^{iλt}V_{−t}.

Noting that e^{iλτ} = \sqrt{−1}, we obtain (4.4) by applying adW to (4.3), i.e.,

W Q_{τ} W^{∗} = \frac{1}{2τ} (U_{τ} + U_{−τ}) = \frac{1}{2τ} cos(τQ)
W P_{τ} W^{∗} = \frac{1}{2τ} (V_{τ} + V_{−τ}) = \frac{1}{2τ} cos(τP).

We may therefore assume that the pair (Q_{τ} , P_{τ}) is defined by (4.4).

For each x = (m, n) ∈ ℤ ⊕ ℤ, define a unitary operator W_{x} by

W_{(m,n)} = e^{imnτ^{2}/2}U_{mτ} V_{nτ} .

A straightforward computation shows that the family of unitaries {W_{x} : x ∈ ℤ⊕ℤ} satisfies

W_{x}W_{y} = ω(x, y)W_{x+y}

ω being the cocycle of (2.1) for the value θ = τ^{2}, and hence there is a representation π of l^{1}(ℤ ⊕ ℤ, τ^{2}) on L^{2}(ℝ) such that

π(w_{x}) = W_{x}, x ∈ ℤ ⊕ ℤ.

It is clear that π carries d_{(1,0)} (resp. d_{(0,1)}) to U_{τ} + U_{−τ} = 2τQ_{τ} (resp. 2τP_{τ}).

It remains to show that the restriction of π to C^{∗}(D_{τ^{2}}) is uniquely defined by its values on the two elements d_{(1,0)}, d_{(0,1)}}, and that Q_{τ} and P_{τ} generate π(C^{∗}(D_{τ^{2}})) as a unital C^{∗}-algebra. We will prove both by showing that the two elements {d_{(1,0)}, d_{(0,1)}} and the identity generate the Banach ∗-algebra D_{τ^{2}}. It is not hard to adapt the results of [5] to prove that these three elements generate D_{τ^{2}}. Instead, we present the following argument since it gives somewhat more structural information.

Actually, we will give a fairly explicit method for calculating each element d_{x} = δ_{x} + δ_{−x} in terms of the self-adjoint elements p = d_{(1,0)} and q = d_{(0,1)}, using a “generating function” for the family {d_{x} : x ∈ ℤ ⊕ ℤ}. Indeed, it suffices to establish the following lemma.

Lemma 4.5. Let θ be a real number such that θ/π is irrational, and consider the real-analytic function F : (−1, 1) × (−1, 1) → D_{θ} defined by

4.6  F (s, t) = ∑_{m,n=−∞}^{+∞} s^{|m|}t^{|n|}e^{−imnθ/2}d_{(m,n)}.

(i) For −1 < u < 1, −2 ≤ x ≤ 2, let

ϕ(u, x) = \frac{1 − u^{2}}{1 + u^{2} − ux}.

Noting that ϕ is separately analytic in each variable, we have

F (s, t) = 2ϕ(s, q)ϕ(t, p), |s|, |t| < 1,

where q, p are the elements of D_{θ} defined by

q = d_{(1,0)}, p = d_{(0,1)}.

(ii) The Banach ∗-algebra D_{θ} is spanned by the set F ∪ F^{∗}, where

F = {F (s, t) : |s|, |t| < 1}.

proof of (i) Let ω be the bicharacter of ℤ ⊕ ℤ defined by

ω((p, q), (m, n)) = e^{i(qm−pn)θ/2}

and let u, v be the following elements of l^{1}(ℤ ⊕ ℤ, ω):

u = δ_{(1,0)}, v = δ_{(0,1)}.

Then w_{(m,n)} = e^{imnθ/2}u^{m}v^{n}, hence

d_{(m,n)} = e^{imnθ/2}(u^{m}v^{n} + u^{−m}v^{−n}).

It follows that

F (s, t) = ∑_{m,n=−∞}^{∞} s^{|m|}t^{|n|}(u^{m}v^{n} + u^{−m}v^{−n}) = 2 ∑_{m,n=−∞}^{∞} s^{|m|}t^{|n|}u^{m}v^{n}

= 2 ∑_{m=−∞}^{∞} s^{|m|}u^{m} ∑_{n=−∞}^{∞} t^{|n|}v^{n}.

An elementary calculation shows that if z is any complex number having absolute value 1 and −1 < s < 1, then

∑_{m=−∞}^{∞} s^{|m|}z^{m} = \frac{1 − s^{2}}{1 + s^{2} − s(z + z̄)} = ϕ(s, z + z̄).

Since q = d_{(1,0)} = u + u^{∗} and p = d_{(0,1)} = v + v^{∗}, the assertion (i) follows from the analytic functional calculus.

To prove (ii), let A_{pq} be the coefficients in the power series expansion of F,

F (s, t) = ∑_{p,q=0}^{∞} A_{pq}s^{p}t^{q}.

