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Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has a unique conditional expectation, and when it has a unique pseudo-expectation (in the sense of Pitts). Likewise for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. As an application, we (slightly) strengthen results of Kishimoto and Archbold-Spielberg concerning $C^*$-simplicity of $\mathcal{A} \rtimes_r G$.

We initiate a study of operator approximate biprojectivity for quantum group algebra $L^{1}({\Bbb{G}})$ , where $\mathbb{G}$ is a locally compact quantum group. We show that if $L^{1}({\Bbb{G}})$ is operator approximately biprojective, then $\mathbb{G}$ is compact. We prove that if $\mathbb{G}$ is a compact quantum group and $\mathbb{H}$ is a non-Kac-type compact quantum group such that both $L^{1}({\Bbb{G}})$ and $L^{1}({\Bbb{H}})$ are operator approximately biprojective, then $L^{1}({\Bbb{G}})\widehat{\otimes}L^{1}({\Bbb{H}})$ is operator approximately biprojective, but not operator biprojective.

We give a short answer to the question in the title: {\em dendrits}. Precisely we show that the $C^{\ast}$-algebra $C(X)$ of all complex-valued continuous functions on a compactum $X$ is projective in the category ${\mathcal C}^{1}$ of all (not necessarily commutative) unital $C^{\ast}$-algebras if and only if $X$ is an absolute retract of dimension $\dim X \leq 1$ or, equivalently, that $X$ is a dendrit., 8 pages

Let A be a locally $m$-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet $A-$module $X=A+/I$ to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that $X$ is strictly flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a locally $m$-convex Fréchet algebra $A$ is amenable if and only if $A$ is isomorphic to a reduced inverse limit of amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and by Helemskii and Sheinberg to the setting of locally $m$-convex Fréchet algebras. As a corollary, we show that Connes and Haagerup's theorem on amenable $C*$-algebras and Sheinberg's theorem on amenable uniform algebras hold in the Fréchet algebra case. We also show that a quasinormable locally $m$-convex Fréchet algebra has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally $m$-convex Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i.

The theory of infinite dimensional oscillatory integrals and some of its applications are discussed, with special attention to the relations with the original work of K. Itô in this area. A recent general approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented, together with some new developments.

It is shown that every continuous homomorphism of Arens-Michael algebras can be obtained as the limit of a morphism of certain projective systems consisting of Fr\'{e}chet algebras. Based on this we prove that a complemented subalgebra of an uncountable product of Fr\'{e}chet algebras is topologically isomorphic to the product of Fr\'{e}chet algebras. These results are used to characterize injective objects of the category of locally convex topological vector spaces. Dually, it is shown that a complemented subspace of an uncountable direct sum of Banach spaces is topologically isomorphic to the direct sum of ({\bf LB})-spaces. This result is used to characterize projective objects of the above category., Comment: 25 pages

In this paper, we investigate the notion $\phi$-injectivity for Banach $A$-modules, where $\phi$ is a character on $A.$ We obtain some hereditary properties of $\phi$-injectivity for certain classes of Banach modules related to closed ideals. These results allow us to study $\phi$-injectivity of certain Banach $A$-modules in commutative case, specially $\ell^{1}$-semilattice algebras. As an application, we give an example of a non-injective Banach module which is $\phi$-injective for each character $\phi.$, Comment: 10 pages, To appear in "Annals of Functional Analysis"

Given an inclusion D $\subseteq$ C of unital C*-algebras, a unital completely positive linear map $\Phi$ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D $\subseteq$ C $\subseteq$ B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' $\cap$ C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D $\subseteq$ C has a unique pseudo-expectation $\Phi$ which is also faithful, then the C*-envelope of any operator space X with D $\subseteq$ X $\subseteq$ C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions., Comment: 26 pages

