Motivated by the definition of a semigroup compactication 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 $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$. Examples include all left Arens product algebras over $A$, but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak$^{\ast }$-continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

In this paper, we study condition $C_{\hat{\ }}^{'}$, which is a projective tensor product analogue of condition $C'$. We show that the finite-dimensional $\text{OLLP}$ operator spaces have condition $C_{\hat{\ }}^{'}$, and ${{M}_{n}}\left( n\,>\,2 \right)$ does not have that property.

We prove that, for operator spaces V and W, the operator space V** ⊗hW** can be completely isometrically embedded into (V ⊗hW)**, ⊗h being the Haagerup tensor product. We also show that, for exact operator spaces V and W, a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V× W**. This paves the way to obtaining a canonical embedding of into with a continuous inverse, where is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of which, in particular, determines the lattice of closed ideals of completely.

We characterize certain properties in a matrix ordered space in order to embed it in a C*-algebra. Let such spaces be called C*-ordered operator spaces. We show that for every self-adjoint operator space there exists a matrix order (on it) to make it a C*-ordered operator space. However, the operator space dual of a (nontrivial) C*-ordered operator space cannot be embedded in any C*-algebra.

In this paper, we give two characterizations of the exactness of operator spaces.

In this paper, we mainly study operator spaces which have the locally lifting property $\left( \text{LLP} \right)$. The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the $\text{LLP}$. Several homological characterizations of the $\text{LLP}$ and weak expectation property are given. We also prove that for any operator space $V$, ${{V}^{**}}$ has the $\text{LLP}$ if and only if $V$ has the $\text{LLP}$ and ${{V}^{*}}$ is exact.

Let $G$ be a locally compact group, $\mathrm{L}^p (G)$ be the usual $\mathrm{L}^p$-space for $1 \leq p \leq \infty$, and $\mathrm{A} (G)$ be the Fourier algebra of $G$. Our goal is to study, in a new abstract context, the completely bounded multipliers of $\mathrm{A} (G)$, which we denote $\mathrm{M_{cb}A} (G)$. We show that $\mathrm{M_{cb}A} (G)$ can be characterised as the ‘invariant part’ of the space of (completely) bounded normal $\mathrm{L}^\infty (G)$-bimodule maps on $\mathcal{B}(\mathrm{L}^2 (G))$, the space of bounded operators on $\mathrm{L}^2 (G)$. In doing this we develop a function-theoretic description of the normal $\mathrm{L}^\infty (X, \mu)$-bimodule maps on $\mathcal{B} (\mathrm{L}^2 (X, \mu))$, which we denote by $\mathrm{V}^\infty (X, \mu)$, and name the {\it measurable Schur multipliers} of $(X, \mu)$. Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to obtaining the functorial properties of $\mathrm{M_{cb} A} (G)$, and a concrete description of a standard predual of $\mathrm{M_{cb} A} (G)$.

The main purpose of this paper is to study the validity of the Hausdorff–Young inequality for vector-valued functions defined on a non-commutative compact group. As we explain in the introduction, the natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some important questions are posed.

In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete.