Ideal Spaces of the Haagerup Tensor Product of Ternary Rings of Operators
We characterize the primal, factorial, and Glimm ideals of the Haagerup tensor product $V\otimes^{h} B$ of a TRO $V$ and a $C^{\ast}$-algebra $B$.
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arXiv subjects
Vandana Rajpal
Publications and source records attributed to Vandana Rajpal.
The ideal structure of the minimal Tensor Product of Ternary Rings of Operators
Let \( V \) be a ternary ring of operator and \( B \) a \( C^* \)-algebra. We study the structure of the ideal space of the operator space injective tensor product \( V \otimes^{\mathrm{tmin}} B \) via two maps: \[ Φ(I, J) = \ker(q_I \otimes^{\mathrm{tmin}} q_J) \quad \text{and} \quad Δ(I, J) = I \otimes^{\mathrm{tmin}} B + V \otimes^{\mathrm{tmin}} J. \] We show that \( Φ\) is continuous with respect to the hull-kernel topology, and that its restriction to primitive and prime ideals defines a homeomorphism onto dense subsets of the respective ideal spaces of \( V \otimes^{\mathrm{tmin}} B \). We prove that if \( Φ= Δ\), then \( Φ\) induces a homeomorphism between the space of minimal primal ideals of \( V \otimes^{\mathrm{tmin}} B \) and the product of the spaces of minimal primal ideals of \( V \) and \( B \)
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Polynomials in operator space theory: matrix ordering and algebraic aspects
We extend the $λ$-theory of operator spaces given by Defant and Wiesner (2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to $λ$ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of $λ$-tensor product of $C^*$-algebras has also been discussed.
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$λ$-tensor product of operator spaces
We propose a theory of $λ$-tensor product of operator spaces which extends the theory of Blecher-Paulsen and Effros-Ruan for the operator space projective tensor product \cite{blecp}, \cite{effros}, \cite{eff} and that of Rajpal-Kumar-Itoh for the Schur tensor product \cite{vandee4} of operator spaces.
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Arens regularity of projective tensor products
For completely contractive Banach algebras $A$ and $B$ (respectively operator algebras $A$ and $B$), the necessary and sufficient conditions for the operator space projective tensor product $A\widehat{\otimes}B$ (respectively the Haagerup tensor product $A\otimes^{h}B$) to be Arens regular are obtained. Using the non-commutative Grothendieck's inequality, we show that, for $C^*$-algebras $A$ and $B$, the Arens regularity of Banach algebras $A\otimes^{h}B$, $A\ot^γ B$, $A\ot^{s} B$ and $A\widehat{\otimes}B$ are equivalent, where $\otimes^h$, $\otimes^γ$, $\ot^s$ and $\widehat{\otimes}$ are the Haagerup, the Banach space projective tensor norm, the Schur tensor norm and the operator space projective tensor norm, respectively.
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Schur tensor product of operator spaces
We develop a systematic study of the schur tensor product both in the category of operator spaces and in that of $C^*$-algebras.
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Projective Tensor Products of $C^*$-algebras
For $C^*$-algebras $A$ and $B$, we study the bi-continuity of the canonical embedding of $A^{**}\ot_γ B^{**}$ ($A^{**}\hat{\ot} B^{**}$) into $(A \ot_γ B)^{**}$ (resp. $(A \hat{\ot} B)^{**}$), and its isomorphism. Ideal structure of $A\hat{\ot} B$ has been obtained in case $A$ or $B$ has only finitely many closed ideals.
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Symmetry and quasi-centrality of operator space projective tensor product
For $C^{*}$-algebras $A$ and $B$, the operator space projective tensor product $A\hat{\otimes}B$ and the Banach space projective tensor product $A\otimes_γB$ are shown to be symmetric. We also show that $A\hat{\otimes}B$ is weakly Wiener algebra. Finally, quasi-centrality, and the unitary group of $A\hat{\otimes}B$ are discussed.
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$^{*}$-Regularity of Operator Space Projective Tensor Product of C$^{*}$-Algebras
The Banach $^{*}$-algebra $A\hat{\otimes}B$, the operator space projective tensor product of $C^{*}$-algebras $A$ and $B$, is shown to be $^{*}$-regular if Tomiyama's property ($F$) holds for $A\otimes_{\min}B$ and $A \otimes_{\min}B=A \otimes_{\max}B$, where $\otimes_{\min}$ and $\otimes_{\max}$ are the injective and projective $C^{*}$-cross norm, respectively. However, $A\hat{\otimes}B$ has a unique $C^{*}$-norm if and only if $A\otimes B$ has. We also discuss the property ($F$) of $A\hat{\otimes}B$ and $A\otimes_{h}B$, the Haagerup tensor product of $A$ and $B$.
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