We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C$^*-$module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator $A-B$-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebras that can act on X are determined by these corners of I(X) and consequently bimodules actions on X extend naturally to actions on I(X). These results give another characterization of the multiplier algebra of an operator space, which was introduced by the first author, and a short proof of a recent characterization of operator modules, and a related result. As another application, we extend Wittstock's module map extension theorem, by showing that an operator $A-B$-bimodule is injective as an operator $A-B$-bimodule if and only if it is injective as an operator space., Comment: Revised version, January 21 2000

The notion of a dual of an operator space which is again an operator space has been introduced independently by Vern Paulsen and the author, and by Effros and Ruan. Its significance in the theory of tensor products of operator spaces has already been partially explored by the aforementioned. Here we establish some other fundamental properties of this dual construction, and examine how it interacts with other natural categorical constructs for operator spaces. We define and study a notion of projectivity for operator spaces, and give a noncommutative version of Grothendieck' s characterization of I1 (I) spaces for a discrete set /.

A duality theorem of projective and inductive limit spaces over a nonspherically complete valued field is obtained under a certain condition, and topologies of spaces of locally analytic functions are ?studied. Introduction. Morita obtained in |5| a duality theorem of projec­ tive limit spaces and inductive limit spaces over a spherically complete nonarchimedean valued field, and Schikhof studied in [8J locally convex spaces over a nonsphorically complete nonarchimedean valued field. In this paper, we use the results of [8j and study the duality of such spaces over a nonspherically complete nonarchimedean valued field. The duality theorem of |5| was obtained as a generalization of the results of Komatsu |8| by Morita using: the theory of van Tiel [10] about locally convex spaces over a spherically complete nonarchimedean valued field. There the following two facts are used essentially: (i) The Mackey topology is the topology of uniform convergence on weakly e-compact sets; (ii) Any absolutely convex weakly c-compact set is strongly closed. Though we can generalize the notion of e-compactness to our ease, it is difficult to obtain good analogues of these two facts over a nonspherically complete valued field. Hence we restrict our attention to a more restricted class than in |5], and prove a duality theorem by making use of van dor Put’s duality theorem of sequence spaces c0 = {(au cl, aa, • • •')} and V" {(&„ L, b„ ■••) X m (m Yn (m n(xn), y j m = (xn, vn_m{ym))n holds for any xn e X a and ym e Ym. Let (X, um) be the locally convex projective limit of {Xm, and let (Y, vm) be the locally convex inductive limit of [Ym, vnim}. We assume further that (iv) the projection map uw: X —> Xm has a dense image for each m. By definition, any element x of the projective limit X can be written as x — (xm) with xm e Xm satisfying umi„(x„) — xm for any m and n with m K by (x, y) = (uM(x), ym)m with such a ym 6 Ym. It is easy to see that this pairing ( , ) is iT-bilinear. Since the projection map um: X —> Xm is con­ tinuous, our pairing ( , ) is bicontinuous on X x Y m for each m. Hence, by the universal mapping property of the inductive limit topology, ( , ) is bicontinuous on X x Y . Let x = (xm) be a nonzero element of X Then xm =£ 0 for some m. Since ( , )m is nondegenerate, (xm, ym)m ^ 0 for some ym 6 Yn. Hence (*, y) = (xm, y j m # 0 for some y = vm(ym) e vm( Y J c Y. Let y = vm( y j (■ym e F J be a nonzero element of Y. Then {xm e Xm; (xm, ym)m ^ 0} is a non-empty open subset of Xm. Since the image of the projection map um: X -* X m is dense, there is an element x = (xm) e X such that (x, y) = (xm, ym)m ¥= 0. Therefore our pairing ( , ) is nondegenerate. Hence we have proved the following: DUALITY OVER A COMPLETE NONARCHIMEDEAN FIELD 389 P r o p o s i t i o n 1. Let X = proj lim and Y = ind lim Y m be as before* Then we have a nondegenerate bicontinuous K'-bilinear fo rm ( , ) : X x Y —> K .