Full Text
MATHEMATICS
V. Ya. Golodets
ON FINITE GROUPS OF AUTOMORPHISMS OF HYPERFINITE FACTORS
(Presented by Academician I. N. Vekua, 10 III 1969)
1. The purpose of the present article is to set forth the main points of the proof of the following theorem.
Theorem 1. Let \(M\) be a factor of type \(\mathrm{II}_1\), whose elements are operators in a separable Hilbert space \(H\). Let, further, \(G\) be a finite group of outer automorphisms of \(M\).
If \(M\) is an approximately finite factor, then \(\mathfrak{M}=G\times M\) is likewise an approximately finite factor. The converse is true.
Since a factor of type \(\mathrm{III}\) which is an inductive limit of hyperfinite factors is itself hyperfinite, Theorem 1 implies the following result.
Theorem 2. Let \(M\) be an approximately finite factor of type \(\mathrm{II}_1\), and let \(G\) be a group of outer automorphisms of \(M\), which can be regarded as an increasing sequence of finite groups of automorphisms of \(M\).
Then the factor \(G\times M\) is also approximately finite.
Theorem 2 is a generalization of the theorem of Murray–von Neumann \((^2)\) on ergodic freely acting automorphisms of commutative weakly closed \(*\)-algebras.
As an example of the group \(G\) referred to in Theorem 2, one may take, for instance, the group of all permutations of a countable number of elements, where each element of this group permutes only a finite number of elements.
2. We shall set forth auxiliary results.
Lemma. Let \(M\) be a factor of type \(\mathrm{II}_1\), the same as in the formulation of Theorem 1, and let \(G\) be a finite group of outer automorphisms of \(M\).
If \(M_0\) is the set of all elements of \(M\) fixed with respect to \(G\), then \(M_0\) is a subfactor of \(M\).
Furthermore, in order that the crossed product \(\mathfrak{M}=G\times M\) be an approximately finite factor, it is necessary and sufficient that \(M_0\) be an approximately finite factor.
A few words concerning the proof of the lemma. \(M_0\) consists of elements \(M\) of the form
\[ \sum_{g\in G} m^g \quad (m\in M). \]
Further,
\[ Q=\frac{1}{n}\sum_{g\in G} g\times 1, \]
where \(n\) is the order of the group \(G\), is a projection from \(\mathfrak{M}\). But then \(Q\mathfrak{M}Q\) is a factor in the space \(Q(G\otimes H)\). The factor \(Q\mathfrak{M}Q\) consists of elements of the form
\[ Q\sum_{g\in G}(1\times m^g)\quad (m\in M). \]
Thus one can establish an algebraic isomorphism between \(M_0\) and \(Q\mathfrak{M}Q\). The further arguments are evident.
We now formulate the theorem whose proof is given in \((^3)\).
Theorem A. Let \(M\) be a factor of type \(\mathrm{II}_1\), the same as in the formula-
of Theorem 1. Let, further, \(G\) be a cyclic group of outer automorphisms of \(M\) of order \(n\).
If \(M\) is an approximately finite factor, then \(\mathfrak{M}=G\times M\) is also an approximately finite factor. The converse is true.
- We proceed to the proof of Theorem 1. Suppose first that \(M\) is an approximately finite factor, and prove that \(\mathfrak{M}=G\times M\) is also approximately finite. For this, by the lemma, it suffices to verify that \(M_0\) is approximately finite.
Let us carry out this program. Let \(n\) be the order of \(G\). Consider the Hilbert space \(H_n\), \(\dim H_n=n\). Denote by \(A_n\) a maximal commutative subring in \(H_n\). Let \(P_0,P_1,\ldots,P_{n-1}\) be minimal projections generating \(A_n\). Consider a group of outer automorphisms of \(A_n\), isomorphic to \(G\); then \(P_i^g=P_j\) \((i\ne j)\).
Now, in the Hilbert space \(\widetilde H=H\otimes H_n\), construct the ring of operators \(\widetilde M=M\otimes A_n\) as the tensor product of the rings \(M\) and \(A_n\). The ring \(\widetilde M\) has a group of outer automorphisms isomorphic to the group \(G\):
\[ (m\otimes 1)^g=m^g\otimes 1;\qquad (1\otimes a)^g=1\otimes a^g, \]
where \(m\in M,\ a\in A_n\). Construct the crossed product \(\widetilde{\mathfrak{M}}=G\times \widetilde M\). Since
\((1\otimes P_0)\widetilde{\mathfrak{M}}(1\otimes P_0)=M\otimes P_0\) is an approximately finite factor of type \(\mathrm{II}_1\) in the space \(P_0\widetilde H\), it follows that \(\widetilde{\mathfrak{M}}\) is also an approximately finite factor. (It is clear that the converse is true.)
Let \(N\) be the set of fixed elements of \(\widetilde M\) with respect to the group \(G\). Using the same device as in the proof of the lemma, we establish that \(N\) is a subfactor of \(\widetilde{\mathfrak{M}}\). Since \(\widetilde{\mathfrak{M}}\) is approximately finite, \(N\) is also approximately finite.
