We classify the gauge-invariant ideals in the C * -algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C * -algebras of arbitrary infinite graphs.

We construct the C * -algebra C(L q (p; m 1 , . . . , m n )) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of Z p on the algebra C(S 2n−1 q ), corresponding to the quantum odd dimensional sphere. We show that C(L q (p; m 1 , . . . , m n )) is isomorphic to the graph algebra C * L (p;m1,...,mn) 2n−1. This allows us to determine the ideal structure and, at least in principle, calculate the Kgroups of C(L q (p; m 1 , . . . , m n )). Passing to the limit with natural imbeddings of the quantum lens spaces we construct the quantum infinite lens space, or the quantum EilenbergMacLane space of type (Z p , 1). Introduction.Classical lens spaces L(p; m 1 , . . . , m n ) are defined as the orbit spaces of suitable free actions of finite cyclic groups on odd dimensional spheres (e.g., see [13]). In the present article, we define and investigate their quantum analogues. The C * -algebras of continuous functions on the quantum lens spaces were introduced earlier by Matsumoto and Tomiyama in [18], but our construction leads to different (in general) algebras. (The very special case of the quantum 3-dimensional real projective space was investigated by Podleś [20] and Lance [17], in the context of the quantum SO(3) group.) The starting point for us is the C * -algebra C(S 2n−1 q ), q ∈ (0, 1), of continuous functions on the quantum odd dimensional sphere. If n = 2 then C(S 3 q ) is nothing but C(SU q (2)) of Woronowicz [27]. The construction in higher dimensions is due to Vaksman and Soibelman [26], and from a somewhat different perspective to Nagy [19]. (See also the closely related construction of representations of the twisted canonical commutation relations due to Pusz and Woronowicz [21].) We define the C * -algebra C(L q (p; m 1 , . . . , m n )) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of the finite cyclic group Z p on C(S 2n−1 q ). This definition depends on the deformation parameter q ∈ (0, 1), as well as on positive integers p ≥ 2 and m 1 , . . . , m n . We normally assume that each of m 1 , . . . , m n is relatively prime to p. On the classical level, this guarantees freeness of the action. In the special case p = 2, m 1 = · · · = m n = 1 we recover 249

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras O n which preserve both the diagonal and the core U HF -subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of Aut(O n ) and leads to numerous new examples. In particular, we completely classify all such automorphisms of O 2 for the permutation unitaries in ⊗ 4 M 2 . We show that the subgroup of Out(O 2 ) generated by these automorphisms contains a copy of the infinite dihedral group Z ⋊ Z 2 .

Abstract. We show that the Cuntz-Krieger algebras of infinite graphs and infinite {0, 1}-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their K-theory. Since the finite approximating graphs have sinks, we have to calculate the K-theory of CuntzKrieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.The Cuntz-Krieger algebras O A were introduced by Cuntz and Krieger in 1980, and have been prominent in operator algebras ever since. At first the algebras O A were associated to a finite matrix A with entries in {0, 1}, but it was quickly realised that they could also be viewed as the C * -algebras of a finite directed graph [33]. Over the past few years, originally motivated by their appearance in the duality theory of compact groups [22], authors have considered analogues of the Cuntz-Krieger algebras for infinite graphs and matrices (see [21] the graph algebra C * (E) is the universal C * -algebra generated by a Cuntz-Krieger E-family {s e , p v }. The equations (0.1) make sense as they stand for row-finite graphs, in which the index set {e ∈ E 1 : s(e) = v} for the sum is always finite. If a vertex v emits infinitely many edges, the sum does not make sense in a C * -algebra, because infinite sums of projections cannot converge in norm. However, it was observed in [13] that the general theory of Cuntz-Krieger algebras carries over to arbitrary countable graphs if one simply removes the relations involving infinite sums from (0.1), and requires instead that the range projections S e S * e are mutually orthogonal and dominated by P s(e) . Exel and Laca have described a different generalisation of the Cuntz-Krieger algebras for infinite matrices A [10]. Their defining relations are complicated: loosely speaking, one has to include a Cuntz-Krieger relation whenever a row-operation