The classical papers by Connes [1] on the classification of factors and their automorphisms and, in particular, the complete classification of injective factors played a significant role in the development of the theory of operator algebras. An analog of the Connes classification for the real factors was obtained in [2,3]. It turns out that the results for real factors of types II~, IIoo, and III1 are completely similar to those in the complex case. However, if a real factor is of the type IIIx, 0 < A < 1, then, in contrast to the case of complex factors, there are exactly two constructions of the crossed product of an algebra by its automorphism. It was proved in [2] that the structure of a real factor R of the type IIIx, 0 < A < 1, is determined either by a pair (Q,0), where Q is a real factor of the type II~ and 0 is its *-automorphism with mod(/9) = A, or by a pair (N, fl), where N is a factor of the type IIoo and fl is its *-antiautomorphism with mod (/3) = v/A (in the sense of [4]). Hence, there exist two nonisomorphic real injective W*-factors of the type IIIx, 0 < A < 1 (see [4,5]).As is known, every real W*-algebra is generated by an involutive *-antiautomorphism on the enveloping W*-algebra, which is unique up to conjugation (for instance, see [5, Theorem 3.1]). Giordano [4] showed that on a complex injective factor of the type IIIx, A E (0, 1), there are two distinct classes of involutive *-automorphisms. Namely, we see that if a real factor is generated by a pair (Q, fl), then we have mod(c~) = 1 (in the sense of [4]) for the corresponding involutive *-antiautomorphism a, and if a real factor is generated by a pair (N, fl), then mod(~) = x/~ (see [2]).In the present paper we study these two classes of nonisomorphic real injective factors of the type IIIx, 0 < A < 1. It is shown that one class contains hyperfinite real factors, and the other contains injective factors that are not hyperfinite. Hence, the notions of hyperfiniteness and injectivity are generally nonequivalent for real algebras, whereas in the complex case these notions coincide. This extends the basic results by Stacey and Giordano that are used substantially in the present paper. Definition [6]. A real factor R is said to be hyperfinite if there is an increasing sequence {R,~) of its finite-dimensional subalgebras such that ]IE R, and (.J,~>l Rn is weakly dense in R.
Preliminaries. Let B(H)beLet Ilxll~ = r x + xz*)/2) 1/2 be the seminorm generated by a normal state r We can readily see that, for each unitary u e R, we have Ilull~ # = 1 and that a sequence {x,) in R is a-strongly convergent x # to an element x if and only if lim~ IIx, -I[r = 0. The following theorem holds. Theorem 1. Let R be a real factor of the type IIIA, 0 < A < 1. Then the following conditions are equivalent:(1) R is a hyperfinite factor;
