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We will give an abstract characterization of an arbitrary self-adjoint weak$^*$-closed subspace of $\mathcal{L}(H)$ (equipped with the induced matrix norm, the induced matrix cone and the induced weak$^*$-topology). In order to do this, we obtain a matrix analogues of a result of Bonsall for $^*$-operator spaces equipped with closed matrix cones. On our way, we observe that for a $^*$-vector $X$ equipped with a matrix cone (in particular, when $X$ is an operator system or the dual space of an operator system), a linear map $\phi:X\to M_n$ is completely positive if and only if linear functional $[x_{i,j}]_{i,j}\mapsto \sum_{i,j=1}^n \phi(x_{i,j})_{i,j}$ on $M_n(X)$ is positive., Comment: It is a pre-refereed version of a paper that will appear in Lin. Alg. Appl. The proof of Lemma 5 are removed in the published version. Some equation numbers and some statement numbers are also altered in the published version

Motivated by quantum states with zero transition probability, we introduce the notion of ortho-set which is a set equipped with a relation $\neq_\mathrm{q}$ satisfying: $x\neq_\mathrm{q} y$ implies both $x\neq y$ and $y \neq_\mathrm{q} x$. For an ortho-set, a canonical complete ortholattice is constructed. Conversely, every complete ortholattice comes from an ortho-set in this way. Hence, the theory of ortho-sets captures almost everything about quantum logics. For a quantum system modeled by the self-adjoint part $B_\mathrm{sa}$ of a $C^*$-algebra $B$, we also introduce a "semi-classical object" called the Gelfand spectrum. It is the ortho-set, $P(B)$, of pure states of $B$ equipped with an "ortho-topology", which is a collection of subsets of $P(B)$, defined via a hull-kernel construction with respects to closed left ideals of $B$. We establish a generalization of the Gelfand theorem by showing that a bijection between the Gelfand spectra of two quantum systems that preserves the respective ortho-topologies is induced by a Jordan isomorphism between the self-adjoint parts of the underlying $C^*$-algebras (i.e. an isomorphism of the quantum systems), when the underlying $C^*$-algebras satisfy a mild condition., Comment: It is the preprint (before peer review) version of a paper to appear in J. Phys. A. The names for some terminologies are changed in the publised version; in particular, "quantum sets" are now called "ortho-sets" to avoid confusion with existing terminologies

This article is to give an infinite dimensional analogue of a result of Choi and Effros. We say that an (not necessarily unital) operator system $T$ is \emph{dualizable} if one can find an equivalent dual matrix norm on the dual space $T^*$ such that under this dual matrix norm and the canonical dual matrix cone, $T^*$ becomes a dual operator system. We show that "a complete" operator system $T$ is dualizable if and only if $M_\infty(T)^\mathrm{sa}$ satisfies a bounded decomposition property. In this case, $$\|f\|^\mathrm{d}:= \sup \big\{\big\|[f_{i,j}(x_{k,l})]\big\|: x\in M_n(T)^+; \|x\|\leq 1; n\in \mathbb{N}\big\},$$ is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on $T^*$ that turns it into a dual operator system, denoted by $T^\mathrm{d}$. $T^\mathrm{d}$ is again dualizable. For every completely positive completely bounded map $\phi:S\to T$ between dualizable operator systems, there is a unique weak-$^*$-continuous completely positive completely bounded map $\phi^\mathrm{d}:T^\mathrm{d} \to S^\mathrm{d}$ which is compatible with the dual map $\phi^*$. This gives a full and faithful functor from the category of dualizable operator systems to that of dualizable dual operator systems. Moreover, we will verify that that if $S$ is either a $C^*$-algebra or a unital operator system, then $S$ is dualizable and the canonical weak-$^*$-homeomorphism from the unital operator system $S^{**}$ to the operator system $(S^\mathrm{d})^\mathrm{d}$ is a completely isometric complete order isomorphism. Furthermore, the category of $C^*$-algebras and that of unital "complete" operator systems can be regarded as full subcategories of the category of dual operator systems., Comment: The completeness assumption is missing from some statements in the published version. This completeness assumption is necessary because preduals of dual SMOS are assumed to be complete. We changed some statements in this revision. However, no change in the proof was made. All changes are marked in the color "magenta"

Let $G$ be a locally compact group and let $C^*(G)$ and $C^*_r(G)$ be the full group $C^*$-algebra and the reduced group $C^*$-algebra of $G$. We investigate the relationship between Property $(T)$ for $G$ and Property $(T)$ as well as its strong version for $C^*(G)$ and $C^*_r(G)$. We show that $G$ has Property $(T)$ if (and only if) $C^*(G)$ has Property $(T)$. In the case where $G$ is a locally compact IN-group, we prove that $G$ has Property $(T)$ if and only if $C^*_r(G)$ has strong Property $(T)$. We also show that $C^*_r(G)$ has strong Property $(T)$ for every non-amenable locally compact group $G$ for which $C^*_r(G)$ is nuclear. Some of these groups (as for instance $G=SL_2(\mathbf{R})$) do not have Property $T$., Comment: 9 pages

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

We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $\alpha$ on a finite nuclear unital $C^*$-algebra $A$ such that the reduced crossed product $A\rtimes_{\alpha,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C^*_u(H)$ has property $T$.

