Imitation product-type actions of UHF algebras
Let A be an algebra given as an infinite tensor product of matrix algebras, say \(A=\otimes_ kM_{n(k)}C\), and let g be an automorphism of A of the form \(g=\otimes Ad u_ k\), where each \(u_ k\) belongs to \(M_{n(k)}C\) and \(u^ 2_ k=1\), \(u_ k=u^*_ k\). Then the fixed point algebra \(A^ g=\{a\in A|\) \(a^ g=a\}\) has at most 2 rationally independent equivalence classes of idempotents (compactly expressed, the \({\mathbb{Z}}\)-rank of \(K_ 0(A^ g)\) is at most 2). In 1977, George Elliott asked me whether the converse holds. (Quoted from the paper.) Specifically, if g is an order 2 \((^*\)-) automorphism of A such that rank \(K_ 0(A^ g)\leq 2\), is g of the form \(\otimes Ad u_ k\) (which may correspond to a different infinite factorization of A), that is, is it of product type ? The problem is solved with counterexamples, and with two positive results, the second of which is: Corollary. Let A be a UHF algebra, and g an order two automorphism that is locally finite dimensional. Suppose the \({\mathbb{Z}}\)-rank of \(K_ 0(A^ g)\) (or of \(K_ 0(A\times <g>))\) is 1 or 2. (a) If A unitally contains a UHF algebra of type \(2^{\infty}q^{\infty}\) (for some odd prime q), then g is conjugate to a product type action. (b) If A contains the UHF algebra \(\otimes M_ pC\) (p varying over all primes), then g is conjugate to a product-type action.
infinite tensor product of matrix algebras
scientific article; zbMATH DE number 3981835