Full Text
Reports of the Academy of Sciences of the USSR
1966. Volume 167, No. 1
UDC 517.433
MATHEMATICS
V. Ya. Golodets
ON FACTOR REPRESENTATIONS OF ANTICOMMUTATION RELATIONS
(Presented by Academician I. M. Vinogradov on 15 VI 1965)
- A representation of the anticommutation relations is a set of bounded linear operators \(\{a_k\}_1^\infty\), acting in a separable Hilbert space \(H\), which satisfy the system of equalities
\[ a_j a_k + a_k a_j = 0,\qquad a_j a_k^* + a_k^* a_j = \delta_{jk}. \tag{1} \]
If the weakly closed ring generated by the operators \(\{a_k, a_k^*\}_1^\infty\) is a factor, then such a representation will be called a factor representation.
Factor representations of the relations (1) were first studied by von Neumann; he constructed examples of factors of types I, II, III, which are generated by the operators \(\{a_k, a_k^*\}_1^\infty\) \((^2)\). In the present paper several general results are obtained on the construction of factor representations and on their properties. We note that the simplest properties are possessed by those factor representations for which the weakly closed ring generated by the operators \(\{a_k, a_k^*\}_1^\infty\) is a factor \(\Pi_1\).
- Our considerations are based on the work \((^1)\). In that work a general construction was given for representations of the relations (1), which was used by the authors to study irreducible representations. We shall apply this construction to the study of factor representations. Let us recall some results of \((^1)\).
By virtue of (1), the operators \(N_k = a_k^* a_k\) form a commuting set of projections. We diagonalize these projections. It is convenient to proceed as follows. Let \(\Gamma\) be the set of all infinite sequences \(\alpha = (\alpha_1, \alpha_2, \ldots)\), where \(\alpha_i = 0,1\). Put \(\Gamma_k = (\alpha;\alpha_k = 0)\). By a Borel set in \(\Gamma\) we shall mean a set constructed from all \(\Gamma_k\) by the usual countable processes. We now represent the space \(H\) in the form of an integral \(\int \oplus H_\alpha d\mu(\alpha)\) over \(\Gamma\) in such a way that if to an element \(f \in H\) there corresponds the vector-function \(f(\alpha)\) with values in the Hilbert space \(H_\alpha\), then \(N_k f(\alpha)=\alpha_k f(\alpha)\); \(\mu(\alpha)\) is a completely additive bounded nonnegative function of Borel sets in \(\Gamma\).
Under componentwise addition modulo 2, \(\Gamma\) is a group. Let \(\delta_k\) be the sequence from \(\Gamma\) with a one in the \(k\)-th place and zeros elsewhere. From the relations (1) it follows that the measures \(\mu(\alpha)\) and \(\mu(\alpha+\delta_k)\) are equivalent. Denote \(\lim H_\alpha=\nu(\alpha)\). For factor representations, \(\nu(\alpha)\) is always constant.
Introduce the operators
\[ A_k = a_k + a_k^*,\qquad B_k = i^{-1}(a_k-a_k^*). \]
In \((^1)\) the formulas
\[ A_k f(\alpha)=j_k(\alpha)c_k(\alpha) \sqrt{\frac{d\mu(\alpha+\delta_k)}{d\mu(\alpha)}}\, f(\alpha+\delta_k), \tag{2} \]
\[ B_k f(\alpha)=i^{-1}j_{k+1}(\alpha)c_k(\alpha) \sqrt{\frac{d\mu(\alpha+\delta_k)}{d\mu(\alpha)}}\, f(\alpha+\delta_k), \]
were found.
where \(j_k(a)=(-1)^{a_i+\cdots+a_{k-1}}\), and \(\{c_k(a)\}_1^\infty\) is a measurable unitary transformation from \(H_a\) into \(H_{a+\delta_k}\), satisfying the relations
\[ c_k(a+\delta_k)=c_k^*(a);\qquad c_k(a)c_l(a+\delta_k)=c_l(a)c_k(a+\delta_l) \tag{3} \]
for almost all \(a\) and all \(k,l\).
Putting \(T_k a=(0,\ldots,0,a_{k+1},a_{k+2},\ldots)\), let us introduce the notation
\[ \gamma_k(a)=c_k(T_k a) \tag{4} \]
and, using (3), we obtain
\[ c_k(a)=\gamma_1^{-a_1}(a)\cdots \gamma_{k-1}^{-a_{k-1}}(a)\gamma_k^{(-1)^{a_k}}(a) \gamma_{k-1}^{a_{k-1}}(a+\delta_k)\cdots \gamma_1^{a_1}(a+\delta_k). \tag{5} \]
Conversely, if \(\{\gamma_k(a)\}_1^\infty\) are arbitrary unitary measurable transformations from \(H_a\) into \(H_a\) that are invariant under shifts by \(\delta_1,\ldots,\delta_k\), then the \(c_k(a)\) defined by formulas (5) satisfy the functional equations (3).