Let $1\leq p \leq +\infty$. We show that the positive part of the closed unit ball of a non-commmutative $L^p$-space, as a metric space, is a complete Jordan $^*$-invariant for the underlying von Neumann algebra., Comment: 8 pages

Let $\Phi:\mathfrak{S}(M_1)\to \mathfrak{S}(M_2)$ be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras $M_1$ and $M_2$, respectively. This paper concerns with the situation when $\Phi$ preserves (or partially preserves) one of the following three notions of "transition probability" on the normal state spaces: the Uhlmann transition probability $P_U$, the Raggio transition probability $P_B$ and an "asymmetric transition probability" $P_0$ as defined in this article. It is shown that the two systems are isomorphic, i.e. $M_1$ and $M_2$ are Jordan $^*$-isomorphic, if $\Phi$ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, i.e., for any normal states $\mu$ and $\nu$, we have $$ P\big(\Phi(\mu),\Phi(\nu)\big) = 0 \quad \text{if and only if} \quad P(\mu,\nu)=0, $$ where $P$ stands for $P_U$ (respectively, $P_R$ or $P_0$). Furthermore, as an extension of Wigner's theorem, it is shown that there is a Jordan $^*$-isomorphism $\Theta:M_2\to M_1$ with $$\Phi = \Theta^*|_{\mathfrak{S}(M_1)}$$ if and only if $\Phi$ preserves the "asymmetric transition probability". This is also equivalent to $\Phi$ preserving the Raggio transition probability. Consequently, if $\Phi$ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or "the metric induced by the self-dual cone" are complete Jordan $^*$-invariants for the underlying von Neumann algebras., Comment: 17 pages

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

We obtained a "decomposition scheme" of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsido), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal $A_{d,1}$, the largest discrete essentially infinite ideal $A_{d,\infty}$, the largest type II finite ideal $A_{II,1}$, the largest type II essentially infinite ideal $A_{II,\infty}$, and the largest type III ideal $A_{III}$ of any C*-algebra $A$ such that $A_{d,1} + A_{d,\infty} + A_{II,1} + A_{II,\infty} + A_{III}$ is an essential ideal of $A$. This "decomposition" extends the corresponding one for $W^*$-algebras. We also give a closer look at C*-algebras with Hausdorff primitive spectrum, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above., Comment: 41 pages; we added a lot of details and some new results

We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group $\Gamma$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(\Gamma)$. Moreover, a countable discrete group $\Gamma$ has property $(T)$ if and only if for any action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$, every $\alpha$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $\alpha$-invariant states., Comment: 10 pages; to appear in Int. Math. Res. Notices

A genuine infinite tensor product of complex vector spaces is a vector space ${\bigotimes}_{i\in I} X_i$ whose linear maps coincide with multilinear maps on an infinite family $\{X_i\}_{i\in I}$ of vector spaces. We give a direct sum decomposition of ${\bigotimes}_{i\in I} X_i$ over a set $\Omega_{I;X}$, through which we obtain a more concrete description and some properties of ${\bigotimes}_{i\in I} X_i$. If $\{A_i\}_{i\in I}$ is a family of unital $^*$-algebras, we define, through a subgroup $\Omega^{\rm ut}_{I;A}\subseteq \Omega_{I;A}$, an interesting subalgebra ${\bigotimes}_{i\in I}^{\rm ut} A_i$. Moreover, it is shown that ${\bigotimes}_{i\in I}^{\rm ut} \mathbb{C}$ is the group algebra of $\Omega^{\rm ut}_{I;\mathbb{C}}$. In general, ${\bigotimes}_{i\in I}^{\rm ut} A_i$ can be identified with the algebraic crossed product of a cocycle twisted action of $\Omega^{\rm ut}_{I;A}$. If $\{H_i\}_{i\in I}$ is a family of inner-product spaces, we define a Hilbert $C^*(\Omega^{\rm ut}_{I;\mathbb{C}})$-module $\bar\bigotimes^{\rm mod}_{i\in I} H_i$, which is the completion of a subspace ${\bigotimes}_{i\in I}^{\rm unit} H_i$ of ${\bigotimes}_{i\in I} H_i$. If $\chi_{\Omega^{\rm ut}_{I;\mathbb{C}}}$ is the canonical tracial state on $C^*(\Omega^{\rm ut}_{I;\mathbb{C}})$, then $\bar\bigotimes^{\rm mod}_{i\in I} H_i\otimes_{\chi_{\Omega^{\rm ut}_{I;\mathbb{C}}}}\mathbb{C}$ is a natural dilation of the infinite direct product $\prod {{\otimes}_{i\in I}} H_i$ as defined by J. von Neumann. We will show that the canonical representation of ${\bigotimes}_{i\in I}^{\rm ut} \mathcal{L}(H_i)$ on $\bar\bigotimes^{\phi_1}_{i\in I} H_i$ is injective. We will also show that if $\{A_i\}_{i\in I}$ is a family of unital Hilbert algebras, then so is ${\bigotimes}_{i\in I}^{\rm ut} A_i$., Comment: 23 pages, to appear in Rev. Mat. Iber

