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Frobenius reciprocity and the Haagerup tensor product

2018
*
Transactions of the American Mathematical Society
*

The characterisation, which builds

doi:10.1090/tran/7203
fatcat:uaud5ylh3zawrghdh326tjkg44
*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. ...##
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Frobenius reciprocity and the Haagerup tensor product
[article]

2017
*
arXiv
*
pre-print

The characterisation, which builds

arXiv:1605.06023v2
fatcat:msg2vr5bcnh3nje5dtkh47egum
*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. ...##
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Measurable Categories
[article]

2004
*
arXiv
*
pre-print

We develop the theory of categories of measurable fields of Hilbert spaces and

arXiv:math/0309185v2
fatcat:hezyftn6snfonlm5gud4ft3mt4
*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. ...##
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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. ...##
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K-Theory of Stable Generalized Operator Algebras

2002
*
K-theory
*

It is proved that algebraic and topological K-

doi:10.1023/a:1021197520756
fatcat:yqh37mns25a75mcgbvx4wzowyy
*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. ...##
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Pseudo effect algebras as algebras over bounded posets
[article]

2019
*
arXiv
*
pre-print

We prove that there is a monadic adjunction between the category of

arXiv:1903.05399v2
fatcat:gwoduvpdszcgbndfydtsrmnpwq
*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. ...##
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Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras

2017
*
Electronic Proceedings in Theoretical Computer Science
*

In short, complete

doi:10.4204/eptcs.236.11
fatcat:qic6uywssrfsbft7jauhkeuobq
*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. ...##
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Expressivity of coalgebraic modal logic: The limits and beyond

2008
*
Theoretical Computer Science
*

We then move

doi:10.1016/j.tcs.2007.09.023
fatcat:yaqkhmwfyrcljagjcvqrthvgde
*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. ...##
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ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

2016
*
Journal of the Institute of Mathematics of Jussieu
*

In this way we shall show that the parabolic induction

doi:10.1017/s1474748016000074
fatcat:2utpmxewrvb5bcjj46lm7dum7u
*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 ...##
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Expressivity of Coalgebraic Modal Logic: The Limits and Beyond
[chapter]

2005
*
Lecture Notes in Computer Science
*

We then move

doi:10.1007/978-3-540-31982-5_28
fatcat:7azakd6xmvbn7h5rcv4ycu67sy
*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. ...##
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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. ...##
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Measurable Categories and 2-Groups
[article]

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 ).) ...

##
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A Categorical Setting for Lower Complexity

2010
*
Electronical Notes in Theoretical Computer Science
*

Polarized

doi:10.1016/j.entcs.2010.08.017
fatcat:e3wsd2dxvnfz7bukgmiznkvd3m
*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. ...##
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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*...##
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C∗-categories, groupoid actions, equivariant KK-theory, and the Baum–Connes conjecture

2004
*
Journal of Functional Analysis
*

The

doi:10.1016/j.jfa.2004.04.016
fatcat:ignx3qrxrfex3h36miz7m4hkjq
*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. ...
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