UDC 519.9

MATHEMATICS

V. Ya. GOLODETS

STUDY OF APPROXIMATELY FINITE VON NEUMANN ALGEBRAS WITH FINITE TRACE

(Presented by Academician A. N. Kolmogorov, 18 XII 1967)

1. Among factors of type \(\mathrm{II}_1\), the simplest properties are possessed by approximately finite factors. Recall that an approximately finite factor of type \(\mathrm{II}_1\) is a factor \(M\) with an exact, normal, and finite trace; moreover, there exists an increasing sequence of factors \(\{M_n\}_1^\infty\) of type \(\mathrm{I}_n\) \((n<\infty)\), generating \(M\).

Let us recall the results of Murray—von Neumann on approximately finite factors \((^2)\).

Let \(A\) be a commutative von Neumann algebra with exact normal and finite trace \(\operatorname{Tr}_A\), and let \(G\) be a discrete group of \(*\)-automorphisms of \(A\) preserving the trace. If \(G\) is ergodic and freely acting, then the crossed product \(G \times A\) is a factor of type \(\mathrm{II}_1\). From the assumption that \(G\) is an increasing sequence of finite groups, it follows that \(G \times A\) is an approximately finite factor.

Murray and von Neumann formulated the theorem:

Let \(G\) be a discrete commutative group of automorphisms of \(A\), preserving the trace; then \(G \times A\) is also an approximately finite factor.

This theorem was proved in full only recently by H. Dye \((^{3,4})\), who obtained profound results in the theory of measure-preserving transformations.

In the present article we have attempted to generalize these facts under the assumption that the commutative \(*\)-ring \(A\) is replaced by an approximately finite weakly closed \(*\)-ring \(M\) with finite trace. The results found are used to study factor representations of type \(\mathrm{II}_1\) of discrete nilpotent groups.

We note that the theory of crossed products as applied to factors of type \(\mathrm{II}_1\) was developed in \((^1)\). We have tried to preserve the notation used in that work.

1. We first study cyclic groups of automorphisms of approximately finite factors.

Theorem 1. Let \(M\) be an approximately finite factor of type \(\mathrm{II}_1\), whose elements are operators in a separable Hilbert space \(H\). Let \(G\) be a cyclic group of order \(n\) \((n=2,3,\ldots)\) of outer automorphisms of \(M\). Then the crossed product \(\mathfrak M = G \times M\) is an approximately finite factor of type \(\mathrm{II}_1\).

The proof of the theorem is based on the following lemmas.

Lemma 1. Let \(M\) be a factor of type \(\mathrm{II}_1\), and let \(G\) be a cyclic group of outer automorphisms of the factor \(M\) of order \(n\) \((n=2,3,\ldots)\). Denote by \(M_0\) the set of fixed elements of \(M\) relative to the group \(G\). Then \(M_0\) is a factor of type \(\mathrm{II}_1\).

If \(M_0\) is an approximately finite factor of type \(\mathrm{II}_1\), then \(\mathfrak M = G \times M\) is also an approximately finite factor of type \(\mathrm{II}_1\). The converse is true.

Lemma 2. Let \(M\) be a Neumann algebra with a faithful, normal and finite trace \(\operatorname{Tr}_M\), whose elements are operators in the Hilbert space \(H\). Let \(\varepsilon\) be a primitive root of unity of the \(n\)-th degree, where \(n\) is a natural number \((n=2,3,\ldots)\).

In order that the ring \(M\) be an approximately finite factor of type \(\mathrm{II}_1\), it is necessary and sufficient that \(M\) be the minimal weakly closed ring generated by a family of operators \(\{w_i\}_{1}^{\infty}\) in \(H\) possessing the following properties:

1) \(w_i^n=1,\; w_i^k\ne 1\;(0<k<n),\; i=1,2,\ldots\);

2) \(w_{i-1}w_i=\varepsilon w_iw_{i-1},\; w_iw_j=w_jw_i\quad (j\ne i-1,i+1)\).

Lemma 3. Let \(M\) be a factor of type \(\mathrm{II}_1\), whose elements are operators in the Hilbert space \(H\). Let \(N_1\) and \(N_2\) be subfactors of \(M\) of type \(\mathrm{I}_n\). Then there exists an automorphism \(\varphi\) of the factor \(M\) which maps \(N_1\) isomorphically onto \(N_2\).

