On Bounded Approximate Identities and Existence of Dense Ideals in Real Locally C*- and Locally JB-Algebras

Characterizations of compact and discrete quantum groups through second duals

A locally compact group $G$ is compact if and only if $L^1(G)$ is an ideal in $L^1(G)^{**}$, and the Fourier algebra $A(G)$ of $G$ is an ideal in $A(G)^{**}$ if and only if $G$ is discrete. On the other hand, $G$ is discrete if and only if $C_0(G)$ is an ideal in $C_0(G)^{**}$. We show that these assertions are special cases of results on locally compact quantum groups in the sense of J. Kustermans and S. Vaes. In particular, a von Neumann algebraic quantum group $(M,Γ)$ is compact if and only if $M_*$ is an ideal in $M^*$, and a (reduced) $C^*$-algebraic quantum group $(A,Γ)$ is discrete if a…

$L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups

Let $G$ be a locally compact group and $1\leq p<\infty$. A continuous unitary representation $π\!: G\to B(\mathcal{H})$ of $G$ is an $L^p$-representation if the matrix coefficient functions $s\mapsto \langle π(s)x,x\rangle$ lie in $L^p(G)$ for sufficiently many $x\in \mathcal{H}$. Brannan and Ruan defined the $L^p$-Fourier algebra $A_{L^p}(G)$ to be the set of matrix coefficient functions of $L^p$-representations. Similarly, the $L^p$-Fourier-Stieltjes algebra $B_{L^p}(G)$ is defined to be the weak*-closure of $A_{L^p}(G)$ in the Fourier-Stieltjes algebra $B(G)$. These are always ideals in the…

通过阶梯技术在不变理想集合之间建立双射

Bijections Between Sets of Invariant Ideals, Via the Ladder Technique

本文提出阶梯技术，在任意局部紧群下建立C*-代数作用与余作用不变理想之间的格同构双射，将可顺从群情形推广到一般群。

Algebraic K-theory and locally convex algebras

We consider functors from the category of locally convex algebras to abelian groups and prove invariance under smooth homotopies for weakly J-stable algebras, where J is a harmonic operator ideal. This applies in particular to negative algebraic K-theory. We develop the technology of Kasparov modules in an algebraic setting in order to obtain the results. The article also includes a computation of the first algebraic K-theory of harmonic operator ideals.

Some Automatic Continuity Theorems for Operator Algebras and Centralizers of Pedersen's Ideal

We prove automatic continuity theorems for "decomposable" or "local" linear transformations between certain natural subspaces of operator algebras. The transformations involved are not algebra homomorphisms but often are module homomorphisms. We show that all left (respectively quasi-) centralizers of the Pedersen ideal of a C*-algebra A are locally bounded if and only if A has no infinite dimensional elementary direct summand. It has previously been shown by Lazar and Taylor and Phillips that double centralizers of Pedersen's ideal are always locally bounded.

The ideal structure of algebraic partial crossed products

Given a partial action of a discrete group $G$ on a Hausdorff, locally compact, totally disconnected topological space $X$, we consider the correponding partial action of $G$ on the algebra $L_c(X)$ consisting of all locally constant, compactly supported functions on $X$, taking values in a given field $K$. We then study the ideal structure of the algebraic partial crossed product $L_c(X)\rtimes G$. After developping a theory of induced ideals, we show that every ideal in $L_c(X)\rtimes G$ may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic ver…

结合代数上线性算子的代数恒等式的一种新分类

A new classification of algebraic identities for linear operators on associative algebras

本研究结合代数operad理论与计算交换代数方法，对结合代数上次数为2、重数为1或2的线性算子代数恒等式进行分类，得到多类已知及新的恒等式，且方法可推广至更一般情形。

On Ideals of $L^1$-algebras of Compact Quantum Groups

We develop a notion of a non-commutative hull for a left ideal of the $L^1$-algebra of a compact quantum group $\mathbb{G}$. A notion of non-commutative spectral synthesis for compact quantum groups is proposed as well. It is shown that a certain Ditkin's property at infinity (which includes those $\mathbb{G}$ where the dual quantum group $\widehat{\mathbb{G}}$ has the approximation property) is equivalent to every hull having synthesis. We use this work to extend recent work of White that characterizes the weak$^*$ closed ideals of a measure algebra of a compact group to those of the measure …

The primitive ideals of some étale groupoid C*-algebras

Consider the Deaconu-Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C*-algebra. For a special class of actions we describe the Jacobson topology.

结合代数的重数3算子恒等式

Operator identities of multiplicity 3 for associative algebras

本研究针对结合代数上次数2、重数3的线性算子代数恒等式，沿用此前的分类方法，通过operad部分合成生成推论、计算部分Smith标准形及Gröbner基分析，发现秩的次最大值为16和19，分别对应6个和8个新恒等式。

关于étale群胚的C*-代数理想结构的一些结果

Some results regarding the ideal structure of C*-algebras of étale groupoids

本文研究了内正合局部紧Hausdorff étale群胚的约化C*-代数理想结构，证明了夹层引理并建立了理想与嵌套开不变集三元组的双射，同时推广了Ara和Lolk的相对强拓扑自由性条件，证明了障碍理想的存在性。