UDC 517.433

MATHEMATICS

V. Ya. GOLODETS

ON FACTOR REPRESENTATIONS OF TYPE \(\mathrm{II}_1\)

FOR THE CLIFFORD ALGEBRA

(Presented by Academician I. M. Vinogradov on VI 6, 1966)

1. A representation of the Clifford (or spinor) algebra with an infinite number of generators is a set of linear self-adjoint operators \(\{A_k\}_1^\infty\), acting in a separable Hilbert space \(H\) and satisfying the relations

\[ A_k^2=1,\qquad A_kA_l+A_lA_k=0\quad (k\ne l). \tag{1} \]

If the weakly closed ring \(M\), generated by the operators \(\{A_k\}_1^\infty\), is a factor in the sense of Murray and von Neumann, then such a representation will be called a factor representation.

Some properties of factor representations were noted in \((^6)\). In the present note we give a complete classification of factor representations of type \(\mathrm{II}_1\), and also consider the question of decomposing a factor representation of type \(\mathrm{II}_1\) into irreducible representations.

2. Let \(\{A_k\}_1^\infty\) and \(\{B_k\}_1^\infty\) be two factor representations acting in the spaces \(H_1\) and \(H_2\); denote the corresponding factors by \(M_1\) and \(M_2\).

Definition. The factor representations \(\{A_k\}_1^\infty\) and \(\{B_k\}_1^\infty\) will be called algebraically isomorphic if one can establish an algebraic isomorphism \(\varphi\) between the factors \(M_1\) and \(M_2\) in such a way that \(\varphi(A_k)=B_k\) \((k=1,2,\ldots)\).

It can be shown that there exist algebraically nonisomorphic factor representations of type III and algebraically nonisomorphic representations of type \(\mathrm{II}_\infty\).

Theorem 1. All factor representations of the Clifford algebra of type \(\mathrm{II}_1\) are algebraically isomorphic.

We outline the proof of the theorem. Introduce in the ring \(M_i\) \((i=1,2)\) a scalar product by putting

\[ \langle A,B\rangle=T_i(AB^*), \]

where \(T_i\) is the relative trace for the factor \(M_i\), and \(A,B\in M_i\). Then \(M_i\) becomes a pre-Hilbert space. We denote by \(Q(M_i)\) the completion of \(M_i\) with respect to this scalar product. We shall assume that \(T_i(I_i)=1\), where \(I_i\) is the identity operator. Consider monomials of the form

\[ A_z=A_{s_1}A_{s_2}\cdots A_{s_n}, \]

where \(z\) denotes a set of indices \(s_1<s_2<\cdots<s_n\); \(n=1,2,\ldots;\ s_i=1,2,\ldots\ (i=1,2,\ldots,n)\). It is not difficult to verify that

\[ T_1(A_z)=0 \]

for all \(z\). Consequently, the monomials of the form \(A_z\) and the identity operator \(I_1\) form a complete orthonormal basis in the space \(Q(M_1)\). Every element \(D\) of \(Q(M_1)\) can be expanded in a series with respect to this basis,

\[ D\sim \sum_z d_z A_z \qquad \left(\sum_z |d_z|^2<\infty\right). \]

Now establish a correspondence between \(Q(M_1)\) and \(Q(M_2)\) by the rule

\[ \varphi(I_1)=I_2;\qquad \varphi(A_z)=B_z . \]

It is not difficult to verify that \(\varphi(C+D)=\varphi(C)+\varphi(D)\) and \(\varphi(C^*)=\varphi(C)^*\), and if \(C\) and \(D\) belong to \(M_1\) (i.e., are bounded operators), then

\[ \varphi(CD)=\varphi(C)\varphi(D). \]

Consequently, if \(C\) is a positive definite operator from \(M_1\), then \(\varphi(C)\) is also a positive definite operator. But then every operator \(D\) from \(M_1\), under the mapping \(\varphi\), falls into \(M_2\), since the norm of the operator \(\|D\|\) is the least number \(a\ge 0\) for which the operator \(a^2I_1-D^*D\) is positive definite. From the considerations given, it follows that \(\varphi(M_1)\subset M_2\); similarly we obtain that \(\varphi^{-1}(M_2)\subset M_1\). Hence, \(\varphi(M_1)=M_2\).

