Symmetric states of infinite tensor products of C∗-algebras

Abstract

Infinite tensor products of C*-algebras, and even more specially of the complex 2 x 2 matrices, have been of great importance in operator theory. For example, the perhaps most fruitful technique for constructing different types of factors, has been to take weak closures of infinite tensor products in different representations. In addition, some of the C*-algebras of main interest, those of the commutation and the anticommutation relations, are closely related to infinite tensor products of C*-algebras. Since we can also apply the theory, when all factors in the infinite tensor product are abelian, to measure theory on product spaces, we see that the theory of infinite tensor products of C*-algebra may have great potential importance. In the present paper we shall study the infinite tensor product VI * of a C*-algebra b with itself, viz. 2l = @ 23{ , where b, = 23, i = 1, 2,..., and then show how information on the C*-algebra ‘$l leads to both new and known results on the different subjects mentioned in the preceding paragraph. If G denotes the group of one-toone mappings of the positive integers onto themselves leaving all but a finite number of integers fixed, then G has a canonical representation as *-automorphisms of 2X, namely as those which permute the factors in the tensor product. Then ‘8 is asymptotically abelian with respect to G in the sense of [Id], h ence the theory of such C*-algebras, as developed in [Id] and [27], is applicable to ‘3. Our main concern will be with the G-invariant states of 2l. Such states are called sym-