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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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