Abstract
In this letter, we prove that the pure state space on the n times n complex Toeplitz matrices converges in the Gromov–Hausdorff sense to the state space on C(S^1) as n grows to infinity, if we equip these sets with the metrics defined by the Connes distance formula for their respective natural Dirac operators. A direct consequence of this fact is that the set of measures on S^1 with density functions c prod _{j=1}^n (1-cos (t-theta _j)) is dense in the set of all positive Borel measures on S^1 in the weak^* topology.
Highlights
In the framework of noncommutative geometry [3], we study the pure states on the Toeplitz matrices as a metric space
With n × n Toeplitz matrices, we mean those matrices in Mn(C) for which each descending diagonal is constant, i.e., the matrices T for which Ti, j = Ti+1, j+1
The reader need only know that these matrices inherit a structure of positivity from Mn(C) which allows us to talk about positive linear functionals
Summary
In the framework of noncommutative geometry [3], we study the pure states on the Toeplitz matrices as a metric space. With n × n Toeplitz matrices, we mean those matrices in Mn(C) for which each descending diagonal is constant, i.e., the matrices T for which Ti, j = Ti+1, j+1 We denote this set by C(S1)(n), as these Toeplitz matrices can be seen as a truncation of C(S1) which we will explain later on. States on the Toeplitz system can be defined as those positive linear functionals φ for which φ(I ) = 1, and pure states as the extreme points in the state space. We denote these spaces as S(C(S1)(n)) and P(C(S1)(n)), respectively. The states on C(S1)(n) can be equipped with a metric that
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