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Frobenius reciprocity and the Haagerup tensor product

Tyrone Crisp
2018 Transactions of the American Mathematical Society  
The characterisation, which builds on previous joint work with N. Higson, exhibits a close connection between the notions of adjoint operators and adjoint functors.  ...  As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H of a locally compact  ...  In [CH16] we showed that the natural extension of Ind G P to a functor on operator modules possesses a left adjoint.  ... 
doi:10.1090/tran/7203 fatcat:uaud5ylh3zawrghdh326tjkg44

Frobenius reciprocity and the Haagerup tensor product [article]

Tyrone Crisp
2017 arXiv   pre-print
The characterisation, which builds on previous joint work with N. Higson, exhibits a close connection between the notions of adjoint operators and adjoint functors.  ...  As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H of a locally compact  ...  By definition, a functor : OM(B) → OM(A) is strongly continuous if the maps : CB B (X , Y ) → CB A ( X , Y ) are strong-operator continuous on bounded subsets.  ... 
arXiv:1605.06023v2 fatcat:msg2vr5bcnh3nje5dtkh47egum

Measurable Categories [article]

D.N. Yetter
2004 arXiv   pre-print
We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators.  ...  Several important technical results are established along the way: First it is shown that all bounded invertible additive functors (and thus a fortiori all invertible *-functors) between categories of  ...  bounded fields of operators on X.  ... 
arXiv:math/0309185v2 fatcat:hezyftn6snfonlm5gud4ft3mt4

Page 4443 of Mathematical Reviews Vol. 58, Issue 6 [page]

1979 Mathematical Reviews  
The principal result here is that there is a one-to-one correspondence between subfunctors of B” and pre-Banach left ideals in B(H), the space of bounded linear operators on H, while the Banach left ideals  ...  The paper deals with a special type of bounded linear operators mapping the space of bounded linear operators L(F,G) into the space L(E,H), where E, F, G and H are Banach spaces.  ... 

K-Theory of Stable Generalized Operator Algebras

Hvedri Inassaridze, Tamaz Kandelaki
2002 K-theory  
It is proved that algebraic and topological K-functors are isomorphic on the category of stable generalized operator algebras which are K i -regular for all i > 0. 2000 Mathematics Subject Classification  ...  Namely it is proved the equality of algebraic and topological K-functors on the category of stable generalized operator algebras (Theorems 9 and 11).  ...  Composing f with the canonical bounded homomorphism B → B α for each C * -seminorm α on B one gets ||f (a)|| α ≤ ||a|| for any a ∈ A.  ... 
doi:10.1023/a:1021197520756 fatcat:yqh37mns25a75mcgbvx4wzowyy

Pseudo effect algebras as algebras over bounded posets [article]

Gejza Jenča
2019 arXiv   pre-print
We prove that there is a monadic adjunction between the category of bounded posets and the category of pseudo effect algebras.  ...  Let us focus on the partial operation / -or, as explained in the previous paragraph, a natural transformation /.  ...  Actually, we equipped the bounded poset Q with a partial binary operation /, that is defined for all comparable pairs of elements of Q. In an analogous way, we may define a partial operation \ on Q.  ... 
arXiv:1903.05399v2 fatcat:gwoduvpdszcgbndfydtsrmnpwq

Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras

Mathys Rennela, Sam Staton, Robert Furber
2017 Electronic Proceedings in Theoretical Computer Science  
In short, complete boundedness is not formalizable in terms of topology. In detail, an abstract operator space is a compatible choice of norms on E, M 2 (E), M 3 (E), · · · for a Banach space E.  ...  A (concrete) operator space is a closed subspace of a C*-algebra, or alternatively a Banach space given together with an isometric embedding into the space of all bounded operators on some Hilbert space  ...  Considering that the composite of two full and faithful functors is full and faithful, one obtains the following corollary. Corollary 1. The functor NPLF is full and faithful.  ... 
doi:10.4204/eptcs.236.11 fatcat:qic6uywssrfsbft7jauhkeuobq

