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UDC 517.4
MATHEMATICS
V. Ya. GOLODETS
ON CERTAIN PROPERTIES OF REPRESENTATIONS OF AN INFINITE-DIMENSIONAL CLIFFORD ALGEBRA
(Presented by Academician I. M. Vinogradov on 7 IV 1967)
A representation of a Clifford algebra is a set of linear self-adjoint bounded operators \(\{A_k\}_1^\infty\), acting in a separable Hilbert space \(H\) and satisfying the commutation rules
\[ A_k^2=I,\qquad A_kA_l+A_lA_k=0 \quad (k\ne l)\quad (k,l=1,2,\ldots). \tag{1} \]
In the present article we study the structure of an operator commuting with the operators \(\{A_k\}_1^\infty\) satisfying relations (1). We also give a necessary and sufficient condition for the weakly closed ring \(M\), generated by the operators \(\{A_k\}_1^\infty\), to be a factor of type \(\mathrm{II}_1\) in the sense of Murray and von Neumann.
- Let \(\Gamma\) be the space of binary sequences \(x=x_1x_2\ldots\) \((x_i=0\) or \(1)\). Under componentwise addition modulo \(2\), \(\Gamma\) becomes a group. Denote by \(\Delta\) the subgroup of \(\Gamma\) whose generators are the sequences \(\delta_k\), in which the \(k\)-th component is equal to \(1\), and the remaining ones to zero. We single out the \(\sigma\)-algebra \(S\) of subsets of \(\Gamma\) with generators \(\Gamma_k=\Gamma(x:x_k=0)\) and \(\Gamma'_k=\Gamma(x:x_k=1)\) \((k=1,2,\ldots)\). A measure \(\mu\), defined on the subsets \(S\), is called quasi-invariant if the measures \(\mu(x)\) and \(\mu(x+\delta)\), for all \(\delta\in\Delta\), are equivalent. A quasi-invariant measure \(\mu\) is called ergodic if an arbitrary bounded measurable function \(f(x)\) on \(\Gamma\), satisfying \(f(x+\delta)=f(x)\) for almost all \(x\in\Gamma\) and all \(\delta\in\Delta\), is constant on a set of measure one.
For convenience, put
\[ A_k=A_{2k-1},\qquad B_k=A_{2k}\quad (k=1,2,\ldots). \]
According to results of L. Gårding and A. Wightman \((^1)\), if a representation of the Clifford algebra \(\{A_k,B_k\}_1^\infty\) is given, then the space \(H\) can be realized in the form of a direct integral of Hilbert spaces \(H_x\) over \(\Gamma\) with respect to a quasi-invariant measure \(\mu\),
\[ H=\int_{\Gamma}^{\oplus} H_x\,d\mu(x), \tag{2} \]
where to each element \(f\in H\) there corresponds a vector-function \(f(x)\) on \(\Gamma\) with values in \(H_x\), and
\[ (B_kA_k f)(x)=i^{-1}(-1)^{x_k}f(x), \]
\[ (A_k f)(x)=i_k(x)C_k(x)f(x+\delta_k)\sqrt{\frac{d\mu(x+\delta_k)}{d\mu(x)}} \tag{3} \]
\[ (k=1,2,\ldots), \]
where \(i_k(x)=(-1)^{x_1+\cdots+x_k}\), and \(\{C_k(x)\}_1^\infty\) is a family of measurable operator-valued functions satisfying the functional equations: for almost
for all \(x\in\Gamma\):
\[ \begin{gathered} C_k(x)C_k^*(x)=I,\qquad C_k^*(x)=C_k(x+\delta_k),\\ C_k(x)C_l(x+\delta_k)=C_l(x)C_k(x+\delta_l)\qquad (k\ne l)\\ (k,l=1,2,\ldots). \end{gathered} \tag{4} \]
Let \(T\) be a self-adjoint bounded operator commuting with the operators of the representation \(\{A_k,B_k\}_1^\infty\). From commutativity with \(B_kA_k\) \((k=1,2,\ldots)\) it follows that \(T\) can be represented in the form of an operator-measurable bounded function \(T(x)\) \(\bigl(T(x)=T^*(x)\) for almost all \(x\in\Gamma\bigr)\)
\[ (Tf)(x)=T(x)f(x), \]
and from commutativity with the operators \(A_k\) \((k=1,2,\ldots)\) it follows that for almost all \(x\)
\[ C_k^*(x)T(x)C_k(x)=T(x+\delta_k)\qquad (k=1,2,\ldots). \tag{5} \]
Theorem 1. Let a representation of the Clifford algebra \(\{A_k,B_k\}_1^\infty\) be given, for which the quasi-invariant measure \(\mu\) is ergodic. If \(T=\{T(x)\}\) is a bounded self-adjoint operator commuting with the operators of the representation, then there exists a measurable mapping \(U(x)\) of the set \(\Gamma\) into the unitary operators of the spaces \(H_x\), for which
\[ T(x)=U(x)CU^*(x) \]
for almost all \(x\), where \(C\) is a bounded self-adjoint operator independent of \(x\).
