Related papers: Eckmann-Hilton arguments in equivariant higher alg…

In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. Evaluating this filtered spectrum on…

Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the…

We introduce a definition of braided tensor product $\operatorname{M}\overline{\boxtimes}\operatorname{N}$ of von Neumann algebras equipped with an action of a quasi-triangular quantum group $\mathbb{G}$ (this includes the case when…

If $G$ is a finite group, the Grothendieck group ${\mathbf{K}}\_G(G)$ of the category of $G$-equivariant ${\mathbb{C}}$-vector bundles on $G$ (for the action of $G$ on itself by conjugation) is endowed with a structure of (commutative)…

A Markov operator $P$ on a probability space $(S,\Sigma,\mu)$, with $\mu$ invariant, is called {\it hyperbounded} if for some $1 \le p<q \le \infty$ it maps (continuously) $L^p$ into $L^q$. We deduce from a recent result of Gl\"uck that a…

We study the relation between certain non-degenerate lower Hessenberg infinite matrices $\mathcal{G}$ and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known…

An equivariant Thom isomorphism theorem in operator K-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a…

We study the "higher algebra" of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal $\infty$-categories and a suitable generalization…

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $\Omega$. Let $E$ be a product system…

We show that the spectral Mackey functors associated to the equivariant algebraic $K$-theory spectra of Guillou-May and Merling (originally constructed using pointset models) can be described purely $\infty$-categorically in terms of the…

We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this…

In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation $R$ imposed on smooth maps of manifolds determines cohomology theories $k^*$ and…

We develop a theory of Goodwillie calculus for functors between $G$-equivariant homotopy theories, where $G$ is a finite group. We construct $J$-excisive approximations of a homotopy functor for any finite $G$-set $J$. These fit together…

Let \( A \subset M \) be an inclusion of von Neumann algebras equipped with a faithful normal semifinite operator valued weight \( E \colon M \to A \). We prove that every positive element \( x \in M \) with \( E(x) < \infty \) satisfies…

We introduce and study the continuous Rokhlin property for actions of compact groups on C*-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence,…

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ of $H$ determines a $G$-equivariant principal bundle ${\mathcal P}$ on $M$ endowed with a $G$-invariant connection. We consider…

Let $k$ be an algebraically closed field. Let $\Lambda$ be a noetherian commutative ring annihilated by an integer invertible in $k$ and let $\ell$ be a prime number different from the characteristic of $k$. We prove that if $X$ is a…

We present a simple extension of the classical Hilton-Eckmann argument classically used to prove that the endomorphism monoid of the unit object in a monoidal category is commutative. It allows us to recover in a uniform way well-known…

Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_\pm \subseteq G$ with unipotent radicals $U_\pm$. The map that sends $x_+x_- \mapsto x_+x_-x_+^{-1}$ for all $x_\pm \in U_\pm$…

It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensor category $\mathcal{C}$ that naturally come with a lift to a braided commutative algebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$. In…