By Luther Pfahler Eisenhart

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Before we proceed, we mention that if a separable C ∗ -algebra has a ﬁnite dual than it is postliminal [9]. From Sect. 4 we know that for any such algebra A, irreducible representations are completely characterized by their kernels so that the structure space A is homeomorphic with the space P rimA of primitive ideals. As we shall see momentarily, the Jacobson topology on P rimA is equivalent to the partial order deﬁned by the inclusion of ideals. This fact in a sense ‘closes the circle’ making any poset, when thought of as the P rimA space of a noncommutative algebra, a truly noncommutative space or, rather, a noncommutative lattice.

The Bratteli diagram associated with the poset for Yn (k) ; the label nk stands Thus, with the notation of Proposition 20, one ﬁnds, K0 K1 K2 K3 K4 K5 .. = {K0 } , = {K0 , K1 } , = {K0 , K1 , K2 } , = {K0 , K1 , K2 , K3 } , = {K0 , K1 , K2 , K3 , K4 } , = {K0 , K1 , K2 , K3 , K4 , K5 } , Y0 (1) = {x1 , x2 , x3 , x4 } , K0 K1 K2 K3 K4 K5 = {K0 } , = {K0 , K1 } , = {K0 , K1 , K2 } , = {K0 , K1 , K2 , K3 , K5 } , = {K0 , K1 , K2 , K3 , K4 , K5 } , = {K0 , K1 , K2 , K3 , K4 , K5 } , F0 (1) = K0 , Y1 (1) = {x1 , x3 , x4 } , Y1 (2) = {x2 } , F1 (1) = K1 , F1 (2) = K0 , Y2 (1) = {x3 } , Y2 (3) = {x1 , x4 } , Y2 (2) = {x2 } , F2 (1) = K2 , F2 (3) = K1 , F2 (2) = K0 , Y3 (1) = {x3 } , Y3 (3) = {x1 } , Y3 (2) = {x2 } , Y3 (4) = {x4 } , F3 (1) = K2 , F3 (3) = K1 , F3 (2) = K0 , F3 (4) = K3 , Y4 (1) = {x3 } , Y4 (3) = {x1 } , Y4 (2) = {x2 } , Y4 (4) = {x4 } , F4 (1) = K2 , F4 (3) = K1 , F4 (2) = K4 , F4 (4) = K3 , Y5 (1) = {x3 } , Y5 (3) = {x1 } , ..

9), the closure V (x) = {x}, of the one point set {x} is given by V (x) =: {y ∈ P | x 2 y} , ∀ x ∈ P . 14) Another equivalent deﬁnition can be given by saying that x y if and only if the constant sequence (x, x, x, · · ·) converges to y. It is worth noticing that in a T0 -space the limit of a sequence need not be unique so that the constant sequence (x, x, x, · · ·) may converge to more than one point. 2 Order and Topology A subset W ⊂ P will be closed if and only if x ∈ W and x Indeed, the closure of W is given by V (x) , V (W ) = 25 y ⇒ y ∈ W.