We outline the proof of Theorem 1. Denote by \(g\) a generator of \(G\), and by \(U_g\) the corresponding unitary operator from \(\mathfrak M=G\times M\). Consider the automorphism \(h\) of the factor \(\mathfrak M\), which we define as follows:

\[ (U_g)^h=\varepsilon U_g;\qquad (1\otimes m)^h=1\otimes m,\qquad m\in M, \]

where \(\varepsilon\) is a primitive root of unity of the \(n\)-th degree. It is easy to prove that \(h\) is an outer automorphism of \(\mathfrak M\) and \(h^n=1\). Denote by \(G_h\) the group generated by \(h\), and construct \(G_h\times\mathfrak M\). Since the set of elements of \(\mathfrak M\) fixed with respect to \(h\) is the approximately finite factor \(M\), it follows, according to Lemma 1, that \(G_h\times\mathfrak M\) is also an approximately finite factor of type \(\mathrm{II}_1\). Consequently, in \(G_h\times\mathfrak M\) there exists a family of operators \(\{w_i\}_{1}^{\infty}\) possessing properties 1) and 2) of Lemma 2 and generating \(G_h\times\mathfrak M\).

Let \(h\) correspond to the unitary operator \(U_h\in G_h\times\mathfrak M\). Consider now the subring of \(G_h\times\mathfrak M\) generated by the operators \(U_g, U_h\). A simple verification shows that this ring is a factor of type \(\mathrm{I}_n\). The ring generated by the operators \(w_1\) and \(w_2\) is also a factor of type \(\mathrm{I}_n\). It follows from Lemma 3 that there exists an automorphism \(\varphi\) of the factor \(G_h\times\mathfrak M\) for which

\[ \varphi(w_1)=U_h,\qquad \varphi(w_2)=U_g. \]

Put

\[ \varphi(w_k)=U_k\qquad (k=3,4,\ldots). \]

Then the family of operators \(U_g, U_k\;(k=3,4,\ldots)\) generates the factor \(\mathfrak M\) and possesses properties 1) and 2) of Lemma 2. From this lemma we conclude that \(\mathfrak M\) is an approximately finite factor, as was required to be established.

As a corollary we obtain the theorem:

Theorem 2. Let \(M\) be an approximately finite factor of type \(\mathrm{II}_1\). Let \(G\) be a solvable finite group of outer automorphisms of \(M\). Then \(G\times M\) is an approximately finite factor of type \(\mathrm{II}_1\).

3. Definition. A Neumann algebra \(M\) with a faithful normal and finite trace \(\operatorname{Tr}_M\) is called approximately finite if, for an arbitrary number \(\varepsilon>0\) and for every finite subset of operators \(x_1,\ldots,x_s\) from \(M\), there exists a subalgebra \(N\) of type I, whose center \(Z_N\) contains the center \(Z_M\) of the algebra \(M\), and which contains operators \(x_1',\ldots,x_s'\) such that

\[ [[x_i-x_i']]=\operatorname{Tr}_M\bigl((x_i-x_i')(x_i-x_i')^*\bigr)^{1/2}<\varepsilon \qquad (i=1,\ldots,s). \]

In what follows we shall assume that the center \(Z_M\) contains no minimal projections.

Theorem 3. Let \(\mathfrak M\) be a factor of type \(\mathrm{II}_1\), whose elements are operators in a separable Hilbert space \(H\). Let \(M\) be an approximately finite subalgebra of \(\mathfrak M\), whose center \(Z_M\) is distinguished from \(\{\lambda I\}\), where \(I\) is the identity operator in \(H\).

Denote by \(N(M)\) the normalizer of \(M\) in \(\mathfrak M\), i.e., the set of unitary operators \(U \in \mathfrak M\) such that \(\varphi_U(M)=U^{-1}MU=M\). We note that \(\varphi_U(Z_M)=Z_M\) for \(U \in N(M)\). Denote the corresponding group of automorphisms of the center \(Z_M\) by \(G\).

Suppose that the weak closure of \(N(M)\) coincides with \(\mathfrak M\), and that to each \(U \in N(M)\), \(U \notin M\), there corresponds a nonidentity automorphism \(\varphi_U\) of the center \(Z_M\). If \(G\) is approximately finite in \(I'\), the group of automorphisms of \(Z_M\), then \(\mathfrak M\) is an approximately finite factor.

G. Dye proved this theorem under the assumption that \(M\) is a commutative subalgebra of \(\mathfrak M\) \((^4)\).

Corollary. Let \(M\) be an approximately finite Neumann algebra with finite trace. Suppose that the center \(Z_M\) is different from \(\{\lambda I\}\). Denote by \(G\) the commutative group of outer automorphisms of \(M\) such that to each \(g \in G\) there corresponds a freely acting automorphism of the center \(Z_M\).

Then the crossed product \(\mathfrak M = G \times M\) is an approximately finite Neumann algebra.

If \(G\) is an ergodic group of automorphisms of the center \(Z_M\), then \(\mathfrak M = G \times M\) is an approximately finite factor of type \(\mathrm{II}_1\).

