In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of... more In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbeck semigroup acting on the Clifford algebra. Our approach is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. With another use of Speicher's central limit theorem, we may also obtain the same bounds for free products of q-deformed von Neumann algebras interpolating between the fermonic and bosonic frameworks. This generalizes the work of Nelson, Gross, Carlen/Lieb and Biane. Our main application yields hypercontractivity bounds for the free Poisson semigroup acting on the group algebra of the free group Fn, uniformly in the number of generators.
In this letter we show that the field of Operator Space Theory provides a general and powerful ma... more In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.
In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented... more In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L_p-space, then it is either a script L_p-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non--Hilbertian complemented operator subspaces of non commutative L_p-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every $2<p< \infty$ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence sigma we prove that most of these spaces are operator script L_p-spaces, not completely isomorphic to previously known such spaces. However it turns out that some column and row versions of our spaces are not operator script L_p-spaces and have a rather complicated local structure which implies that the Lindenstrauss--Rosenthal alternative does not carry over to the non-commutative case.
We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutati... more We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.
We prove that the Kalton-Peck twisted sum $Z_2^n$ of $n$-dimensional Hilbert spaces has GL-l.u.st... more We prove that the Kalton-Peck twisted sum $Z_2^n$ of $n$-dimensional Hilbert spaces has GL-l.u.st.\ constant of order $\log n$ and bounded GL constant. This is the first concrete example which shows different explicit orders of growth in the GL and GL-l.u.st.\ constants. We discuss also the asymmetry constants of $Z_2^n$.
Let $1 < p < \infty$. It is shown that if $G$ is a discrete group with the approximation property... more Let $1 < p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by Haagerup and Kraus, then the non-commutative $L_p(VN(G))$ space has the operator space approximation property. If, in addition, the group von Neumann algebra $VN(G)$ has the QWEP, i.e. is a quotient of a $C^*$-algebra with Lance's weak expectation property, then $L_p(VN(G))$ actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on $L_p(VN(G))$. Finally, we show that if $G$ is a countable discrete group having the approximation property and $VN(G)$ has the QWEP, then $L_p(VN(G))$ has a very nice local structure, i.e. it is a $\mathcal C\OL_p$ space and has a completely bounded Schauder basis.
We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von... more We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated $BMO$ space.
In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is impose... more In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\M_n)_{n \ge 1}$. Let $\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \in L_p(\M,\tau)$ one can find $a, b \in L_p(\M,\tau)$ and contractions $u_n, v_n \in \M$ such that $$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, it turns out that $a u_n$ and $v_n b$ converge in the row/column Hardy spaces $\H_p^r(\M)$ and $\H_p^c(\M)$ respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case $p=1$, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in $L_1$, w...
This paper is devoted to the study of rigid local operator space structures on non-commutative Lp... more This paper is devoted to the study of rigid local operator space structures on non-commutative Lp-spaces. We show that for 1 ≤ p = 2 < ∞, a non-commutative Lp-space Lp(M) is a rigid OLp space (equivalently, a rigid COLp space) if and only if it is a matrix orderly rigid OLp space (equivalently, a matrix orderly rigid COLp space). We also show that Lp(M) has these local properties if and only if the associated von Neumann algebra M is hyperfinite. Therefore, these local operator space properties on non-commutative Lp-spaces characterize hyperfinite von Neumann algebras.
In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-space... more In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometry T : Lp(M) → Lp(N ) between arbitrary noncommutative Lp-spaces can always be written in the form
First we show that the Carl-Maurey inequality for entropy numbers e.(S) ~_ c. ( l + ln.h_(n/h) ) ... more First we show that the Carl-Maurey inequality for entropy numbers e.(S) ~_ c. ( l + ln.h_(n/h) ) l-1/" ,[S[ ' just characterizes weak type p spaces (S: i~ ---* X, 1 < k < n, 1 _~ p < 2).
Suppose A is a von Neumann algebra with a normal faithful normal- ized trace ¿. We prove that if ... more Suppose A is a von Neumann algebra with a normal faithful normal- ized trace ¿. We prove that if E is a homogeneous Hilbertian subspace of Lp(¿) (1 • p &amp;amp;amp;amp;amp;amp;amp;amp;lt; 1) such that the norms induced on E by Lp(¿) and L2(¿) are equivalent, then E is completely isomorphic to the subspace of Lp((0,1)) spanned by Rademacher functions. Consequently,
Let E, F be exact operator spaces (for example subspaces of the C*-algebra K(H) of all the compac... more Let E, F be exact operator spaces (for example subspaces of the C*-algebra K(H) of all the compact operators on an infinite dimensional Hilbert space H). We study a class of bounded linear maps u : E -~ F* which we call tracially bounded. In particular, we prove that every completely bounded (in short c.b.) map u : E --~ F* factors boundedly through a Hilbert space. This is used to show that the set OS, of all n-dimensional operator spaces equipped with the c.b. version of the Banach Mazur distance is not separable ifn ~> 2.
