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Register a new userMorita embeddings for dual operator algebras and dual operator spaces
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Authors
G. K. Eleftherakis
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We define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case.
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We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Delta $-equivalent if they have completely isometric representations $alpha $ and $beta $ respectively and there exists a ternary ring of operators $M$ such that $alpha (A)$ (resp. $beta (B)$) is equal to the norm closure of the linear span of the set $M^*beta (B)M, $ (resp. $Malpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Delta $-equivalent, then the operator algebras $Aotimes cl K$ and $Botimes cl K$ are isomorphic. Here $cl K$ is the set of compact operators on an infinite dimensional separable Hilbert space and $otimes $ is the spatial tensor product. Conversely, if $Aotimes cl K$ and $Botimes cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Delta $-equivalent.
We introduce the notion of $Delta$ and $sigma,Delta-$ pairs for operator algebras and characterise $Delta-$ pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of $Delta$-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that $sigmaDelta$-Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to $Delta$-Morita equivalence of their algebraic extensions.
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Aglers theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.
In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($mathcal{B},Psi)$ consisting of a reflexive Banach spaces $mathcal{B}$ of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $Psi$. We prove that there exist a dual pair ($mathcal{B}^prime,Psi^prime)$ such that the space $mathcal{B}^prime$ is unitarily equivalent to the space $mathcal{B}^*$ and the following intertwining relations hold begin{equation*} mathscr{L} mathcal{U} = mathcal{U}mathscr{M}_z^* quadtext{and}quad mathscr{M}_zmathcal{U} = mathcal{U} mathscr{L}^*, end{equation*} where $mathcal{U}$ is the unitary operator between $mathcal{B}^prime$ and $mathcal{B}^*$. In addition we show that $Psi$ and $Psi^prime$ are connected through the relationbegin{equation*} langle(Psi^prime( bar{z}) e_1) (lambda),e_2rangle= langle e_1,(Psi( bar{ lambda}) e_2)(z)rangle end{equation*} for every $e_1,e_2in E$, $zin varOmega$, $lambdain varOmega^prime$. If a left-invertible operator $T$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^prime$ can be modelled as a multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $mathscr{H}$ and $mathscr{H}^prime$, respectively. We prove that Hilbert space of the dual pair of $(mathscr{H},Psi)$ coincide with $mathscr{H}^prime$, where $Psi$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $mathscr{H}$ and $mathscr{H}^prime$ obtained by identifying them with $mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.
We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.
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