1 Introduction

The resolvent algebra \(\mathfrak {R}\) of a single particle on the real line is the C*-algebra generated by all resolvents of linear combinations of its position and momentum operators,

$$\begin{aligned} (aQ + bP - ir )^{-1} \, , \quad a,b \in \mathbb R\, , \ r \in \mathbb R\setminus \{ 0 \} \, . \end{aligned}$$
(1)

It was introduced in [2] and shown to have properties of physical interest that are not shared by the Weyl algebra, being generated by unitary exponentials of those linear combinations. Many operators, such as physically significant Hamiltonians, are affiliated with the resolvent algebra; moreover, in contrast to the Weyl algebra, it is stable under the action of a large family of dynamics, describing interaction [2, 3].

It is frequently argued that the Weyl algebra is more convenient for computations of states since it suffices to determine the states on individual Weyl operators, making use of the fact that their span generates the algebra. It is the aim of the present note to show that this feature is also shared by the resolvent algebra. Namely the linear span of the basic resolvents is norm dense in \(\mathfrak {R}\).

It has been shown in [2] that the algebra \(\mathfrak {R}\) acts faithfully in the Schrödinger representation on \(L^2(\mathbb R)\), where Q acts as a multiplication operator and P by differentiation. We restrict our attention in the following to this representation and make use of the notation

$$\begin{aligned} R_{(a,b)}(r) {:}{=}(a Q + b P - i r)^{-1} \, , \quad (a,b) \in \mathbb R^2 \, , \ \ r \in \mathbb R\setminus \{ 0\} \, . \end{aligned}$$
(2)

Thus, \(R_{(a,b)}(r)^* = R_{(a,b)}(-r)\) and \(R_{(0,0)}(r) = i/r\).

Turning to the products of resolvents that are different from multiples of the identity, \(R_{(a,b)}(r) R_{(a',b')}(r') \in \mathfrak {R}\), there appear two cases. Either \(a b' = a' b\). Then, the resolvents commute and are elements of the abelian C*-algebra \(\mathfrak {A}_{(a,b)} \subset \mathfrak {R}\) generated by \(R_{(a,b)}(r)\) for arbitrary \(r \in \mathbb R\setminus \{ 0\}\). It coincides with the norm closure of the linear span generated by these resolvents. This can be seen by noticing that arbitrary products of the resolvents for different values of r are obtained by making use of the resolvent equation

$$\begin{aligned} R_{(a,b)}(r) R_{(a,b)}(r') = i(r' - r)^{-1} (R_{(a,b)}(r) - R_{(a,b)}(r')) \, . \end{aligned}$$
(3)

Thus, since \(r \mapsto R_{(a,b)}(r)\) is norm continuous for \(r \ne 0\), the linear combinations of these resolvents are norm dense in \(\mathfrak {A}_{(a,b)}\), as stated.

The second possibility is \(ab' \ne a'b\). Then, \(R_{(a,b)}(r) R_{(a',b')}(r') \in \mathfrak {R}\) is a compact operator on \(L^2(\mathbb R)\) since the underlying generators are canonically conjugate, cf. [5, Thm. XI.20]. The C*-algebra \(\mathfrak {K}\) of compact operators does not contain any finite linear combinations of basic resolvents, however, apart from 0. A total set of compacts is obtained by weak integrals of the functions \((a,b) \mapsto R_ {(a,b)}(r)\) with test functions \(f \in \mathcal{S}(\mathbb R^2)\). Since the resolvents are discontinuous in norm, it is not clear from the outset that these integrals can be approximated in norm by linear combinations of resolvents. That this is possible will be shown in the subsequent section, thereby establishing our main result.

Theorem

The resolvent algebra \(\mathfrak {R}\) coincides with the norm closure of the span of its generators,

$$\begin{aligned} \mathfrak {R}= \overline{\text {span}} \, \{ R_{(a,b)}(r) : (a,b) \in \mathbb R^2 \, , \ r \in \mathbb R\setminus \{ 0 \} \} \, . \end{aligned}$$
(4)

We restrict our attention here to a single particle in one spatial dimension. But, as outlined in the concluding remarks, our results extend to finite and infinite systems in any number of dimensions. They complement and simplify an argument, alluded to in [7], that is based on analogous results for the classical theory in [7] and on Berezin quantization [6].

2 Proof of the theorem

The missing step in the proof of the theorem is the demonstration that compact operators can be approximated in norm by linear combinations of the basic resolvents. Since \(a,b \mapsto R_{(a,b)}(r)\) is continuous in the strong operator topology, we can proceed to the integrals

$$\begin{aligned} \overline{R}_f(r) {:}{=}\int \! da db \, f(a,b) R_{(a,b)}(r) \, , \quad f \in \mathcal{S}(\mathbb R^2) \, , \ \ r \in \mathbb R\backslash \{ 0 \} \, . \end{aligned}$$
(5)

By arguments in [1, Lem. 3.2], they can be shown to define compact operators. We need to show here that these integrals can be approximated in norm by linear combinations of the basic resolvents \(R_{(a,b)}(r)\) and that one obtains in this manner a dense set of compact operators.