Abstract. We prove that the Kalton-Peck twisted sum Zn 2 of n-dimensional Hilbert spaces has a GL... more Abstract. We prove that the Kalton-Peck twisted sum Zn 2 of n-dimensional Hilbert spaces has a GL-lu st. constant of order log n and bounded GL constant. This is the first concrete example which shows different explicit orders of growth in the GL and GL-lu st. constants. ...
The operator space analogue of the strong form of the principle of local reflexivity is shown to ... more The operator space analogue of the strong form of the principle of local reflexivity is shown to hold for any von Neumann algebra predual, and thus for any C * -algebraic dual. This is in striking contrast to the situation for C *algebras, since, for example, K(H) does not have that property. The proof uses the Kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces.
Developing the theory of COL p spaces (a variation of the non-commutative analogue of L p spaces)... more Developing the theory of COL p spaces (a variation of the non-commutative analogue of L p spaces), we provide new tools to investigate the local structure of non-commutative L p spaces. Under mild assumptions on the underlying von Neumann algebras, non-commutative L p spaces with Grothendieck's approximation property behave locally like the space of matrices equipped with the p-norm (of the sequences of their singular values). As applications, we obtain a basis for non-commutative L p spaces associated with hyperfinite von Neumann algebras with separable predual von Neumann algebras generated by free groups, and obtain a basis for separable nuclear C Ã -algebras. r
We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume that there exists a sem... more We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume that there exists a semigroup of positive operators on $L^p(\Omega, \mu)$ satisfying a monotone property but do not assume any geometric/metric structure on $\Omega$. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's $H^1$-BMO duality theory. We also get a $H^1$-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative $L^p$ spaces.
In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of... more In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbeck semigroup acting on the Clifford algebra. Our approach is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. With another use of Speicher's central limit theorem, we may also obtain the same bounds for free products of q-deformed von Neumann algebras interpolating between the fermonic and bosonic frameworks. This generalizes the work of Nelson, Gross, Carlen/Lieb and Biane. Our main application yields hypercontractivity bounds for the free Poisson semigroup acting on the group algebra of the free group Fn, uniformly in the number of generators.
In this letter we show that the field of Operator Space Theory provides a general and powerful ma... more In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.
In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented... more In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L_p-space, then it is either a script L_p-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non--Hilbertian complemented operator subspaces of non commutative L_p-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every $2<p< \infty$ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence sigma we prove that most of these spaces are operator script L_p-spaces, not completely isomorphic to previously known such spaces. However it turns out that some column and row versions of our spaces are not operator script L_p-spaces and have a rather complicated local structure which implies that the Lindenstrauss--Rosenthal alternative does not carry over to the non-commutative case.
We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutati... more We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.
We prove that the Kalton-Peck twisted sum $Z_2^n$ of $n$-dimensional Hilbert spaces has GL-l.u.st... more We prove that the Kalton-Peck twisted sum $Z_2^n$ of $n$-dimensional Hilbert spaces has GL-l.u.st.\ constant of order $\log n$ and bounded GL constant. This is the first concrete example which shows different explicit orders of growth in the GL and GL-l.u.st.\ constants. We discuss also the asymmetry constants of $Z_2^n$.
Let $1 < p < \infty$. It is shown that if $G$ is a discrete group with the approximation property... more Let $1 < p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by Haagerup and Kraus, then the non-commutative $L_p(VN(G))$ space has the operator space approximation property. If, in addition, the group von Neumann algebra $VN(G)$ has the QWEP, i.e. is a quotient of a $C^*$-algebra with Lance's weak expectation property, then $L_p(VN(G))$ actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on $L_p(VN(G))$. Finally, we show that if $G$ is a countable discrete group having the approximation property and $VN(G)$ has the QWEP, then $L_p(VN(G))$ has a very nice local structure, i.e. it is a $\mathcal C\OL_p$ space and has a completely bounded Schauder basis.
