Prime tensor ideals in some triangulated categories of C*-algebras

iii Abstract The purpose of this thesis is to inaugurate the application of algebraico-geometric ideas, in the form of Paul Balmer’s tensor triangular geometry [Bal05], to the study of such tensor triangulated categories as arise in relation to the topological K-theory of operator algebras. More precisely, we are interested in Balmer’s spectrum Spc of the equivariant Kasparov category KK associated to a (second countable Hausdorff) locally compact group G. Using work of Ralf Meyer and Ryszard Nest [MN06], we show (Theorem 4.3.7) that if the space Spc(KK) is covered by the union ⋃ H Spc(KK ), with H varying among the compact subgroups of G, then a strong form of the Baum-Connes conjecture (which is known to hold for all groups with the Haagerup property) holds for G. As a first step towards understanding the spectrum of these tensor triangulated categories, we turn our attention to T G := 〈11〉loc, the localizing subcategory of KK generated by the tensor unit 1 ∈ T . If G = {1} is the trivial group, then T G is better known as the Bootstrap category Boot ⊆ KK. We prove (Theorem 5.1.11) that the spectrum of its subcategory Bootc of compact objects (i.e., separable C∗algebras in the Bootstrap class having finitely generated K-theory) is isomorphic to the Zariski spectrum of the integers: Spc(Bootc) ∼= Spec(Z). We conjecture the analog result at least for finite groups, namely, that there is a homeomorphism Spc(T G c ) ∼= Spec(R(G)), where R(G) = EndT G c (11) is the complex representation ring of the finite group G. In order to tackle this conjecture we first provide, speaking quite roughly, an abstract criterion (Theorem 2.5.2) for a ‘continuous generalized support datum’ (X,σ) of a certain nice form defined on a compactly generated tensor triangulated category T , to be classifying for the subcategory Tc of compact objects, i.e. – equivalently – to provide a homeomorphism Spc(Tc) ∼= X. In Section 5.4 we construct such a support (Spec(R(G)), σ) for T G when G is finite, but we fail – for the moment – to show that it enjoys all the properties required by our criterion. We can use it though to show that the natural map ρ : Spec(T G c )→ Spec(R(G)) is a retraction. Along the way to constructing σ we define, for G a compact Lie group and p ∈ Spec(R(G)), a ‘p-local’ subcategory T G p ⊆ T , and we show that it enjoys several nice properties (see Theorem 5.2.1). Moreover, we use relative homological algebra [MN07] [Mey08] to prove (a new version of) N. Christopher Phillips’s [Phi85] Künneth Theorem for certain important localized K-theory groups (see Theorem 5.3.11). The purpose of this thesis is to inaugurate the application of algebraico-geometric ideas, in the form of Paul Balmer’s tensor triangular geometry [Bal05], to the study of such tensor triangulated categories as arise in relation to the topological K-theory of operator algebras. More precisely, we are interested in Balmer’s spectrum Spc of the equivariant Kasparov category KK associated to a (second countable Hausdorff) locally compact group G. Using work of Ralf Meyer and Ryszard Nest [MN06], we show (Theorem 4.3.7) that if the space Spc(KK) is covered by the union ⋃ H Spc(KK ), with H varying among the compact subgroups of G, then a strong form of the Baum-Connes conjecture (which is known to hold for all groups with the Haagerup property) holds for G. As a first step towards understanding the spectrum of these tensor triangulated categories, we turn our attention to T G := 〈11〉loc, the localizing subcategory of KK generated by the tensor unit 1 ∈ T . If G = {1} is the trivial group, then T G is better known as the Bootstrap category Boot ⊆ KK. We prove (Theorem 5.1.11) that the spectrum of its subcategory Bootc of compact objects (i.e., separable C∗algebras in the Bootstrap class having finitely generated K-theory) is isomorphic to the Zariski spectrum of the integers: Spc(Bootc) ∼= Spec(Z). We conjecture the analog result at least for finite groups, namely, that there is a homeomorphism Spc(T G c ) ∼= Spec(R(G)), where R(G) = EndT G c (11) is the complex representation ring of the finite group G. In order to tackle this conjecture we first provide, speaking quite roughly, an abstract criterion (Theorem 2.5.2) for a ‘continuous generalized support datum’ (X,σ) of a certain nice form defined on a compactly generated tensor triangulated category T , to be classifying for the subcategory Tc of compact objects, i.e. – equivalently – to provide a homeomorphism Spc(Tc) ∼= X. In Section 5.4 we construct such a support (Spec(R(G)), σ) for T G when G is finite, but we fail – for the moment – to show that it enjoys all the properties required by our criterion. We can use it though to show that the natural map ρ : Spec(T G c )→ Spec(R(G)) is a retraction. Along the way to constructing σ we define, for G a compact Lie group and p ∈ Spec(R(G)), a ‘p-local’ subcategory T G p ⊆ T , and we show that it enjoys several nice properties (see Theorem 5.2.1). Moreover, we use relative homological algebra [MN07] [Mey08] to prove (a new version of) N. Christopher Phillips’s [Phi85] Künneth Theorem for certain important localized K-theory groups (see Theorem 5.3.11).