We note the most essential point of the proof. Denote by \(\sigma\) the measure belonging to the maximal spectral type of the operator \(T\). Consider the weakly closed commutative \(*\)-ring generated by the operators \(B_kA_k\) \((k=1,2,\ldots)\) and \(T\). It is known that there exists a certain self-adjoint operator \(R\) which generates this commutative ring. Denote by \(\rho\) the measure belonging to the maximal spectral type of \(R\). It turns out that \(\rho\) is equivalent to \(\mu\times\sigma\) \((\rho\sim\mu\times\sigma)\). From this result Theorem 1 follows easily.
As a consequence of Theorem 1 we obtain:
Every quasi-invariant ergodic measure on \(\Gamma\) with respect to the group \(\Delta\) is equivalent to the direct product of two quasi-invariant ergodic measures on \(\hat\Gamma\).
2. Theorem 2. Let a representation of the Clifford algebra \(\{A_k\}_1^\infty\) be given. Denote by \(M\) the minimal weakly closed ring generated by the operators \(\{A_k\}_1^\infty\). Suppose that \(M\) is a factor in the sense of Murray and von Neumann. In order that \(M\) have type \(\mathrm{II}_1\), it is necessary and sufficient that on the elements of \(M\) one can define a linear homogeneous positive functional \(T\), continuous with respect to the weak topology in \(M\), for which
\[ T(I)=1,\qquad T(A_{i_1}\ldots A_{i_k})=0, \tag{6} \]
where \(i_1<\cdots<i_k\); \(i_s=1,2,\ldots\ (s=1,\ldots,k)\); \(k=1,2,\ldots\).
Necessity is obvious. Let us dwell on the proof of sufficiency. According to the results stated in (5), it is enough to show that:
1) \(T(C_1C_2)=T(C_2C_1)\) for \(C_i\in M\) \((i=1,2)\);
2) if \(P\) is a projection operator in \(M\) and \(T(P)=0\), then \(P=0\).
Suppose that the operators \(C_i\) \((i=1,2)\) can be represented in the form of a finite polynomial in the \(A_k\) \((k=1,2,\ldots)\). Select all operators \(A_k\) which enter into the expansion for \(C_1\) and \(C_2\). Construct the finite-dimensional Clifford algebra \(M_1\), whose generators are these operators. It is well known (4) that \(M_1\) is algebraically isomorphic to the full matrix algebra. By the uniqueness of the trace in \(M_1\), it coincides with \(T\); from the properties of the trace it follows that
\[ T(C_2C_1)=T(C_1C_2). \tag{7} \]
For arbitrary operators \(C_1, C_2\) from \(M\), (7) remains valid by virtue of the continuity of the functional \(T\).
Let us turn to the consideration of property 2). Suppose that \(T\) does not have this property; then \(M\) cannot have type \(\mathrm{II}_1\). Denote by \(N\) the set of all elements \(A\) of \(M\) for which \(T(A^*A)=0\). Obviously, \(N\) is a two-sided closed ideal in \(M\). By assumption, \(N\) is nonempty. Since a factor of type III is simple, the only remaining possibility is to suppose that \(M\) has either type \(\mathrm{II}_\infty\) or type \(\mathrm{I}_\infty\).