We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von... more We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated $BMO$ space.
In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is impose... more In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\M_n)_{n \ge 1}$. Let $\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \in L_p(\M,\tau)$ one can find $a, b \in L_p(\M,\tau)$ and contractions $u_n, v_n \in \M$ such that $$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, it turns out that $a u_n$ and $v_n b$ converge in the row/column Hardy spaces $\H_p^r(\M)$ and $\H_p^c(\M)$ respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case $p=1$, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in $L_1$, w...
This paper is devoted to the study of rigid local operator space structures on non-commutative Lp... more This paper is devoted to the study of rigid local operator space structures on non-commutative Lp-spaces. We show that for 1 ≤ p = 2 < ∞, a non-commutative Lp-space Lp(M) is a rigid OLp space (equivalently, a rigid COLp space) if and only if it is a matrix orderly rigid OLp space (equivalently, a matrix orderly rigid COLp space). We also show that Lp(M) has these local properties if and only if the associated von Neumann algebra M is hyperfinite. Therefore, these local operator space properties on non-commutative Lp-spaces characterize hyperfinite von Neumann algebras.
In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-space... more In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometry T : Lp(M) → Lp(N ) between arbitrary noncommutative Lp-spaces can always be written in the form
First we show that the Carl-Maurey inequality for entropy numbers e.(S) ~_ c. ( l + ln.h_(n/h) ) ... more First we show that the Carl-Maurey inequality for entropy numbers e.(S) ~_ c. ( l + ln.h_(n/h) ) l-1/" ,[S[ ' just characterizes weak type p spaces (S: i~ ---* X, 1 < k < n, 1 _~ p < 2).
Suppose A is a von Neumann algebra with a normal faithful normal- ized trace ¿. We prove that if ... more Suppose A is a von Neumann algebra with a normal faithful normal- ized trace ¿. We prove that if E is a homogeneous Hilbertian subspace of Lp(¿) (1 • p &amp;amp;amp;amp;amp;amp;amp;amp;lt; 1) such that the norms induced on E by Lp(¿) and L2(¿) are equivalent, then E is completely isomorphic to the subspace of Lp((0,1)) spanned by Rademacher functions. Consequently,
Let E, F be exact operator spaces (for example subspaces of the C*-algebra K(H) of all the compac... more Let E, F be exact operator spaces (for example subspaces of the C*-algebra K(H) of all the compact operators on an infinite dimensional Hilbert space H). We study a class of bounded linear maps u : E -~ F* which we call tracially bounded. In particular, we prove that every completely bounded (in short c.b.) map u : E --~ F* factors boundedly through a Hilbert space. This is used to show that the set OS, of all n-dimensional operator spaces equipped with the c.b. version of the Banach Mazur distance is not separable ifn ~> 2.
Abstract. We prove that the Kalton-Peck twisted sum Zn 2 of n-dimensional Hilbert spaces has a GL... more Abstract. We prove that the Kalton-Peck twisted sum Zn 2 of n-dimensional Hilbert spaces has a GL-lu st. constant of order log n and bounded GL constant. This is the first concrete example which shows different explicit orders of growth in the GL and GL-lu st. constants. ...
The operator space analogue of the strong form of the principle of local reflexivity is shown to ... more The operator space analogue of the strong form of the principle of local reflexivity is shown to hold for any von Neumann algebra predual, and thus for any C * -algebraic dual. This is in striking contrast to the situation for C *algebras, since, for example, K(H) does not have that property. The proof uses the Kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces.
Developing the theory of COL p spaces (a variation of the non-commutative analogue of L p spaces)... more Developing the theory of COL p spaces (a variation of the non-commutative analogue of L p spaces), we provide new tools to investigate the local structure of non-commutative L p spaces. Under mild assumptions on the underlying von Neumann algebras, non-commutative L p spaces with Grothendieck's approximation property behave locally like the space of matrices equipped with the p-norm (of the sequences of their singular values). As applications, we obtain a basis for non-commutative L p spaces associated with hyperfinite von Neumann algebras with separable predual von Neumann algebras generated by free groups, and obtain a basis for separable nuclear C Ã -algebras. r
We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume that there exists a sem... more We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume that there exists a semigroup of positive operators on $L^p(\Omega, \mu)$ satisfying a monotone property but do not assume any geometric/metric structure on $\Omega$. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's $H^1$-BMO duality theory. We also get a $H^1$-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative $L^p$ spaces.
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Papers by Marius Junge