Charles A. Akemann, Joel Anderson
Given an n-tuple {a_1, ..., a_n} of self-adjoint operators on an infinite dimensional Hilbert space H and a positive integer k, there exists a projection p of rank k such that, for each for j = 1, ..., n, pa_jp is a scalar multiple of p. Assuming that {a_1, ...,a_n, 1} is a linearly independent set in the Calkin algebra, then p can be chosen of infinite rank.
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Let $\mathbb{G}$ be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in $L^\infty(\mathbb{G})$ preserved by the scaling group and idempotent states on the dual quantum group $\widehat{\mathbb{G}}$. As a byproduct we give a simple proof that normal integrable coideals in $L^\infty(\mathbb{G})$ which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of $\mathbb{G}$.
The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions. We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these results is the co-duality for coideals in algebras of functions on locally compact quantum groups and the concept of a normal coideal.
A one to one correspondence between shifts of group-like projections on a locally compact quantum group ${\mathbb{G}}$ which are preserved by the scaling group and contractive idempotent functionals on the dual $\hat{\mathbb{G}}$ is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group $G$ and cosets of open subgroups of $G$. We also establish a one to one correspondence between non-degenerate, integrable, ${\mathbb{G}}$-invariant ternary rings of operators $X\subset L^\infty({\math…
Ring isomorphisms of type II$_\infty$ locally measurable operator algebras
本文证明了II$_\infty$型von Neumann代数的局部可测算子代数间的环同构相似于实$^*$-同构，从而完整描述了一般无有限型I直和项的von Neumann代数对的环同构与格同构。
Self-similar states and projections in noncommutative metric spaces
本文将迭代函数系统与分形理论推广至非交换几何，并探讨局部紧非交换度量空间的相关概念。
We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study properties. Our approach is slightly different from the one introduced in the thesis of S.~Roy and, unlike the investigations of Roy, we work within the von Neumann algebraic picture. This allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups. In particular we show existence of a "braided comultiplication" on the algebra spanned by slices of "braided multiplicative unitary".
We study universal properties of locally compact G-spaces for countable infinite groups G. In particular we consider open invariant subsets of the β-compactification of G (which is a G-space in a natural way), and their minimal closed invariant subspaces. These are locally compact free G-spaces, and the latter are also minimal. We examine the properies of these G-spaces with emphasis on their universal properties. As an example of our resuts, we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non-compact Cantor set.
Gelfand - Naĭmark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theorem states that complex topological $K$-theory coincides with $K$-theory of $C^*$-algebras. This article devoted to the noncommutative generalization of infinite covering projections. Infinite covering projections of spectral triples are also discussed. …
We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mat…
We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$. We also define and study a natural generalization of the Fell topology to arbitrary Banach space representations of a locally compact group. We give several applications in terms of stability of rigidity under perturbations. Among them, we show a Banach-space version of the Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent for $σ$-compact locally compact groups. Another kind of applications…
Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $π:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $π(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal…
Let ${\cal A}$ be a von Neumann algebra and ${\cal P}_{\cal A}$ the manifold of projections in ${\cal A}$. There is a natural linear connection in ${\cal P}_{\cal A}$, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of $\mathbb{C}^n$. In this paper we show that two projections $p,q$ can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ${\cal A}$), if and only if $$ p\wedge q^\perp\sim p^\perp\wedge q, $$ where $\sim$ stands for the Murray-von Neumann equivalence of projections. …
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