Note that \(N\) contains the factor \(1\times (M_0\otimes 1)\), isomorphic to \(M_0\), as a subfactor. We shall show that the approximate finiteness of \(N\) implies the approximate finiteness of \(M_0\).
Denote the operators \(1\times(1\otimes P_i)\) \((i=0,1,\ldots,n-1)\) simply by \(P_i\), and denote the commutative algebra which they generate, as before, by \(A_n\). Next, denote the operators \([g\times(1\otimes 1)]\) from \(\widetilde{\mathfrak{M}}\) by \(U_g\). Now consider the subalgebra \(\widetilde{\mathfrak{M}}_n\subset\widetilde{\mathfrak{M}}\) generated by the operators \(P_i\) \((i=0,1,\ldots,n-1)\) and \(U_g\) \((g\in G)\). It is clear that \(\widetilde{\mathfrak{M}}_n\) is isomorphic to the full matrix algebra of order \(n\). Then \(\widetilde{\mathfrak{M}}_n\) contains a unitary operator \(U_h\) such that
\[ U_h^*P_iU_h=P_{i+1}\quad \text{(indices are taken mod } n) \tag{1} \]
and \(U_h^n=1\). A more detailed analysis shows that \(U_h\) can be represented in the form (see (4))
\[ U_h=\sum_{g\in G} P_gU_g, \tag{2} \]
where the \(P_g\) are mutually disjoint projections from \(A_n\), and moreover
\[ \sum_{g\in G} P_g=I,\qquad \sum_{g\in G} P_g^g=I. \tag{3} \]
It is clear that the operator \(U_h\) determines an outer automorphism \(h\) of the algebra \(A_n\):
\[ U_h^*aU_h=a^h\quad (a\in A_n). \]
Let us verify that \(U_h\) determines an outer automorphism of \(1\times\widetilde M\). Indeed, if \(m\in 1\times\widetilde M\), then
\[ m^h=U_h^*mU_h =\sum_{g_1,g_2} U_{g_1}^*P_{g_1}mP_{g_2}U_{g_2} =\sum_{g\in G} U_g^*(mP_g)U_g =\sum_{g\in G} m^gP_g^g. \tag{4} \]
But
\[ \sum_{g\in G} m^gP_g^g \in 1\times \widetilde M. \]
We shall show that \(\widetilde{\mathfrak{M}}\) is the crossed product \(G_h\times\widetilde M\), where \(G_h\) is the cyclic group of order \(n\) generated by \(U_h\). First, \(\widetilde{\mathfrak{M}}\) is generated by the operators from \(1\times\widetilde M\) and \(G_h\). Second, the ope-
the operators \(U_h^i\) \((i=0,1,\ldots,n-1)\) are orthogonal to the operators from \(I\times \widetilde M\). Indeed, let \(m\in I\times \widetilde M\) and let \(\operatorname{Tr}\) be the trace in \(\mathfrak M\); then
\[ \operatorname{Tr}(mU_h)=\sum_{g\in G}\operatorname{Tr}(mP_gU_g)=0, \]
since the \(U_g\) \((g\in G)\) are orthogonal to the operators from \(I\times \widetilde M\) by the definition of the crossed product.
Denote by \(N_1\) the subset of \(\widetilde M\) invariant with respect to \(h\). From (2) and (4) it follows that \(N\subset N_1\). Since (2) is invertible, \(N_1\subset N\), and \(N=N_1\).
Let us note that the factor \(N_1\) is generated by elements of the form
\[ \sum_{i=1}^{n-1}(mP_k)^{h^i}, \qquad m\in I\times(M\otimes I). \]
Define the mapping \(\sigma:N\to N\)
\[ \sigma:\ \sum_{i=1}^{n-1}(mP_k)^{h^i}\to \sum_{i=1}^{n-1}(mP_{k+1})^{h^i} \quad\text{(the indices are taken modulo \(n\)).} \]
An elementary verification shows that \(\sigma\) is an outer automorphism of \(N\) of order \(n\). Denote by \(G_\sigma\) the group generated by \(\sigma\). Since \(N\) is approximately finite, \(G_\sigma\times N\) is also an approximately finite factor (Theorem A). Further, the set of elements of \(N\) invariant with respect to \(\sigma\) coincides with \(I\times(M_0\otimes I)\). From the lemma we conclude that \(I\times(M_0\otimes I)\), and consequently \(M_0\), is an approximately finite factor. This completes the first part of the proof of Theorem 1.
The proof of the second part of the theorem now presents no difficulties.
Physical-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR
Kharkov
Received
3 II 1969
REFERENCES
- M. Nakamura, Z. Takeda, Proc. Nat. Acad., 34, 489 (1958).
- F. Murray, J. von Neumann, Ann. Math., 44, 716 (1943).
- V. Ya. Golodets, DAN, 181, 1307 (1968).
- H. Choda, Proc. Japan. Acad., 41, 280 (1965).
- M. A. Naimark, Normed Rings, 2nd ed., “Nauka,” 1968.