In \((^1)\) von Neumann constructed an example of a factor-representation of type \(\mathrm{II}_1\). From Theorem 1 it follows that all factor-representations of type \(\mathrm{II}_1\) for the Clifford algebra are algebraically isomorphic to this representation.

1. Let us turn to the question of the unitary classification of factor-representations of type \(\mathrm{II}_1\).

Let \(M\) be a factor of type \(\mathrm{II}_1\), whose elements act in the space \(H\). By \(M'\), as usual, denote the commutant of the factor \(M\). Let \(f\in H\); then by \(H_f^{M'}\) denote the closure of the set \(M'f\) with respect to the norm of \(H\), and by \(H_f^M\) the closure of \(Mf\). If \(P_1^f\) is the projector onto \(H_f^{M'}\), and \(P_2^f\) the projector onto \(H_f^M\), then

\[ T_M(P_1^f)=cT_{M'}(P_2^f), \]

where the constant \(c\) does not depend on \(f\), and \(T_M\) and \(T_{M'}\) are the traces for \(M\) and \(M'\), respectively. Introduce for consideration the number \((^2)\)

\[ \theta=T_{M'}(I)/cT_M(I). \tag{2} \]

Under the assumption that \(M\) is a finite factor, \(\theta\) varies in the range \(0<\theta\le \infty\).

Theorem 2. For every number \(\theta\) \((0<\theta\le \infty)\) there exists a factor-representation of the Clifford algebra of type \(\mathrm{II}_1\), and this number \(\theta\) determines the factor-representation uniquely up to unitary equivalence.

For other representations of the Clifford algebra with a continuous ergodic measure \((^3)\), such a simple classification is unknown.

1. Analyzing the example of von Neumann, which was already mentioned, one can easily verify that the factor-representation of the Clifford algebra of type \(\mathrm{II}_1\) decomposes into a direct integral of irreducible representations for which \(\nu=1\), according to the classification of Gårding and Wightman \((^3)\). It is of interest to determine whether irreducible representations for which \(\nu\ne 1\) will enter into the decomposition of a factor-representation of type \(\mathrm{II}_1\).

Theorem 3. A factor-representation of type \(\mathrm{II}_1\) for the Clifford algebra can be decomposed into a direct integral of irreducible representations for which \(\nu=2^n\), where \(n=0,1,2,\ldots\).

The proof of the theorem consists in constructing, for \(\nu=2^n\) \((n=0,1,\ldots)\), a factor-representation of type \(\mathrm{II}_1\) for the Clifford algebra that has the properties of interest to us. We note that the very method of constructing factors of type \(\mathrm{II}_1\) was previously unknown.

Let \(\Gamma\) be the space of sequences \(x=x_1x_2\ldots\), where \(x_i=0\) or \(1\). One may regard \(\Gamma\) as a group if componentwise addition modulo 2 is introduced for sequences. By \(\delta_k\) \((k=1,2,\ldots)\) denote the sequence from \(\Gamma\) with a one in the \(k\)-th place and zeros in all the others. The Haar measure \(\mu\) of the group \(\Gamma\) coincides with Lebesgue measure under the natural mapping of \(\Gamma\) onto the unit interval.

Let \(H\) be the Hilbert space of vector-functions \(f(x,y)\) on \(\Gamma\times\Gamma\), whose values belong to a \(2^n\)-dimensional complex pro-

space \(R_n\). We define the scalar product in \(\tilde H\) by the formula

\[ \langle f,g\rangle=\int_{\Gamma\times\Gamma} (f(x,y),g(x,y))\,d\mu(x)\,d\mu(y), \]

where \((\cdot,\cdot)\) is the scalar product in \(R_n\).

We define a representation of the anticommutation relations \(\{A_k,B_k\}_1^\infty\) in \(\tilde H\) as follows:

\[ \begin{aligned} A_k f(x,y)&=j_k(x)c_k(x+y)f(x+\delta_k,y),\\ B_k f(x,y)&=i^{-1}j_{k+1}(x)c_k(x+y)f(x+\delta_k,y) \qquad (k=1,2,\ldots), \end{aligned} \tag{3} \]

where \(j_k(x)=(-1)^{x_1+\cdots+x_{k-1}}\), and \(\{c_k(x)\}_1^\infty\) are measurable functions on \(\Gamma\), whose range consists of unitary operators in the space \(R_n\). Similarly we define the representation \(\{\tilde A_l,\tilde B_l\}_1^\infty\):

\[ \begin{aligned} \tilde A_l f(x,y)&=j_l(y)c_l(x+y)f(x,y+\delta_l),\\ \tilde B_l f(x,y)&=i^{-1}j_l(y)c_l(x+y)f(x,y+\delta_l) \qquad (l=1,2,\ldots). \end{aligned} \tag{3'} \]