In (1) it was shown that a representation of the anticommutation relations is determined by specifying a measure \(\mu\), a dimension function \(\nu\), and a set of operators \(c_k(a)\) satisfying (3). Therefore it is sometimes convenient to denote a representation by \((\mu,\nu,\{c_k(a)\}_1^\infty)\).
A quasi-invariant measure is called ergodic if every bounded measurable function \(f(a)\) satisfying the condition \(f(a)=f(a+\delta_k)\) for all \(k\) and almost all \(a\) is constant.
The simplest examples of quasi-invariant ergodic measures can be constructed as follows. We shall regard the set \(\Gamma\) as the direct product of a countable set of copies of two-point sets. Put \(\mu(\Gamma)=1\), \(\mu(\Gamma_k)=p_k\), \(0<p_k<1\), and define the measure \(\mu\) on \(\Gamma\) as the product measure. If \(p_k=1/2\) for all \(k\), then such a measure is called Lebesgue.
- We proceed to the study of factor representations.
Theorem 1. If a representation is a factor representation, then the measure \(\mu(a)\) is ergodic.
Theorem 2. A factor representation of the anticommutation relations for which \(\nu<\infty\) is a direct sum of \(q\) copies of some irreducible representation, where \(q\) divides \(\nu\).
- Let \(H\) be the Hilbert space of vector-functions \(f(x,y)\) on \(\Gamma\times\Gamma\), whose values belong to a Hilbert space \(R\), and
\[ \int_{\Gamma\times\Gamma}\|f(x,y)\|^2\,d\mu_1(x)\,d\mu_2(y)<\infty, \]
where \(\|\cdot\|\) is the norm of a vector in \(R\), and \(\mu_1(x)\), \(\mu_2(y)\) are quasi-invariant ergodic measures.
We define representations of the anticommutation relations by the formulas
\[ A_k f(x,y)=j_k(x)c_k(x,y)f(x+\delta_k,y) \sqrt{\frac{d\mu_1(x+\delta_k)}{d\mu_1(x)}}, \tag{6} \]
\[ B_k f(x,y)=i^{-1}j_{k+1}(x)c_k(x,y)f(x+\delta_k,y) \sqrt{\frac{d\mu_1(x+\delta_k)}{d\mu_1(x)}}. \]
Define the dual representation
\[ \widetilde A_k f(x,y)=j_k(y)\,\widetilde c_k(x,y)\,f(x,y+\delta_k) \sqrt{\frac{d\mu_2(y+\delta_k)}{d\mu_2(y)}} , \]
\[ \widetilde B_k f(x,y)=i^{-1}j_{k+1}(y)\,\widetilde c_k(x,y)\,f(x,y+\delta_k) \sqrt{\frac{d\mu_2(y+\delta_k)}{d\mu_2(y)}} . \tag{7} \]
Here \(j_k(x), j_k(y)\) are defined by formula (2), and \(c_k(x,y)\) \((\widetilde c_k(x,y))\) is a measurable unitary transformation in the space \(R\), satisfying relations (3) for fixed \(y\) (respectively, for fixed \(x\)). Suppose that each operator \(A_k, B_k\) commutes with each operator \(\widetilde A_l, \widetilde B_l\). For this it is necessary and sufficient that the relations
\[ c_k(x,y)\widetilde c_l(x+\delta_k,y) = \widetilde c_l(x,y)c_k(x,y+\delta_l) \tag{8} \]
hold for all \(k,l\) and almost all \((x,y)\in \Gamma\times\Gamma\).
The following theorem shows that the construction just described is of a general character.
Theorem 3. Let \(\{A_k,B_k\}_1^\infty\) and \(\{\widetilde A_k,\widetilde B_k\}_1^\infty\) be factor representations of relations (1), and let the operators of both representations act in the same Hilbert space \(H\). If each operator \(A_k,B_k\) commutes with each operator \(\widetilde A_l,\widetilde B_l\), then the space \(H\) can be realized in such a way that the operators \(A_k,B_k\) \((k=1,2,\ldots)\) have the form (6), while the operators \(\widetilde A_k,\widetilde B_k\) \((k=1,2,\ldots)\) have the form (7).
Let us return to our construction. Denote by \(M\) the weakly closed ring generated by the operators (6), and by \(\widetilde M\) the weakly closed ring generated by the operators (7).
Theorem 4. If, for almost every fixed \(y\), relations (6) define an irreducible representation, and, for almost every fixed \(x\), relations (7) also define an irreducible representation and, moreover, the operators \(A_k,B_k\) \((k=1,2,\ldots)\) commute pairwise with the operators \(\widetilde A_l,\widetilde B_l\) \((l=1,2,\ldots)\), then \(M\) and \(\widetilde M\) are factors, and the weakly closed ring generated by the operators from \(M\cup \widetilde M\) is the ring of all bounded linear operators in the space \(H\).
