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Symmetries of the CAR algebra. (English) Zbl 0718.46024
This paper gives (for the first time) a negative answer to the question whether the fixed-point algebra of a *-automorphism of period 2 of the CAR algebra is an AF algebra. The construction of the counter-example is done by appropriately modifying the wound around embedding that is used to define Bunce-Deddens algebras.
In this way the 0-dimensional CAR algebra is obtained as a direct $$C^*$$-limit of 1-dimensional $$C^*$$-algebras in fact of the form $$C(S^ 1,M(r,{\mathbb{C}})))$$ while the automorphism is designed to make the $$K_ 1$$-group of the fixed-point algebra non-trivial.
The same construction gives also more insight into the structure of the CAR algebra, e.g. existence of (uncountably many non-conjugate) diagonal maximal commutative $$C^*$$-subalgebras with totally disconnected spectrum.

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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