The model theory of metric structures ([3]) was successfully applied to analyze ultrapowers of C*-algebras in [13] and [12]. Since important classes of separable C*-algebras, such as UHF, AF, or nuclear algebras, are not elementary (i.e., not characterized by their theory-see [12, §6.1]), for a moment it seemed that model theoretic methods do not apply to these classes of C*-algebras. We prove results suggesting that this is not the case.Many of the prominent problems in the modern theory of C*-algebras are concerned with the extent of the class of nuclear C*-algebras. We have the bootstrap class problem (see [5, IV3.1.16]), the question of whether all nuclear C*-algebras satisfy the Universal Coefficient Theorem, UCT, (see [21, §2.4]), and the Toms-Winter conjecture (to the effect that the three regularity properties of nuclear C*-algebras discussed in [9] are equivalent; see [23]). If one could characterize classes of algebras in question-such as nuclear algebras, algebras with finite nuclear dimension, or algebras with finite decomposition rank-as algebras that omit certain sets of types (see §1) then one might use the omitting types theorem ([3, §12]) to construct such algebras, modulo resolving a number of nontrivial technical obstacles. This paper is the first, albeit modest, step in this project.Recall that a unital C*-algebra is UHF (Uniformly HyperFinite) if it is a tensor product of full matrix algebras, M n (C). Non-unital UHF algebras are direct limits of full matrix algebras, and in the separable, unital case the two definitions are equivalent. A C*-algebra is AF (Approximately Finite) if it is a direct limit of finite-dimensional C*-algebras. These three classes of C*-algebras were the first to be classified -by work of Glimm, Dixmier and Elliott (building on Bratteli's results), respectively. Elliott's classification of separable AF algebras by the ordered K 0 group was a prototype for Elliott's program for classification of nuclear, simple, separable, unital C*-algebras by their K-theoretic invariants (see [21] or [9]). Types are defined in §1.Theorem 1. There are countably many types such that a separable C*algebra A is UHF if and only if it omits all of these types.Theorem 2. There are countably many types such that a separable C*algebra A is AF if and only if it omits all of these types.

We present a classification of all spherical indecomposable representations of classical and exceptional Lie superalgebras. We also include information about stabilizers, symmetric algebras, and Borels for which sphericity is achieved. In one such computation, the symmetric algebra of the standard module of osp(m|2n) is computed, which in particular gives the representation-theoretic structure of polynomials on the complex supersphere.Lemma 3.10. If g is basic, then an irreducible representation V is spherical if and only if V * is. If (V, g) is spherical, then (V * , g) is equivalent to it. 9

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We give a definition of the notion of spherical varieties in the world of complex supervarieties with actions of algebraic supergroups. A characterization of affine spherical supervarieties is given which generalizes a characterization in the classical case. We also explain some general properties of the monoid of highest weights. Several examples are discussed that are interesting in their own right and highlight differences with the classical case, including the regular representation, symmetric supervarieties, and actions of graded supergroups.Résumé. -Nous introduisons les supervariétés sphériques, une généralisation des variétés sphériques. Nous prouvons une caractérisation des supervariétés sphériques affines qui généralise une caractérisation classique des variétés sphériques affines. De plus, nous montrons quelques propriétés du monoïde des plus grands poids. Nous discutons plusieurs exemples intéressants qui montrent des différences avec le cas classique parmi lesquels la représentation régulière, les supervariétés symétriques, et les actions de super groupes gradués.

We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $({\mathfrak{g}},{\mathfrak{k}})$ and $({\mathfrak{g}},{\mathfrak{k}}^{\prime})$ admit Iwasawa decompositions. For such pairs, we define a ghost algebra, generalizing the subalgebra of ${\mathcal{U}}{\mathfrak{g}}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the “special” ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $({\mathfrak{g}},{\mathfrak{k}})$ is interlaced.