Since \(\Vert \overline{R}_f(r) \Vert \le \Vert f \Vert _1 / |r|\), where \(\Vert f \Vert _1\) denotes the norm of f in \(L^1(\mathbb R^2)\), it suffices to consider the dense set of test functions f with compact support. We can then approximate the integrals by Riemann sums of the form

$$\begin{aligned} \overline{R}_{f,n}(r) {:}{=}\frac{1}{n^2} \sum _{l,m = -nL}^{nL} f\left( \tfrac{l}{n}, \tfrac{m}{n}\right) \, R_{\left( \frac{l}{n}, \frac{m}{n}\right) }(r) \, , \quad n \in \mathbb N\, , \end{aligned}$$
(6)

where \(L \in \mathbb N\) is sufficiently large such that the support of f is contained in a square about 0 of side length 2L. These sums converge for large n in the strong operator topology to \(\overline{R}_f(r)\). Now

$$\begin{aligned}&\overline{R}_{f, n}(r)^* \overline{R}_{f, n}(r) \nonumber \\&= \frac{1}{n^4} \sum _{l,m,l',m' = -nL}^{nL} \delta _{\, lm', l'm} \, f(\tfrac{l}{n}, \tfrac{m}{n})^* f(\tfrac{l'}{n}, \tfrac{m'}{n}) R_{(\frac{l}{n}, \frac{m}{n})}(r)^* R_{(\frac{l'}{n}, \frac{m'}{n})}(r) + C_n \, , \end{aligned}$$
(7)

where \(C_n \in \mathfrak {K}\) is a sum involving products of non-commuting resolvents, hence of compact operators, \(n \in \mathbb N\). An estimate of the first term yields

$$\begin{aligned} \Vert \overline{R}_{f, n}(r)^* \overline{R}_{f, n}(r) - C_n \Vert \le \frac{(2nL + 1)^3}{n^4 r^2} \, \Vert f \Vert _\infty ^2 \, , \quad n \in \mathbb N\, , \end{aligned}$$
(8)

where \(\Vert f \Vert _\infty \) denotes the supremum norm of f. It follows from this estimate by standard arguments (making use of continuity properties of the square root and the polar decomposition of operators) that there is a sequence of compacts \(K_n \in \mathfrak {K}\) such that \(\Vert \overline{R}_{f, n}(r) - K_n \Vert \rightarrow 0\) in the limit of large \(n \in \mathbb N\). Thus, these compacts also converge in this limit to the operator \(\overline{R}_f(r) \in \mathfrak {K}\) in the strong operator topology.

The dual space of the Banach space \(\mathfrak {K}\) consists of normal functionals \(K \mapsto \text{ Tr } \, K \tau \), where \(\tau \) are trace class operators. Since the bounded sequence \(K_n \in \mathfrak {K}\), \(n \in \mathbb N\), converges in the strong operator topology to \(\overline{R}_f(r) \in \mathfrak {K}\), it is also convergent in the weak Banach space topology of \(\mathfrak {K}\). This puts us into the position to apply Mazur’s lemma [4, p. 6]. Namely, there exist convex combinations of these compact operators,

$$\begin{aligned} \varvec{K}_{n} {:}{=}\sum _{l = n}^{N_n} c_{l,n} \, K_{l} \, , \quad c_{l,n} \ge 0 \, , \ \ \sum _{l = n}^{N_n} c_{l,n} = 1 \, , \ \ n \in \mathbb N\, , \end{aligned}$$
(9)

which approximate \(\overline{R}_f(r)\) in norm, \(\lim _n \Vert \varvec{K}_{n} - \overline{R}_f(r) \Vert = 0\). We proceed now to the corresponding sums of the basic resolvents,

$$\begin{aligned} \overline{\varvec{R}}_{f, n}(r) {:}{=}\sum _{l = n}^{N_n} c_{l,n} \overline{R}_{f, l}(r) \, , \quad c_{l,n} \ge 0 \, , \ \ \sum _{l = n}^{N_n} c_{l,n} = 1 \, , \ \ n \in \mathbb N\, . \end{aligned}$$
(10)

It follows that

$$\begin{aligned} \Vert \overline{\varvec{R}}_{f, n}(r) - \overline{R}_f(r) \Vert&\le \Vert \overline{\varvec{R}}_{f, n}(r) - \varvec{K}_{n} \Vert + \Vert \varvec{K}_{n} - \overline{R}_f(r) \Vert \nonumber \\&\le \sup _{l \ge n} \Vert \overline{R}_{f, \, l}(r) - K_l \Vert + \Vert \varvec{K}_{n} - \overline{R}_f(r) \Vert \, . \end{aligned}$$
(11)

So the linear combinations of the basic resolvents entering in \(\overline{\varvec{R}}_{f, n}(r)\) converge in norm to the compact operator \(\overline{R}_f(r)\) in the limit of large \(n \in \mathbb N\).