Obviously, {F (s, t) : s, t ∈ (−1, 1)} and {A_{pq} : p, q ≥ 0} have the same closed linear span. Using the fact that d_{(−m,−n)} = d_{(m,n)}, a straightforward computation shows that

A_{pq} = 2e^{−ipqθ/2}d_{(p,q)} + 2e^{ipqθ/2}d_{(−p,q)}.

Thus,

A_{0q} = 2(d_{(0,q)} + d_{(0,q)}) = 4d_{(0,q)},  and
A_{p0} = 2(d_{(p,0)} + d_{(−p,0)}) = 4d_{(p,0)}.

In the remaining cases where pq ≠ 0, the determinant of the coefficients of the 2×2 system of operator equations

A_{pq} = 2e^{−ipqθ/2}d_{(p,q)} + 2e^{ipqθ/2}d_{(−p,q)})
(4.7)  A_{pq}^{∗} = 2e^{ipqθ/2}d_{(p,q)} + 2e^{−ipqθ/2}d_{(−p,q)})

is 4(e^{−ipqθ} − e^{ipqθ}) ≠ 0, and in particular we can solve (4.7) for d_{(p,q)} as a complex linear combination of A_{pq} and A_{pq}^{∗}. This argument shows that the closed linear span of F ∪ F^{∗} contains {d_{(p,q)} : p, q ∈ ℤ}, and (ii) follows. That completes the proof of Theorem 4.2 □

Remark. In some very recent work [7], Bratteli and Kishimoto have established the striking result that B_{θ} is an AF -algebra.

Appendix: Failure of Extensions. We present a simple example of a pair of commutative unital Banach ∗-algebras B ⊆ A such that A is a subalgebra of its enveloping C^{∗}-algebra, but such that the natural morphism γ_{B} : C^{∗}(B) → C^{∗}(A) is not injective. Let A be the algebra of all complex-valued continuous functions defined on the annulus {1 ≤ |z| ≤ 2} which are analytic in its interior. With norm and involution defined by

∥f∥ = sup_{1≤|z|≤2} |f(z)|, f^{∗}(z) = f̄(z̄),

f̄ denoting the complex conjute of f, A is a unital Banach ∗-algebra. C^{∗}(A) is the commutative C^{∗}-algebra C(X),

X = [−2, −1] ∪ [+1, +2]

denoting the intersection of the annulus {1 ≤ |z| ≤ 2} with the real axis, and the completion map γ : A → C(X) is defined by restriction to X. Let B be the norm closure of all holomorphic polynomials in A. Then B is a self-adjoint subalgebra whose enveloping C^{∗}-algebra is C(Y), Y being the intersection of the polynomially convex hull of the annulus with the real axis, namely

Y = [−2, +2].

The morphism γ_{B} : C(Y) → C(X) is given by restriction to X, and hence there is a nontrivial kernel. Put differently, for every real λ ∈ (−1, +1), the complex homomorphism of B defined by

ω_{λ}(f) = f(λ), f ∈ B

is a bounded positive linear functional on B which cannot be extended to a positive linear functional on A.

References

1. Arveson, W., Non-commutative spheres and numerical quantum mechanics, preprint.

2. Avron, J., v. Mouche, P. H. M. and Simon, B, On the measure of the spectrum for the almost Mathieu equation, Comm. Math. Phys. 132 (1990), 103–118.

3. Bellissard, J.,Lima, R., and Testard, D., On the spectrum of the almost Mathieu Hamiltonian, preprint (1983).

4. Bellissard, J. and Simon, B., Cantor spectrum for the almost Mathieu equation, J. Funct. Anal. 48 (1982), 408–419.

5. Bratteli, O., Elliott, G., Evans, D. and Kishimoto, A., Non-commutative spheres I, preprint.

6. , Non-commutative spheres II, J. Op. Th. (to appear).

7. Bratteli, O. and Kishimoto, A., Non-commutative spheres III, manuscript.

8. Carmona, R. and Lacroix, J., Spectral theory of random Schrödinger operators, Birkhäuser, Boston, 1990.

9. Choi, M.-D. and Elliott, G., Gauss polynomials and the rotation algebra, Invent. Math. 99, 225–246.

10. Connes, A., An analogue of the Thom isomorphism for crossed products of a C^{∗}-algebra by an action of ℝ, Adv. Math. 39 (1981), 31–55.

11. Hulanicki, A., On the symmetry of group algebras of discrete nilpotent groups, Studia Math. 35 (1970), 207–219.

12. Jones, V. F. R., Index for subfactors, Inv. Math 72 (1983), 1–25.

13. Leptin, H., Lokal Kompakte Gruppen mit Symmetrischen algebren, Istituo Naz. di Alta Matematica, symposia mathematica XXII (1977).

14. Podles̀, Quantum spheres, Letters in Math. Phys. 14 (1987), 193–203.

15. Rickart, C., Banach Algebras, van Nostrand, Princeton, 1960.

16. Riedel, N., Point spectrum for the almost Mathieu equation, C. R. Math. Rep. Acad. Sci. Canada VIII 6 (1986), 399-403.

17. Schweitzer, L., Dense subalgebras of C^{∗}-algebras with applications to spectral invariance, Thesis, U.C. Berkeley (1991).