Form the space of cosets \(M/N\). If \(A\in M\), but \(A\notin N\), then by \(\widetilde A\) we denote the coset that contains the element \(A\). Define in the linear space \(M/N\) the scalar product
\[ \langle \widetilde A,\widetilde B\rangle=T(AB^*), \]
where \(A\in \widetilde A,\ B\in \widetilde B\). Obviously, \(\langle \widetilde A,\widetilde B\rangle\) does not depend on the choice of the representatives \(A\) and \(B\).
Thanks to the scalar product \(\langle\cdot,\cdot\rangle\), the space \(M/N\) becomes a pre-Hilbert space; denote its completion by \(\widehat H\). Define in \(\widehat H\) an involution \(J\), putting \(J\widetilde A=\widetilde {A^*}\) for elements of \(M/N\), and on the remaining elements of \(\widehat H\) define the operator \(J\) by continuity. Then \((\widehat H,J,M/N)\) is a Hilbert algebra in the sense of I. Segal \((^2)\).
Denote by \(L\) the minimal weakly closed ring generated by the operators \(L_{\widetilde A}\):
\[ L_{\widetilde A}\widetilde X=\widetilde A\widetilde X \qquad (\widetilde A,\widetilde X\in M/N). \]
I. Segal showed that the ring \(L\) possesses a measure \(m\) on projections, and if a projection has the form \(L_{\widetilde C}\), where \(\widetilde C\in M/N\), then \(m(L_{\widetilde C})=\langle \widetilde C,\widetilde C\rangle=T(C^*C)\).
Consider in \(M\) the projections \(P_1=\frac12(I+iA_1A_2)\) and \(P_2=\frac12(I-iA_1A_2)\). Obviously, \(P_1+P_2=I\) and \(P_2=A_2P_1A_2\). Since, by assumption, \(M\) has type either \(\mathrm{II}_\infty\) or \(\mathrm{I}_\infty\), \(P_1\) is an infinite projection; moreover, there exists a partially isometric operator \(U\) in \(M\) for which
\[ UU^*=I,\qquad U^*U=P_1. \]
Consequently,
\[ I=L_{\widetilde U\widetilde U^*}=L_{\widetilde U^*}L_{\widetilde U^*}=L_{\widetilde U}\cdot L_{\widetilde U}^*,\qquad L_{P_1}=L_{\widetilde U}^*L_{\widetilde U}. \tag{8} \]
In the cited work, I. Segal proved that if \(V\) is a partially isometric operator belonging to some weakly closed ring \(R\) with a measure \(m'\) on projections, then \(m'(V^*V)=m'(VV^*)\). Therefore, in our case (see (6))
\[ m(L_{P_1})=m(L_I)=T(I)=1; \]
on the other hand (see (6)),
\[ m(L_{P_1})=T(P_1)=T\bigl(\tfrac12(I+iA_1A_2)\bigr)=\tfrac12. \]
The contradiction obtained proves that \(M\) cannot have type \(\mathrm{II}_\infty\) or \(\mathrm{I}_\infty\).
Thus, if \(T\) does not have property 2), then the factor \(M\) belongs neither to type \(\mathrm{I}_\infty\), nor to \(\mathrm{II}_\infty\), nor to III; but this contradicts the Murray–von Neumann theory. Thus, property 2) is proved.
Theorem 2 shows that a factor representation of type \(\mathrm{II}_1\) for the Clifford algebra preserves important properties of the finite-dimensional Clifford algebra. The theorem also gives a simple method for establishing the type of the factor.
Physical-Technical Institute
of Low Temperatures
Academy of Sciences of the Ukrainian SSR
Received
5 IV 1967
CITED LITERATURE
- L. Gårding, A. Wightman, Proc. Nat. Acad. Sci. U. S. A., 40, No. 7, 617 (1954).
- I. Segal, Ann. Math., 57, 401 (1953).
- R. Blattner, Pacific J. Math., 8, 665 (1958).
- G. Weil, Classical Groups…, IL, 1947.
- M. A. Naimark, Normed Rings, Moscow, 1958.
- M. A. Naimark, S. V. Fomin, UMN, 10, 111 (1955).