arXiv subjects

Jason Crann

Publications and source records attributed to Jason Crann.

26 records · Page 2Linked to original sources

arXiv2018

A new duality via the Haagerup tensor product

We initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert $C^*$-modules, it also captures quantum group duality in a fundamental way. We compute the so-called Haagerup dual for various operator algebras arising from $\ell^p$ spaces. In particular, we show that the dual of $\ell^1$ under any operator space structure is $\min\ell^\infty$. In the setting of abstract harmonic analysis we generalize a result of Varopolous by showing that $C(\mathbb{G})$ is an operator algebra under convolution for any compact Kac algebra $\mathbb{G}$. We then prove that the corresponding Haagerup dual $C(\mathbb{G})^h=\ell^\infty(\widehat{\mathbb{G}})$, whenever $\widehat{\mathbb{G}}$ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.

math.OA↗

arXiv2018

Mapping ideals of quantum group multipliers

We study the dual relationship between quantum group convolution maps $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ and completely bounded multipliers of $\widehat{\mathbb{G}}$. For a large class of locally compact quantum groups $\mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(\widehat{\mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coinciding with $\ell^1(\widehat{b\mathbb{G}})$, where $b\mathbb{G}$ is the quantum Bohr compactification of $\mathbb{G}$. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with $C(b\mathbb{G})$. Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of quasi-SIN quantum group, which we introduce and exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras $VN(G)$ for quasi-SIN locally compact groups $G$.

math.OA↗

arXiv2016

Fourier algebras of hypergroups and central algebras on compact (quantum) groups

This paper concerns the study of regular Fourier hypergroups through multipliers of their associated Fourier algebras. We establish hypergroup analogues of well-known characterizations of group amenability, introduce a notion of weak amenability for hypergroups, and show that every discrete commutative hypergroup is weakly amenable with constant 1. Using similar techniques, we provide a sufficient condition for amenability of hypergroup Fourier algebras, which, as an immediate application, answers one direction of a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564, 2009] on the amenability of $ZL^1(G)$ for compact groups $G$. In the final section we consider Fourier algebras of hypergroups arising from compact quantum groups $\mathbb{G}$, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact $\mathbb{G}$ of Kac type.

math.FA↗

arXiv2016

Amenability and covariant injectivity of locally compact quantum groups II

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}})$-module maps on $L^{\infty}(\widehat{\mathbb{G}})$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory and the Powers--Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

math.OA↗

arXiv2015

Private algebras in quantum information and infinite-dimensional complementarity

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

quant-ph↗

arXiv2014

Amenability and covariant injectivity of locally compact quantum groups

As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.

math.OA↗

arXiv2014

An uncertainty principle for unimodular quantum groups

We present a generalization of Hirschman's entropic uncertainty principle for locally compact abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to normal central states of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the central state.

math-ph↗

arXiv2012

Quantum channels arising from abstract harmonic analysis

We present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups $G$. Beginning with a representation of the measure algebra $M(G)$, we unify and elaborate on recent counter-examples to fixed point subalgebras in infinite dimensions, as well as present an application to the noiseless subsystems method of quantum error correction. Using a representation of the completely bounded Fourier multiplier algebra $McbA(G)$, we provide a new class of counter-examples to the recently solved asymptotic quantum Birkhoff conjecture, along with a systematic method of producing the examples using a geometric representation of Schur maps. Further properties of our channels including duality, quantum capacity, and entanglement preservation are discussed along with potential applications to additivity conjectures.

math-ph↗