In this paper, we investigate the structure of the magic square C*algebra A(4) of size 4. We show that a certain twisted crossed product of A( 4) is isomorphic to the homogeneous C*-algebra M 4 (C(RP 3 )). Using this result, we show that A(4) is isomorphic to the fixed point algebra of M 4 (C(RP 3 )) by a certain action. From this concrete realization of A(4), we compute the K-groups of A(4) and their generators. 2020 Mathematics Subject Classification. Primary 46L05; Secondary 46L55, 46L80. Key words and phrases. C*-algebra, magic square C*-algebra, twisted crossed product, Ktheory.1. Definitions of and basic facts on magic square C*-algebras 2. General results on magic square C*-algebras Proposition 2.1. For n = 1, 2, 3, A(n) is commutative. Hence the surjection A(n) ։ A ab (n) is an isomorphism for n = 1, 2, 3.Proof. For n = 1 and n = 2, it is easy to see A(1) ∼ = C and A(2) ∼ = C 2 . To show that A(3) is commutative, it suffices to show p 1,1 commutes with p 2,2 . In fact if p 1,1 commutes with p 2,2 , we can see that p 1,1 commutes with p 2,3 , p 3,2 and p 3,3 using the action α defined in Definition 1.3. Then p 1,1 commutes with every generators because p 1,1 is orthogonal to and hence commutes with p 1,2 , p 1,3 , p 2,1 and p 3,1 . Using the action α again, we see that every generators commutes with every generators. Now we are going to show that p 1,1 commutes with p 2,2 . We have= p 2,2 − (1 − p 2,3 − p 3,3 )p 2,2 = p 3,3 p 2,2 .By symmetry, we have p 2,2 p 3,
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