Before explicitly writing the expression for \(c_k(x)\) \((k=1,2,\ldots)\), we make a remark. Denote by \(\{p_s\}_1^{2n}\) an irreducible representation of the Clifford algebra in \(R_n\) with \(2n\) generators
\(p_k=p_k^*\), \(p_k^2=1\) \((k=1,2,\ldots,n)\), \(p_kp_l+p_lp_k=0\) \((l\ne k)\) \({}^4\). Now put

\[ c_{2nk+r}(x)=s_{2nk+r}(x)p_r, \tag{4} \]

where \(k=0,1,2,\ldots;\ 0\le r\le 2n\),

\[ s_{2nk+r}(x)=(-1)^{\sum_{i=1}^{2nk+r}x_i-\sum_{j=0}^{k}x_{2nj+r}}. \]

It is not hard to verify that the relations

\[ \begin{aligned} c_k(x+\delta_k)&=c_k^*(x),\\ c_k(x)c_l(x+\delta_k)&=c_l(x)c_k(x+\delta_l)\qquad (l\ne k) \end{aligned} \tag{5} \]

are satisfied.

From the results of (3) it follows that, for almost every \(y\), \(\{A_k,B_k\}_1^\infty\) in (3) define an irreducible representation, and for almost every \(x\), \(\{\tilde A_k,\tilde B_k\}_1^\infty\) also define an irreducible representation. On the basis of Theorem 3 \({}^6\) we assert that the weakly closed ring \(M\) generated by the operators \(\{A_k,B_k\}_1^\infty\), and also the ring \(\tilde M\) generated by \(\{\tilde A_k,\tilde B_k\}_1^\infty\), are factors. Moreover, it turns out that \(\tilde M=M'\) and \(M=\tilde M'\).

Let us verify that \(M\) and \(\tilde M\) are factors of type \(\mathrm{II}_1\). For this it is enough to show \({}^5\) that on all operators from \(M\) one can define a linear homogeneous positive functional \(T\), satisfying the conditions:
1) \(T(I)=1\), where \(I\) is the identity operator; 2) \(T(AB)=T(BA)\) \((A,B\in M)\); 3) if \(P\) is a projector from \(M\) and \(T(P)=0\), then \(P=0\).

Let \(e_1,\ldots,e_{2^n}\) be an orthonormal basis in \(R_n\), whose elements permute the operators \(p_1,\ldots,p_{2n}\). Denote \(\varphi_0(x,y)\equiv 1\) and consider the vector-function \(f(x,y)=\varphi_0(x,y)e_1\). Then the functional \(T(A)=\langle Af,f\rangle\), where \(A\in M\), has all the properties listed.

We note that if in our construction \(n=0\), then we obtain an example of von Neumann \({}^1\). The case \(n\ne0\) has not been considered previously.

It seems interesting that if, as the measure \(\mu(y)\), one takes an arbitrary measure not equivalent to Lebesgue measure, but quasi-invariant and ergodic with respect to shifts by \(\delta_k\) \((k=1,2,\ldots)\), then we obtain a factor-representation of type III for the Clifford algebra. For the case \(n\ne0\) these representations, apparently, were not known.

The author expresses sincere gratitude to Prof. M. A. Naimark for his attention to the work.

Physical-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR

Received
23 V 1966

CITED LITERATURE

\({}^1\) S. von Neumann, Compositio Math., 6, 1, 1 (1938).
\({}^2\) F. S. Murray, S. von Neumann, Ann. Matt., 44, 716 (1943).
\({}^3\) L. Gårding, A. Wightman, Proc. Nat. Acad. Sci. U.S.A., 40, No. 7, 617 (1954).
\({}^4\) T. Weyl, Classical Groups, Their Invariants and Representations, IL, 1947.
\({}^5\) M. A. Naimark, Normed Rings, M., 1956.
\({}^6\) V. Ya. Golodets, DAN, 167, No. 1 (1966).