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We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of B\'edos, Conti and Tuset. As a corollary, we extend to locally compact quantum groups a result of the first-named author, which characterizes amenability of a locally compact group $G$ by nuclearity of the reduced group $C^{*}$-algebra $C_{r}^{*}(G)$ and an additional condition., Comment: 9 pages; a new co-author; added to Theorem 3.2 a second part about the case of a trivial scaling group, and gave an alternative proof of one of the implications; to appear in the Bulletin of the London Mathematical Society

In this short article, we obtained some equivalent formulations of property $T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type, we show that $\mathbb{G}$ has property $T$ if and only if every finite dimensional irreducible $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ is an isolated point in the spectrum of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property $T$ discrete quantum groups using bicrossed products., Comment: 12 pages; accepted for publication in Int. J. Math

For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$, we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of continuous functions on a second countable locally compact group $G$ with compact supports, equipped with the convolution $^*$-algebra structure and a certain inductive topology. We show that $(C_c(G), \varepsilon_G)$ has property $(FH)$ if and only if $G$ has property $(T)$. On the other hand, many Banach algebras equipped with canonical characters have property $(FH)$ (e.g., those defined by a nice locally compact quantum group). Furthermore, through our studies on both property $(FH)$ and character amenablility, we obtain characterizations of property $(T)$, amenability and compactness of $G$ in terms of the vanishing of one-sided cohomology of certain topological algebras, as well as in terms of fixed point properties. These characterizations can be regarded as analogues of one another. Moreover, we show that $G$ is compact if and only if the normed algebra $\big\{f\in C_c(G): \int_G f(t)dt =0\big\}$ (under $\|\cdot\|_{L^1(G)}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set., Comment: 24 pages, accepted for publication in Studia Mathematica

We define type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ as well as C*-semi-finite C*-algebras. It is shown that a von Neumann algebra is a type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ or C*-semi-finite C*-algebra if and only if it is, respectively, a type I, type II, type III or semi-finite von Neumann algebra. Any type I C*-algebra is of type $\mathfrak{A}$ (actually, type $\mathfrak{A}$ coincides with the discreteness as defined by Peligrad and Zsido), and any type II C*-algebra (as defined by Cuntz and Pedersen) is of type $\mathfrak{B}$. Moreover, any type $\mathfrak{C}$ C*-algebra is of type III (in the sense of Cuntz and Pedersen). Furthermore, any purely infinite C*-algebra (in the sense of Kirchberg and Rordam) with real rank zero is of type $\mathfrak{C}$, and any separable purely infinite C*-algebra with stable rank one is also of type $\mathfrak{C}$. We also prove that type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ and C*-semi-finiteness are stable under taking hereditary C*-subalgebras, multiplier algebras and strong Morita equivalence. Furthermore, any C*-algebra $A$ contains a largest type $\mathfrak{A}$ closed ideal $J_\mathfrak{A}$, a largest type $\mathfrak{B}$ closed ideal $J_\mathfrak{B}$, a largest type $\mathfrak{C}$ closed ideal $J_\mathfrak{C}$ as well as a largest C*-semi-finite closed ideal $J_\mathfrak{sf}$. Among them, we have $J_\mathfrak{A} + J_\mathfrak{B}$ being an essential ideal of $J_\mathfrak{sf}$, and $J_\mathfrak{A} + J_\mathfrak{B} + J_\mathfrak{C}$ being an essential ideal of $A$. On the other hand, $A/J_\mathfrak{C}$ is always C*-semi-finite, and if $A$ is C*-semi-finite, then $A/J_\mathfrak{B}$ is of type $\mathfrak{A}$., Comment: 28 pages. The appendix in the original version of this paper was removed, and was then expanded to another paper (in Chinese). The current version of this paper is appeared in the proceedings of the conference: "Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics", took place in Herrnhut, Germany, in honor of Charles Batty for his 60th birthday

Let $��$ be a locally compact Hausdorff space. We show that any local $\mathbb{C}$-linear map (where "local" is a weaker notion than $C_0(��)$-linearity) between Banach $C_0(��)$-modules are "nearly $C_0(��)$-linear" and "nearly bounded". As an application, a local $\mathbb{C}$-linear map $��$ between Hilbert $C_0(��)$-modules is automatically $C_0(��)$-linear. If, in addition, $��$ contains no isolated point, then any $C_0(��)$-linear map between Hilbert $C_0(��)$-modules is automatically bounded. Another application is that if a sequence of maps $\{��_n\}$ between two Banach spaces "preserve $c_0$-sequences" (or "preserve ultra-$c_0$-sequences"), then $��_n$ is bounded for large enough $n$ and they have a common bound. Moreover, we will show that if $��$ is a bijective "biseparating" linear map from a "full" essential Banach $C_0(��)$-module $E$ into a "full" Hilbert $C_0(��)$-module $F$ (where $��$ is another locally compact Hausdorff space), then $��$ is "nearly bounded" (in fact, it is automatically bounded if $��$ or $��$ contains no isolated point) and there exists a homeomorphism $��: ��\rightarrow ��$ such that $��(e\cdot ��) = ��(e)\cdot ��\circ ��$ ($e\in E, ��\in C_0(��)$).

In this article, we give an abstract characterization of the ``identity'' of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $\mathcal{L}(H)$ sending $u$ to ${\rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems'' using an idea arose from the definition of $n_{cb}(V,u)$., Comment: Some remarks were added

By giving an interesting characterisation of amenable multiplicative unitaries in term of one dimensional representations, we show in a simple way that bicrossproducts of amenable locally compact groups is both amenable and coamenable.

In this paper, we will give a natural definition for morphisms between multiplicative unitaries. We will then discuss some equivalences of this definition and some interesting properties of them. Moreover, we will define normal sub-multiplicative unitaries for multiplicative unitaries of discrete type and prove an imprimitivity type theorem for discrete multiplicative unitaries., Comment: 21 pages, Latex; to appear in the Journal of Operator Theory