If the conditions of Theorem 4 are fulfilled, then the functions \(c_k(x,y)\) and \(\widetilde c_k(x,y)\) satisfy two systems of functional equations (3) and (8). We indicate some solutions of these equations.
The simplest solution is obtained if one assumes that \(c_k(x,y)=c_k(x)\) and \(\widetilde c_k(x,y)=\widetilde c_k(y)\). Then (8) reduces to the requirement that \(c_k(x)\) and \(\widetilde c_l(y)\), for all \(k,l\) and almost all \(x\) and \(y\), commute with one another.
Consider another solution. Suppose that \(\widetilde c_k(x,y)=c_k(x,y)\). Then it follows from (8) that, for almost all \(x\) and \(y\) and all \(k\),
\[ c_k(x+\delta_k,y)=c_k(x,y+\delta_k). \tag{9} \]
If the measures \(\mu_1\) and \(\mu_2\) are product measures, then it follows from (9) that \(c_k(x,y)=c_k(x+y)\). Conversely, if this condition is fulfilled, then from it and from (3) condition (8) follows.
Theorem 5. For every irreducible representation \((\mu,\nu,\{c_k(a)\}_1^\infty)\) one can construct a factor representation which will have the form (6), with
\(c_k(x,y)=\widetilde c_k(x,y)=c_k(x+y)\), \((k=1,2,\ldots)\).
The theorem makes it possible to construct a large class of factor representations. For example, to every irreducible representation with \(\nu<\infty\), constructed in (1), there corresponds a nontrivial factor representation. It is relatively easy to determine whether it is of type I or not. However, the exact type can be established only in the simplest cases, which we shall now consider.
Let in (6) and (7) \(c_k(x,y)=\tilde c_k(x,y)=i(-1)^{x_k+y_k}\); \(\mu_1,\mu_2\) are Lebesgue measures; \(f(x,y)\) are complex-valued functions with integrable square of the modulus. Then \(M\) and \(\widetilde M\) are factors of type \(\mathrm{II}_1\). If one assumes that \(\mu_2\) is not equivalent to Lebesgue measure, then \(\widetilde M\) will have type III.
- In this section we shall consider factor representations of the relations (1) in connection with factors of type \(\mathrm{II}_1\).
Theorem 6. If the operators \(\{A_k,B_k\}_1^\infty\) generate a factor of type \(\mathrm{II}_1\), then
\[ \operatorname{Tr}\left(B_{i_1}^{\alpha_1}A_{i_1}^{\beta_1},\ldots,B_{i_n}^{\alpha_n}A_{i_n}^{\beta_n}\right)=0, \]
where \(\operatorname{Tr}\) is the relative trace in the factor of type \(\mathrm{II}_1\), \(\alpha_i,\beta_i=0,1\).
Theorem 7. If the condition of Theorem 6 is satisfied, then the measure \(\mu(\alpha)\) is equivalent to Lebesgue measure.
Theorem 8. There exist representations of the anticommutation relations whose operators generate spatially nonisomorphic factors of type \(\mathrm{II}_1\).
- We give one more method for constructing representations.
Theorem 9. Let a free commutative group with a finite number of generators \(a_1,\ldots,a_t\) be given. Let \(U\) be a unitary projective representation of the group \(G\). Construct a representation of the anticommutation relations for which: 1) \(\gamma_{nt+i}=U(a_i)\), where \(0<i\le t\), \(n=1,2,\ldots\); 2) \(\mu(\alpha)\) is a product of measures, with \(\mu(\Gamma_k)=p_k\) having neither 0 nor 1 as limit points.
Then, for this representation of the anticommutation relations, the commutant is algebraically isomorphic to the commutant of the representation \(U\) of the group \(G\).
With the aid of this theorem one can construct irreducible representations of the relations (1) for \(\nu=\infty\) \((^{1,4})\), as well as representations for which the ring generated by \(\{A_k,B_k\}_1^\infty\) is a factor of type \(\mathrm{II}_\infty\), and the measure \(\mu(\alpha)\) is not equivalent to Lebesgue measure.
The author expresses deep gratitude to F. A. Berezin, under whose supervision the work was carried out.
Physical-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR
Received
15 VI 1965
REFERENCES
- L. Gårding, A. Wightman, Proc. Nat. Acad. Sci. U.S.A., 40, No. 7 (1954).
- J. von Neumann, Compositio Math., 6, 1 (1938).
- M. A. Naimark, Normed Rings, Moscow, 1956.
- V. Ya. Golodets, UMN, 20, no. 2 (1965).