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

As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $\Phi : E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$ \langle \Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0, $$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$ \langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E). $$ As a consequence, $\Phi$ is automatically bounded, the induced map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$., Comment: 15 pages

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

Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $\theta$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if $\left<\theta(x),\theta(y)\right> =0$ whenever $\left =0$. We prove that if $\theta$ is an orthogonality preserving map from a full Hilbert $C_0(\Omega)$-module $E$ into another Hilbert $C_0(\Omega)$-module $F$ that satisfies a weaker notion of $C_0(\Omega)$-linearity (known as "localness"), then $\theta$ is bounded and there exists $\phi\in C_b(\Omega)_+$ such that $$ \left<\theta(x),\theta(y)\right>\ =\ \phi\cdot\left, \quad \forall x,y \in E. $$ On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(\Delta)$, we show that a "bi-orthogonality preserving" bijective map $\theta$ with some "local-type property" will be bounded and satisfy $$ \left<\theta(x),\theta(y)\right>\ =\ \phi\cdot\left\circ\sigma, \quad \forall x,y \in E $$ where $\phi\in C_b(\Omega)_+$ and $\sigma: \Delta \rightarrow \Omega$ is a homeomorphism.

In this paper, we will give a thorough study of the notion of Property $(T)$ for $C^*$-algebras (as introduced by M.B. Bekka in \cite{Bek-T}) as well as a slight stronger version of it, called "strong property $(T)$" (which is also an analogue of the corresponding concept in the case of discrete groups and type $\rm II_1$-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property $(T)$ and strong property $(T)$. We will also relate them to certain $(T)$-type properties of the unitary group of the underlying $C^*$-algebra.

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

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C*-algebras. The stronger one of the two is called ``property (T)'' and the weaker one is called ``property (T_{e})''. It is shown that all non-unital C*-algebras do not have property (T) (neither do their unitalizations). Moreover, all non-unital $\sigma$-unital C*-algebras do not have property (T_e)., Comment: 7 pages; to appear in J. Math. Anal. Appl

In this article, we will give a characterization of Banach bimodules over $C^*$-algebras of compact operators that arises from operator spaces as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. These two characterizations are concerned with whether certain natural map from a Banach bimodule to its canonical bidual is isometric (we call such bimodule \emph{regular})., Comment: to appear in Journal of Operator Theory

In this article, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion $B^*$-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real $C^*$-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend two results of Kulkarni (namely, we will give the full versions of the Gelfand-Naimark theorem and the Gelfand theorem for quaternion $B^*$-algebras). On our way to these results, we compare, clarify and unify the term "quaternion Hilbert spaces" in the literatures., Comment: to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

We prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on $C^*$-algebras., Comment: A mistake in the sentance after the proof of Lemma 2 was corrected. We would like to thank Prof. Echterhoff for pointing out the mistake to us

In this paper, we study the duality theory of Hopf $C^*$-algebras in a general ``Hilbert-space-free'' framework. Our particular interests are the ``full duality'' and the ``reduced duality''. In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf $C^*$-algebra., Comment: It is a shortened version of a paper with the same title that was in circulation since 1998. To be published in the International Journal of Mathematics

In this article, we will define two canonical cohomology theories for Hopf $C^*$-algebras and for Hopf von Neumann algebras (with coefficients in their bicomodules). We will then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups will be considered in more details., Comment: The first statement of Remark 1.9(a) in the original published article (in Proc. LMS) does not follow directly from Lemma 1.7(a) but it does follow very easily from the argument of Proposition 1.10. Therefore, we change the presentations of Remark 1.9 and Proposition 1.10 (one line is added in the proof of Proposition 1.10)

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 define the reduced cross-sectional $C^*$-algebras of $C^*$-algebraic bundles over locally compact groups and show that if a $C^*$-algebraic bundle has the approximation property (defined similarly as in the discrete case), then the full cross-sectional $C^*$-algebra and the reduced one coincide. Moreover, if a semi-direct product bundle has the approximation property and the underlying $C^*$-algebra is nuclear, then the cross-sectional $C^*$-algebra is also nuclear. We will also compare the approximation property with the amenability of Anantharaman-Delaroche in the case of discrete groups., Comment: 13 pages, Latex

Three natural definitions for amenability of general Hopf C^*-algebras (all of them being generalizations of the case of locally compact groups) were given and the relations between them were studied. Moreover, amenability in the situation of duality of Hopf C^*-algebras was also studied., Comment: some changes in major terminology to avoid confusion

In this paper, we will prove that if $A$ is a $C^*$-algebra with an effective coaction $\epsilon$ by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac $C^*$-algebras., Comment: 14 pages, Latex; to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

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