A Computation with the Connes–Thom Isomorphism

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Abstract: Let

A i n M_{n} (m a t h b b R)

be an invertible matrix. Consider the semi-direct product

m a t h b b R^{n} t i m e s m a t h b b Z

where

m a t h b b Z

acts on

m a t h b b R^{n}

by matrix multiplication. Consider a strongly continuous action

(a l p h a, a u)

of

m a t h b b R^{n} t i m e s m a t h b b Z

on a

C^{*}

-algebra

B

where

a l p h a

is a strongly continuous action of

m a t h b b R^{n}

and

a u

is an automorphism. The map

a u

induces a map

w i d e t i l d e a u

on

B t i m e s_{a l p h a} m a t h b b R^{n}

. We show that, at the

K

-theory level,

a u

commutes with the Connes-Thom map if

d e t (A) > 0

and anticommutes if

d e t (A) < 0

. As an application, we recompute the

K

-groups of the Cuntz-Li algebra associated to an integer dilation matrix.

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