Theorem 3 has an interesting application in the theory of representations of discrete groups.

Theorem 4. Let a discrete nilpotent group \(G\) have a factor representation \(g \to U_g\) of type \(\mathrm{II}_1\) in the Hilbert space \(H\). Then the factor \(M\), generated by the operators \(U_g\), \(g \in G\), is approximately finite.

The proof of the theorem relies essentially on the following result.

Lemma 4. Let a discrete nilpotent group \(G\) have a commutative normal divisor \(K\), which is a maximal commutative subgroup of \(G\), i.e., the centralizer \(C(K)\) of the group \(K\) in \(G\) coincides with \(K\). If \(g \to U_g\) is an exact factor representation of type \(\mathrm{II}_1\) of the group \(G\) in the Hilbert space \(H\), then the factor \(M\), generated by the operators \(U_g\), \(g \in G\), is approximately finite.

We outline the proof of the lemma. Recall that a representation is called exact if operators of the form \(\lambda I\) correspond only to elements of the center \(Z_0\) of the group \(G\). From the assumption of nilpotency of the group \(G\) it follows that there exists a sequence of subgroups:

\[ G=C(Z_0)\supset C(Z_1)\supset \ldots \supset C(Z_t), \]

where \(Z_k\) is a subgroup of \(K\) and \(Z_0 \subset Z_1 \subset \ldots \subset Z_t=K\), while \(C(Z_k)\) is the centralizer of \(Z_k\) in \(C(Z_{k-1})\), and the commutator group \((C(Z_{i-1}), Z_i)\subset Z_{i-1}\) \((i=1,\ldots,t-1)\). From the assumption on the group \(K\) it follows that \(C(Z_t)=K\). Further, each group \(C(Z_i)\) is a normal divisor of \(C(Z_{i-1})\). Since \((C(Z_{i-1}), Z_i)\subset Z_{i-1}\), the group \(C(Z_{i-1})/C(Z_i)\) induces a commutative group of automorphisms of the center \(Z_i\) of the group \(C(Z_i)\). Indeed, if \(k \in Z_i\), \(\tilde x,\tilde y \in C(Z_{i-1})/C(Z_i)\), then

\[ x^{-1}kx=k_xk,\qquad y^{-1}ky=k_yk\qquad (x\in\tilde x,\ y\in\tilde y), \]

where \(k_x,k_y\in Z_{i-1}\), and therefore

\[ y^{-1}(x^{-1}kx)y=y^{-1}k_xky=k_xk_yk=k_yk_xk=x^{-1}(y^{-1}ky)x. \]

Denote by \(M_i\) \((i=0,1,\ldots,t)\) the weakly closed ring generated by the operators \(U_g\), where \(g\in C(Z_i)\). Then \(M_t\) is a commutative subring of \(M\); consequently, \(M_t\) is approximately finite. Suppose that \(M_i\) is approximately finite, and prove the approximate finiteness of \(M_{i-1}\), where \(M_i\subset M_{i-1}\). We note that the center of \(M_i\) is generated by the operators \(U_g\), where \(g\in Z_{i-1}\), and, as we have seen, the operators \(U_g\), where \(g\in C(Z_{i-1})\), induce a commutative group \(C(Z_{i-1})/C(Z_i)\) of automorphisms of the cent-

From Theorem 3 we conclude that \(M_{i-1}\) is approximately finite. This completes the proof of the lemma.

The proof of Theorem 4 requires only minor complications.

Example. Let \(G\) be the group of triangular matrices of order \(n\) \((n=2,3,\ldots)\) with ones on the main diagonal over the field of rational numbers or over the ring of integers. It is not difficult to show that the regular representation of this group decomposes into a direct integral of factor representations of type \(\mathrm{II}_1\). It follows from Theorem 4 that these factors are approximately finite.

Theorem 5. Let \(G\) be a discrete group. Then its regular representation generates a Neumann algebra with a finite, faithful, and normal trace.

This result is well known for discrete commutative groups (Pontryagin’s theorem) and for finite groups.

In particular, the regular representation of a discrete nilpotent group generates an approximately finite Neumann algebra with a finite trace.

Institute for Low Temperature Physics
Academy of Sciences of the Ukrainian SSR

Received
4 XII 1967

REFERENCES

1. M. Nakamura, Z. Takeda, Proc. Japan. Acad., 34, 489 (1958).
2. F. S. Murray, J. von Neumann, Ann. of Math., 44, 716 (1943).
3. H. A. Dye, Am. J. Math., 81, 119 (1959).
4. H. A. Dye, Am. J. Math., 55, 551 (1963).
5. H. Choda, Proc. Japan. Acad., 41, 280 (1965).
6. M. A. Naimark, Normed Rings, Moscow, 1956.