Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles.

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson.

Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most $12$ . We extend their classification to dimension $13$ and $14$ . As predicted by Donovan’s conjecture, we obtain only finitely many such Morita equivalence classes.

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

In a recent article, Barrett & Halvorson (2016) define a notion of equivalence for first-order theories, which they call “Morita equivalence.” To argue that Morita equivalence is a reasonable measure of “theoretical equivalence,” they make use of the claim that Morita extensions “say no more” than the theories they are extending. The goal of this article is to challenge this central claim by raising objections to their argument for it and by showing why there is good reason to think that the claim itself is false. In light of these criticisms, this article develops a natural way for the advocate of Morita equivalence to respond. I prove that this response makes her criterion a special case of bi-interpretability, an already well-established barometer of theoretical equivalence. I conclude by providing reasons why the advocate of Morita equivalence should opt for a notion of theoretical equivalence that is defined in terms of interpretability rather than Morita extensions.

The Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra. We show that any Peter–Weyl Iwahori algebra is Morita equivalent to the Iwahori–Hecke algebra. Both the Iwahori–Hecke algebra and a Peter–Weyl Iwahori algebra have a natural conjugate linear anti-involution $\star$, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for $\bullet$.

We prove two results on the tube algebras of rigid C*-tensor categories. The first is that the tube algebra of the representation category of a compact quantum group G is a full corner of the Drinfeld double of G. As an application, we obtain some information on the structure of the tube algebras of the Temperley–Lieb categories 𝒯ℒ(d) for d > 2. The second result is that the tube algebras of weakly Morita equivalent C*-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the 2-category defining the Morita context.

We characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order theories. This provides a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos.

We study the C*-algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer–Raeburn ‘stabilization trick’, we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C*-algebras of any saturated upper semi-continuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C*-algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C*-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C*-algebra of a bundle over G in terms of an action, described by Ionescu and Williams, of G on the primitive-ideal space of the C*-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.

We show that every subset of vertices of a directed graph $E$ gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of $E$ can be contracted to a new graph $G$ such that the Leavitt path algebras of $E$ and $G$ are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.

We define an equivalence relation between bimodules over maximal abelian self-adjoint algebras (MASA bimodules), which we call spatial Morita equivalence. We prove that two reflexive MASA bimodules are spatially Morita equivalent if and only if their (essential) bilattices are isomorphic. We also prove that if are bilattices that correspond to reflexive MASA bimodules , and is an onto bilattice homomorphism, then

(i) if is synthetic, then is synthetic;

(ii) if contains a non-zero compact (or a finite or a rank 1) operator, then also contains a non-zero compact (or a finite or a rank 1) operator.

C*-algebras form a 2-category with *-homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby–Smith twisted actions and the equivalence of such actions, covariant representations and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green, and Busby and Smith.

The Packer–Raeburn Stabilization Trick implies that all Busby–Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalize this to actions of strict 2-groupoids.

We analyse Hecke pairs (G,H) and the associated Hecke algebra when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

Vincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.

The Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

Let $C^*$-algebras $A$ and $B$ be Morita equivalent and let $X$ be an $A$–$B$-imprimitivity bimodule. Suppose that $A$ or $B$ is unital. It is shown that $X$ has the weak Banach–Saks property if and only if it has the uniform weak Banach–Saks property. Thus, we conclude that $A$ or $B$ has the weak Banach–Saks property if and only if $X$ does so. Furthermore, when $C^*$-algebras $A$ and $B$ are unital, it is shown that $X$ has the Banach–Saks property if and only if it is finite dimensional.

We develop a notion of Morita equivalence for general C$^{\ast}$-correspondences over C$^{\ast}$-algebras. We show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz--Pimsner algebras. Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C$^{\ast}$-algebras, then the correspondences are Morita equivalent. This generalizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure-preserving transformations. The notion of aperiodicity, which also generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated. 1991 Mathematics Subject Classification: 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.

We give a new proof of the classification of ℵ0-categorical quasivarieties by using Morita equivalence to reduce to term minimal quasivarieties.