Tracial Positive Linear Maps of C ∗ -Algebras

Abstract

A positive linear map 0: 21 -• 33 between two C*-algebras is said to be tracial if (%) is analyzed as the composition of a tracial positive linear map 21 — C(X) followed by a positive linear map C(X) — ®(0C). Tracial positive linear maps are the natural generalizations of tracial states on C*-algebras. We invite special attention to the natural occurrence of tracial positive linear maps in the study of finite von Neumann algebras, Toeplitz operators, as well as others (see Examples 1-5 in the context). In consideration of the general global structure, we are concerned with two familiar classes of tracial positive linear maps: The first is the class of tracial positive linear maps from a C*-algebra 31 into a commutative C*-algebra C(X)—actually, each such map can be described as a continuous (with respect to the compact Hausdorff space A) family of finite traces on 21. The second class consists of positive linear maps from a commutative C*-algebra C(X) into (%). The main theorem asserts that the compositions of these two classes exhaust all; namely, each tracial positive linear map 21 -» %(%) admits a factorization 21 -> C(X) -» (%) through a commutative C*-algebra C(X). Therefore, every tracial positive linear map is completely positive, and consequently, each contractive tracial positive linear map 4>: 21 -> 33 satisfies the Schwarz inequality $( A*A) > $(A*)$(A). This answers a question raised in (4). Throughout this paper, general C*-algebras are written in the German type 21, 33. We denote by %(%) (resp. %(%)) for the C*-algebra of all bounded operators (resp. all compact operators) on a Hubert space 3C. A linear map $: 21 -» 33 is said to be tracial if $(AXA2) = : 2I -» 33 is said to be positive if &(A) is positive for every positive A E 21. For each operator A, we write C*(A) for the C*-algebra generated by A. We begin with several examples to illustrate the natural occurrence of tracial linear maps in structure theory. Example 1. 7/21 is a unital C*-algebra with a unique tracial state r (in particular, if 21 is a finite factor), then every tracial positive linear map : 21 -* %(%) is of the form _