A note on positivity of elementary operators
نویسنده
چکیده
We show that operators on n×n matrices which are representable in the form T (X) = ∑` i=1 aiXbi (for ai and bi n×n matrices) and are k-positive for k = [ √ `] must be completely positive. As a consequence, elementary operators on a C*-algebra with minimal length ` which are k-positive for k = [ √ `] must be completely positive. For A a C*-algebra, an operator T :A→ A is called an elementary operator if T can be expressed in the form Tx = ∑` i=1 aixbi with ai and bi (1 ≤ i ≤ `) in the multiplier algebra M(A) of A. (We will mainly be concerned with the case where A is the algebraMn(C) of n×nmatrices and thenM(A) = A. IndeedM(A) = A ifA is unital.) Such representations of T may not be unique. The smallest ` in such representations of T is called the minimal length of T . If A is a prime C*-algebra (Mn(C) is prime) and the collections {ai : 1 ≤ i ≤ `} and {bi : 1 ≤ i ≤ `} are each linearly independent, then ` is the minimal length of T [4]. If A is prime and such an elementary operator T sends the set Ah = {x ∈ A : x = x∗} of hermitian elements of A into itself, then T is representable as
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