tailieunhanh - Báo cáo toán học: "Nonnuclear subalgebras of C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nonnuclear subalgebras của C *- đại số. | C Copyright by INCREST Ỉ985 X OPERATOR THEORY 14 1935 347-350 NONNUCLEAR SUBALGEBRAS OF C -ALGEBRAS BRUCE BLACKADAR Ĩ. INTRODUCTION There has been much work in the last decade or so attempting to shed light on the internal structure of non-type-I c -algebras. Much of this work has concerned the structure of nuclear c -algebras. Nuclear c -algebras are characterized by a large number of equivalent structure properties which are of great technical use in applications and which make the algebras relatively tractable objects forrstudy at the same time the class of nuclear c -algebras appears to be broad enough to include almost all c -algebras which arise naturally . See 4 for a survey of the structure of nuclear c -algebras. The class of nuclear c -algebras is closed under many standard operations such as quotients extensions inductive limits tensor products and crossed products by amenable groups. It was long an open question however whether a c -subalgebra of a nuclear C S-algebra is necessarily nuclear. Some experts believed this to be the case until Choi 1 produced a counterexample the nonnuclear Choi algebra c can be embedded in the nuclear Cuntz algebra O2. The main result of this note is Theorem 1. Let A be a c -algebra which is not type 1. Then A contains a nonnuclear C - -subalgebra. Of course a type I c -algebra cannot contain a nonnuclear subalgebra since any subalgebra of a type I c -algebra is type I. Theorem 1 is a consequence of Theorem 2 which is of independent interest. Theorem 2. Let A be a non-type-l C algebra. Then A contains a -subalgebra which has 02 as a quotient. Proof of Theorem 1 from Theorem 2. Let B be a subalgebra of A and 7Ĩ jB - - ỡ2 a surjective homomorphism. Regard the Choi algebra c as a subalgebra of 02. Then 7t-1 C is a nonnuclear c -subalgebra of A. 348 BRLCE BLACKADAR To prove Theorem 2. it suffices to find an example of such a subalgebra for A a UHF algebra since every non-type-I c -algebra has a c - ubalgebra with a UHF quotient