Symmetry in the Cuntz Algebra on Two Generators

We investigate the structure of the automorphism of O2 which exchanges the two canonical isometries. Our main observation is that the fixed point C*-subalgebra for this action is isomorphic to O2 and we detail the relationship between the crossed-product and fixed point subalgebra. This paper studies the structure of the fixed point C*-algebra of the action of Z2 which switches the canonical generators of the Cuntz algebra O2. We show that both the C*-crossed-product and the fixed point C*-algebra for this action are *-isomorphic to O2. This action is an example of an action of a finite group on a noncommutative C*-algebra, and in general the structures associated to such actions can be quite difficult to describe. To any action α of a finite group G on a unital C*-algebra A, one can associate two new related C*-algebras: the fixed point C*-algebra A1 and the C*-crossed-product A oα G [6]. By construction, A1 is a C*-subalgebra of A while A is in fact the fixed point C*-subalgebra of A oα G for the dual action of the Pontryagin dual of G — so A is itself a subalgebra of A oα G. In [5], Rieffel shows that A1 and A oα G are in fact Morita equivalent. In general, however, understanding the structure of A1 or A oα G can be quite complex, as demonstrated for instance in [2]. In this paper, when A is chosen to be O2 and the group is Z2, for the natural action swapping the generators of O2, we obtain a complete picture of the relative positions of these three C*-algebras, which we prove are all *-isomorphic to O2. We shall say that two isometries S1 and S2 on some Hilbert space satisfy the Cuntz relation when: (0.1) S1S 1 + S2S ∗ 2 = 1. By [3][4, Theorem V.4.6 p. 147], the Cuntz relation defines, up to *-isomorphism, a unique simple C*-algebra denoted by O2. Moreover, by universality, there is a unique *-automorphism σ of O2 which satisfies: σ(S1) = S2 and σ(S2) = S1. Since σ is the identity, we can define the C*-crossed-product O2 oσ Z2 as the universal C*-algebra generated by two isometries S1 and S2 and a unitary w such that w = 1 and wS1 = S2w [6]. We also can define the fixed point C*-subalgebra [O2]1 of O2 as {a ∈ O2 : σ(a) = a}. In the first section of this paper, we show that Date: February 22nd, 2010. 1991 Mathematics Subject Classification. 46L55; 46L80.