Tracial states on crossed products associated with Furstenberg transformations on the 2-torus

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Title

Authors ¹

Department of Mathematics, College of Science, Ryukyu University, Nishihara-Cho, Okinawa, 903-01 Japan

Let

ϕ_{f}

be a Furstenberg transformation on the 2-torus

^2

defined by

ϕ_{f} (x, y) = (e^{2 π i θ} x, e^{2 π if (x)} x y) f or a n y x, y \in, w h e r e θ i s a n i r r a t i o n a l ν m b e r and f i s a r e a l - v a l u e d c o n t \in u o u s f u n c t i o n o n t h e 1 - \to r u s . L e t

A(ϕ_{f})

b e t h e c r o s s e d \prod u c t a s s o c i a t e d w i t h

ϕ_{f}

. W e s h o w t \hat{}

A(ϕ_{f})!$! has a unique tracial state for any irrational number θ and any real-valued continuous function f on .

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