The notion of (m,n)-pure C*-algebras was introduced by Winter in his seminal work on separable, simple, unital C*-algebras of finite nuclear dimension. Although a lot of effort has been put on understanding (0,0)-pureness (often simply called pureness), much less is known about the apparently weaker notion of (m,n)-pureness for m,n>0. This is especially the case in the non-simple setting.

I will begin the talk by recalling the Toms-Winter conjecture and the importance of pureness in its study. I will then discuss results from two different ongoing projects: Pure *-homomorphisms and their properties (joint with J. Bosa), and (m,n)-pure C*-algebras (joint with R. Antoine, F. Perera, and H. Thiel).

Nonclosed ideals of bounded operators play a prominent role in the theory of singular traces as developed by Dixmier, Connes and many others, and the Calkin correspondence is a powerful tool that can be used to answer many questions about nonclosed ideals in this context. For general C*-algebras, a systematic study of nonclosed ideals was initiated by Pedersen in the late 1960s, but much less is known in this broader setting.

We show that a not necessarily closed ideal in a C*-algebra is semiprime (that is, an intersection of prime ideals) if and only if it is closed under roots of positive elements. Quite unexpectedly, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint. This can be viewed as a generalization of the well-known result that closed ideals in C*-algebras are semiprime and self-adjoint.

This is joint work with Eusebio Gardella and Kan Kitamura.

An important family of Banach algebras are the L^1(G) algebras for a locally compact group. In the 50s, Wendel proved that an isometric isomorphism between two such L^1-algebras implies that the underlying groups are topologically isomorphic. Hence, in some sense, all information on the group can be recovered from L^1(G) and regarding amenability Johnson proved that it is equivalent to a completely Banach algebraic property of L^1(G), which therefore was baptized to be the definition of amenability of Banach algebras.

Concretely, a Banach algebra is amenable if it admits a bounded approximate diagonal. An approximate diagonal of a Banach algebra B is an asymptotically central net in B \otimes B so that the image net in B, when we collapse tensors to multiplication signs, is a right approximate identity for B. 

There are two Banach algebra constructions that generalize the interlude of G and L^1(G): For groupoids with a Haar system, the analogue is the completion of compactly supported functions in the I-norm L^I(\mathcal{G}) and for C*-dymanical systems G \curvedrightarrow A the analogue is the twisted convolution algebra L^1(G,A). In this talk, I will discuss ongoing research with Eusebio Gardella and Eduard Ortega on the generalizations of Johnson’s theorem in those cases.

Ultimately, the question is how amenability for those Banach algebras can be characterized in terms of the underlying groupoid or the C*-dynamical system, respectively.

We develop the theory of general quotients of Cuntz semigroups that includes the quotients by ideals. To this end, we introduce the concept of admissible pair and quotients by admissible pairs. This is similar to the way normal subgroups arise as kernels of group homomorphisms. This study allows us to construct the dynamical Cuntz semigroup as a universal object built from an action of a group $G$.

This is part of a joint work with Joan Bosa, Jianchao Wu, and Joachim Zacharias.

A C*-algebra is residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations. The property of a C*-algebra of being RFD is central in C*-algebra theory and has connections with other important notions and problems. The topic of this talk will be the RFD property in dynamical context, namely we will discuss the RFD property of crossed products by amenable actions and, if time permits,  of C*-algebras of amenable etale groupoids. We will present consequences of our results to residual properties of groups and to approximations of representations in spirit of Exel and Loring, and we will discuss examples. Joint work with Adam Skalski.