Expressivity of coalgebraic modal logic: The limits and beyond

Lutz Schröder
2008 Theoretical Computer Science  
We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic.  ...  Expressivity results stating that, conversely, logically indistinguishable states are behaviourally equivalent depend on the existence of separating sets of predicate liftings for the signature functor  ...  Acknowledgements The author wishes to thank Till Mossakowski, Markus Roggenbach, and Horst Reichel for collaboration on COCASL, Erwin R.  ... 
doi:10.1016/j.tcs.2007.09.023 fatcat:yaqkhmwfyrcljagjcvqrthvgde

ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

Pierre Clare, Tyrone Crisp, Nigel Higson
2016 Journal of the Institute of Mathematics of Jussieu  
In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al.  ...  The purpose of this paper is to study adjunctions between functors of this sort.  ...  Being a bounded operator between Hilbert spaces, δ X has an adjoint operator (3.20) η X : X −→ F ⊗ B F * ⊗ A X, and we obtain a natural transformation from the identity functor on A H to the tensor product  ... 
doi:10.1017/s1474748016000074 fatcat:2utpmxewrvb5bcjj46lm7dum7u

Expressivity of Coalgebraic Modal Logic: The Limits and Beyond [chapter]

Lutz Schröder
2005 Lecture Notes in Computer Science  
We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic.  ...  Expressivity results stating that, conversely, logically indistinguishable states are behaviourally equivalent depend on the existence of separating sets of predicate liftings for the signature functor  ...  Acknowledgements The author wishes to thank Till Mossakowski, Markus Roggenbach, and Horst Reichel for collaboration on CoCasl, Erwin R.  ... 
doi:10.1007/978-3-540-31982-5_28 fatcat:7azakd6xmvbn7h5rcv4ycu67sy

Page 652 of Mathematical Reviews Vol. 49, Issue 2 [page]

1975 Mathematical Reviews  
Our goal is to describe the structure of the functors between these categories. For functors 6: BW and F: W->B there are defined natural transformations 7:6—>F and %: F—>6, the Banach space n.t.  ...  OPERATOR THEORY Let T be a bounded linear operator in Hilbert space, W(T)={(Tx, x): ||a|| =1} its numerical range. The author reproves a theorem of 8. Hildebrandt [Math.  ... 

Measurable Categories and 2-Groups [article]

L. Crane, D.N. Yetter
2003 arXiv   pre-print
bounded fields of bounded operators on X.  ...  A measurable field of bounded operators is bounded if x → φ x x is a bounded real-valued function (Here x denotes the operator norm on B(H x , K x ).)  ... 
arXiv:math/0305176v1 fatcat:nfm7a6jbcvhldgjduwu6objvvi

A Categorical Setting for Lower Complexity

Robin Cockett, Brian F. Redmond
2010 Electronical Notes in Theoretical Computer Science  
Polarized operators A polarized operator F on an X-strong category consists of a pair of strong functors F p : Y n − → Y, and F o : X n − → X, and a map of cross maps F op : (X 1 , 1) f 1 − − → Y 1 · ·  ...  Polarized operators compose as operations on a polarized strong category. Thus, further examples can be generated from the above basic examples.  ... 
doi:10.1016/j.entcs.2010.08.017 fatcat:e3wsd2dxvnfz7bukgmiznkvd3m

Page 7137 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
Then a Janus functor from Q to abelian groups is two functors M, and M”*, one covariant and one contravariant, which have the same value on objects.  ...  The (Kock-Zoberlein) monad of bounded ideals on the category of orders is studied and an order admitting bounded suprema satisfies the above-mentioned infinite distributive law if the map sending a bounded  ... 

C∗-categories, groupoid actions, equivariant KK-theory, and the Baum–Connes conjecture

Paul D. Mitchener
2004 Journal of Functional Analysis  
The operator T is called the adjoint of the operator T . An operator T is called bounded if the norm T = sup{ T A | A ∈ Ob(B)} is finite. The adjoint of a bounded operator is bounded.  ...  It is shown in [19] that an operator T : E → F is a natural transformation, each map T A : E(A) → F(A) is bounded and linear, and the collection of maps T A defines an operator T .  ...  Then by proposition 6.10, and naturality of the Kasparov product and descent map, we have a commutative diagram Taking direct limits, the desired result follows.  ... 
doi:10.1016/j.jfa.2004.04.016 fatcat:ignx3qrxrfex3h36miz7m4hkjq
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