We propose a notion of isometric coaction of a compact quantum group on a compact quantum metric space in the framework of Rieffel where the metric structure is given by a Lipnorm. We prove the existence of a quantum isometry group for finite metric quantum spaces, preserving a given state.
局部紧群G在C* 代数A上的余作用d是极大的，如果从A × d G times_{d hat} G到A × K（L^2（G））上的某个自然映射是同构.群作用的全交叉积上的所有对偶余作用都是极大的;离散余作用是极大的当且仅当A是相应的Fell丛的全截面代数。对于G在A上的每一个非退化余作用，存在G在A的扩张上的极大余作用，使得商映射诱导交叉积的同构。
We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coa…
A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittade…
If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products, and each has a coaction of G. We investigate "exotic" coactions in between, that are determined by certain ideals E of the Fourier-Stieltjes algebra B(G) -- an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wid…
Partial actions of groups on C*-algebras and the closely related actions and coactions of Hopf algebras received much attention over the last decades. They arise naturally as restrictions of their global counterparts to non-invariant subalgebras, and the ambient eveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of C*-bialgebras, focussing on C*-quantum, and prove existence of an enveloping global coaction under mild technical assumptions. The construction of the latter provides a left adjoint to the…
This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,α)$ and $(B,β)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_αG,\widehatα)$ and $(B\rtimes_βG,\widehatβ)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respect…
The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreov…
Suppose that a compact quantum group $\clq$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^*$ (co)-action $α$ on $C(M)$, such that the action $α$ is isometric in the sense of \cite{Goswami} for some Riemannian structure on $M$. We prove that $\clq$ must be commutative as a $C^{\ast}$ algebra i.e. $\clq\cong C(G)$ for some compact group $G$ acting smoothly on $M$. In particular, the quantum isometry group of $M$ (in the sense of \cite{Goswami}) coincides with $C(ISO(M))$.
Motivated by the definition of a semigroup compactification of a locally compact group and a large collection of examples, we introduce the notion of an (operator) "homogeneous left dual Banach algebra" (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over $A$ has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^*$ with a compatible (matrix) norm and a type of left Arens product ${\scriptscriptstyle \square}$. Examples include all left Arens product alg…
Existence and examples of quantum isometry group for a class of compact metric spaces
给出了紧致度量空间上紧致量子群（CQG）的等距作用的定义，推广了Banica在有限度量空间上的定义。对于可以等距嵌入到某个欧氏空间中的度量空间$(X,d)$，证明了在等距作用于$(X,d)$上的紧量子群范畴中存在泛对象.事实上，我们的存在性定理适用于更大的一类，即对于任何紧致度量空间$(X,d)$，它允许一个一对一连续映射$f : X \raro \IR^n$对于某些$n$，使得$d_0(f(x),f(y))=ϕ(d(x,y))$（其中$d_0$是欧几里得度量）对于$\IR^+$的某些同胚$ϕ$。作为具体的例子，我们得到了Wang的量子置换群$\cls_n^+$和$\IZ_2$与$\cls_n^+$的自由圈积作为在\cite{huang1}中取区间拓扑并构造的紧致连通度量空间的量子等距群.
The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine…
We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).
