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.
We determine the injective envelope and local multiplier algebra of a continuous trace C*-algebra that arises from a continuous Hilbert bundle over an arbitrary locally compact Hausdorff space. In addition, we show that the second-order local multiplier algebra of any such algebra is injective.
A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Delta is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show to be a Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A. In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A.
Given a unital C*-subalgebra of B(H), we study the set of all possible images of its injective envelope that are contained in B(H) and their position relative to the double commutant of the algebra in order to obtain more information about the existence or non-existence of weak expectations. We study the subset of B(H) that is the intersection of all possible images of the injective envelope and show that it is simultaneously a reflexive cover and a new type of order completion of the algebra.
We consider the ideal structure of a reduced crossed product of a unital $C^*$-algebra equipped with an action of a discrete group. More specifically we find sufficient and necessary conditions for the group action to have the intersection property, meaning that non-zero ideals in the reduced crossed product restrict to non-zero ideals in the underlying $C^*$-algebra. We show that the intersection property of a group action on a $C^*$-algebra is equivalent to the intersection property of the action on the equivariant injective envelope. We also show that the centre of the equivariant injective…
We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and $C^*$-envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.
This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the groupoid). We study bundles of compact topological spaces equipped with an action of a groupoid. We show that any groupoid has a minimal boundary (the Furstenberg boundary of the groupoid). We prove that the Hamana and Furstenberg boundaries are the same, for (discrete) groupoids. We find the relation between the reduced crossed product of the $\G$-injective env…
We study the issue of issue of purity (as a completely positive linear map) for identity maps on operators systems and for their completely isometric embeddings into their C$^*$-envelopes and injective envelopes. Our most general result states that the canonical embedding of an operator system $\mathcal R$ into its injective envelope $\mathcal I(\mathcal R)$ is pure if and only if the C$^*$-envelope of $\mathcal R$ is a prime C$^*$-algebra. To prove this, we also show that the identity map on any AW$^*$-factor is a pure completely positive linear map. For embeddings of operator systems into th…
A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.
Given C$^*$-algebras $A$ and $B$ acting cyclically on Hilbert spaces $\h$ and $\k$, respectively, we characterize completely isometric $A,B$-bimodule maps from $\bkh$ into operator $A,B$-bimodules. We determine cogenerators in some classes of operator bimodules. For an injective cogenerator $X$ in a suitable category of operator $A,B$-bimodules we show: if $A$, regarded as a C$^*$-subalgebra of $\al(X)$ (adjointable left multipliers on $X$), is equal to its relative double commutant in $\al(X)$, then $A$ must be a W$^*$-algebra.
A left ideal of any C*-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in a `pre-quel' to this paper [B], that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C*-algebra. We study operator algebras and their multiplier algebras from the perspective of `Hamana theory' and using the multiplier algebras introduced by the first author.
We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope fo…
We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the Banach-Stone theorem.
This paper is a continuation of the program started by Ruan in 2003, of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, C*-envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided M-ideal theory and characterize one-sided M-ideals in real C*-algebras and real operator algebras with contractive approximate identity.
Let $\A$ be a unital operator algebra and let $α$ be an automorphism of $\A$ that extends to a *-automorphism of its $\ca$-envelope $\cenv (\A)$. In this paper we introduce the isometric semicrossed product $\A \times_α^{\is} \bbZ^+ $ and we show that $\cenv(\A \times_α^{\is} \bbZ^+) \simeq \cenv (\A) \times_α \bbZ$. In contrast, the $\ca$-envelope of the familiar contractive semicrossed product $\A \times_α \bbZ^+ $ may not equal $\cenv (\A) \times_α \bbZ$. Our main tool for calculating $\ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator al…
We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related…
We introduce a matricial analogue of an Archimedean order unit space, which we call a $k$-AOU space. We develop the category of $k$-AOU spaces and $k$-positive maps and exhibit functors from this category to the category of operator systems and completely positive maps. We also demonstrate the existence of injective envelopes and C*-envelopes in the category of $k$-AOU spaces. Finally, we show that finite-dimensional quantum correlations can be characterized in terms of states on finite-dimensional $k$-AOU spaces. Combined with previous work, this yields a reformulation of Tsirelson's conjectu…
Given Hilbert spaces $H_1,H_2,H_3$, we consider bilinear maps defined on the cartesian product $S^2(H_2,H_3)\times S^2(H_1,H_2)$ of spaces of Hilbert-Schmidt operators and valued in either the space $B(H_1,H_3)$ of bounded operators, or in the space $S^1(H_1,H_3)$ of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras $M_i\subset B(H_i)$, $i=1,2,3$, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps $u\colon S^2(H_2,H_3)\times S^2(H_1,H_2)\to B(H_1,H_3)$ by…
We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna-Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to $*$-representations of the ambient $C^*$-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna-Pick property. We also deduce operator theoretic dilation results which are in the spirit of…
Motivated by the recent interest in the examination of unital completely positive maps and their effects in C*-theory, we revisit an older result concerning the existence of the Šilov ideal. The direct proof of Hamana's theorem for the existence of an injective envelope for a unital operator subspace X of some B(H) that we provide implies that the Šilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in B(H). As an immediate consequence we deduce that the Šilov ideal is the biggest boundary operator subsystem for X in C*(X). The new proof of the existence of the…
In this paper we give characterizations of essential left ideals of a C*-algebra $A$ in terms of their properties as operator $A$-modules. Conversely, we seek C*-algebraic characterizations of those ideals $J$ in $A$ such that $A$ is an essential extension of $J$ in various categories of operator modules. In the case of two-sided ideals, we prove that all the above concepts coincide. We obtain results, analogous to {M. Hamana's} results, which characterize the injective envelope of a C*-algebra as a maximal essential extension of the C*-algebra, but with completely positive maps replaced by co…
We present a systematic development of inductive limits in the categories of ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator C*-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, a…