The proof that one obtains in this manner a dense set of compact operators is accomplished by showing that suitable linear combinations of the compact operators \(\overline{R}_f(r)\) for \(r \in \mathbb R{\setminus } \{ 0 \}\) approximate in norm the smoothed Weyl operators

$$\begin{aligned} \overline{W}_f {:}{=}\int \! da db \, f(a,b) \, e^{-i(aQ + bP)} \, , \quad f \in \mathcal{S}(\mathbb R^2) \, . \end{aligned}$$
(12)

From a well-known theorem of von Neumann [8], it follows that the span of the latter operators forms a dense subalgebra of the compacts \(\mathfrak {K}\). These operators can be approximated in the strong operator topology by the sequence

$$\begin{aligned} \overline{W}_{f,n} {:}{=}\int \! da db \, f(a,b) \, (1 + (i/n)(a Q + bP))^{-n} \, , \quad n \in \mathbb N\, . \end{aligned}$$
(13)

The operators \((1 + (i/n)(a Q + bP))^{-n}\), \(n \in \mathbb N\), are elements of the abelian algebra \(\mathfrak {A}_{(a.b)}\). As was shown in the preceding section, cf. relation (3), they can be approximated in norm by linear combinations of the basic resolvents \(R_{(a,b)}(r)\) for \(r \in \mathbb R\backslash \{ 0 \}\). This approximation is uniform in \((a,b) \in \mathbb R^2\). It implies that the operators \(\overline{W}_{f,n}\) can be approximated in norm by linear combinations of the compact operators \(\overline{R}_{f}(r)\), \(r \in \mathbb R\backslash \{ 0 \}\); whence, they are also elements of \(\mathfrak {K}\). Since the sequence \(\overline{W}_{f,n} \in \mathfrak {K}\), \(n \in \mathbb N\), is convergent in the strong operator topology to \(\overline{W }_f \in \mathfrak {K}\), it follows from Mazur’s lemma that suitable convex combinations of these operators converge in norm. Hence, \(\overline{W }_f\) is the norm limit of linear combinations of the basic resolvents, completing the proof of the theorem.

3 Concluding remarks

Restricting attention to the simple example of a single particle in one dimension, we have shown that the corresponding resolvent algebra coincides with the norm closure of the span of the underlying basic resolvents. As in case of the Weyl algebra, this simplifies the computation of states on this algebra.

Let us briefly illustrate this point for quasifree states on \(\mathfrak {R}\) [2, Sect. 4]. In the case at hand, these are determined by suitable scalar products on \(\mathbb C^2\). For vectors \(\varvec{c}= (c_1,c_2) \in \mathbb C^2\) they are of the form

$$\begin{aligned} \langle \varvec{c}, \varvec{c}\rangle = \alpha |c_1|^2 + \beta |c_2|^2 + (i/2)(c_1^* c_2 - c_1 c_2^*) \, , \end{aligned}$$
(14)

where \(\alpha , \beta > 0\) and \(\alpha \beta \ge 1/4\). According to the preceding results, the corresponding quasifree states \(\omega \) on \(\mathfrak {R}\) are fixed by linear extension of the expectation values of the basic resolvents. They are given by

$$\begin{aligned} \omega (R_{(a,b)}(r)) = i \, \text {sign}(r) \int _0^\infty \! dt \, e^{-t |r|} e^{-(t^2/2) \langle \varvec{c}, \, \varvec{c}\rangle } \, , \end{aligned}$$
(15)

where \(\varvec{c}= (a,b) \in \mathbb R^2\) and \(r \in \mathbb R\backslash \{ 0 \}\).

The preceding arguments can be extended to arbitrary finite and infinite systems in any number of dimensions. As is outlined below, the linear span of the basic resolvents is norm dense in the resolvent algebras of any finite quantum system. Since for infinite systems (quantum fields) the corresponding resolvent algebras are the C*-inductive limit of nets of such finite subalgebras [2], the span of the underlying basic resolvents is norm dense in those cases as well.

We briefly indicate the straightforward proof for canonically conjugate position and momentum operators \((Q_m, P_m)\) that commute for different values of \(m = 1, \dots , n\). The corresponding resolvent algebra \(\mathfrak {R}_n\) on \(L^2(\mathbb R^n)\) is the C*-algebra generated by the resolvents

$$\begin{aligned} \big (\sum _{m = 1}^n (a_m Q_m + b_m P_m) -ir \big )^{-1} \, , \quad a_m, b_m \in \mathbb R\, , \ m = 1, \dots , n \, , \ \ r \in \mathbb R\setminus \{ 0 \} \, . \end{aligned}$$
(16)

The essential step is the demonstration that the product of any two such resolvents can be approximated in norm by linear combinations of basic resolvents. The proof that the linear span of all resolvents is norm dense in \(\mathfrak {R}_n\) is then obtained by an obvious inductive procedure. There are the following three possibilities.

  1. (1)

    The generators \(G {:}{=}\sum _{m = 1}^n (a_m Q_m + b_m P_m)\) of the two resolvents differ by a constant factor. Then, the product of the resolvents is defined by the resolvent equation, as discussed in the introduction.

  2. (2)

    The generators \(G_1, G_2\) of the two resolvents commute, but are not proportional. In that case one can show that the product of the respective resolvents can be approximated in norm by linear combinations of the resolvents of \((a G_1 + b G_2)\) with \((a,b) \in \mathbb R^2\). Proceeding to the joint spectral resolution of the generators, it amounts to proving that the functions

    $$\begin{aligned} \mathbb R^2 \ni (x,y) \mapsto (x -i r)^{-1} (y -i r')^{-1} \, , \quad r, r' \in \mathbb R\setminus \{ 0 \} \, , \end{aligned}$$
    (17)

    being elements of \(C_0(\mathbb R^2)\), can be approximated in the supremum norm by the span of the basic resolvents

    $$\begin{aligned} (x,y) \mapsto (a x + b y - i s)^{-1} \, , \quad (a, b) \in \mathbb R^2 \, , \ \ s \in \mathbb R\backslash \{ 0 \} \, . \end{aligned}$$
    (18)

    Similarly to the case of compact operators, discussed in the preceding section, this is accomplished by integrating the basic resolvents with regard to the parameters (a, b) with test functions \(f \in \mathcal{S}(\mathbb R^2)\). The resulting integrals are elements of \(C_0(\mathbb R^2)\). They can be approximated by Riemann sums which converge to the integrals in the weak topology of \(C_0(\mathbb R^2)\), recalling that its dual space consists of finite complex measures. Note that the resolvents are continuous with regard to (a, b), uniformly for (x, y) varying in compact subsets of \(\mathbb R^2\). By applying again the procedure used in case of compact operators (passage to squares of Riemann sums, determination of the contributions in \(C_0(\mathbb R^2)\), estimates of the remainders, and Mazur’s lemma), it follows that convex combinations of the Riemann sums approximate the integrals in norm. That one obtains by this procedure a norm dense set of elements of \(C_0(\mathbb R^2)\) can be seen by noting that the Fourier transforms of \(f \in \mathcal{S}(\mathbb R^2)\),

    $$\begin{aligned} x,y \mapsto \hat{f}(x,y) = \int \! da db \, f(a,b) \, e^{-i(ax +by)} \, , \end{aligned}$$
    (19)

    can be approximated in the weak topology of \(C_0(\mathbb R^2)\) by linear combinations of integrals involving the basic resolvents. As before, one replaces the exponential function by the sequence \(a,b \mapsto (1 + (i/n)(ax +by))^{-n}\), \(n \in \mathbb N\). The latter functions are norm limits of finite linear combinations of the basic resolvents \(x,y \mapsto (ax +by -is)^{-1}\), \(s \in \mathbb R\backslash \{ 0 \}\), uniformly for \((a,b) \in \mathbb R^2\). It then follows by another application of Mazur’s lemma that linear combinations of the basic resolvents approximate the Fourier transforms of \(f \in \mathcal{S}(\mathbb R^2)\) in the supremum norm. So their span is dense in \(C_0(\mathbb R^2)\) and the product of the resolvents of \(G_1, G_2\) can be approximated in this manner.

  3. (3)

    If the generators \(G_1, G_2\) do not commute, their commutator is a multiple of the imaginary unit i. The C*-algebra generated by the basic resolvents of all linear combinations of \(G_1, G_2\) is then isomorphic to the C*-algebra \(\mathfrak {R}\) of a one-dimensional system, as discussed in the preceding sections. The product of the resolvents of \(G_1\) and \(G_2\) is contained in its compact ideal, so it can also be approximated in norm by linear combinations of the basic resolvents.

Thus, all finite and infinite resolvent algebras are generated by the span of the underlying basic resolvents. We hope that these results will help to make progress in the structural analysis of the resolvent algebras, such as the determination of their automorphism groups and of the affiliated operators.