Mathematics

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete.

We prove an atomic type decomposition for the noncommutative martingale Hardy space $\h_p$ for all 0<p<2 by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of $\h_p$ for all 0<p<1, and provide a constructive proof of the atomic decomposition for p=1 . We also study $(p,\8)_c$-atoms, and show that every (p,2 ) c -atom can be decomposed into a sum of $(p,\8)_c$-atoms; consequently, for every 0<p≤1 , the (p,q ) c -atoms lead to the same atomic space for all $2\le q\le\8$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space $\h_p$ ( 0<p<1 ) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to proving some sharp martingale inequalities.

In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new perspectives and interesting observations; therefore the style is expository.

Three canonical decompositions concerning commuting pair of isometries, power partial isometries, and contractions are reassessed. They have already been proved in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. Ignoring topological structures, we give an algebraic approach to obtain them in the larger category of Baer ∗ -rings.

We introduce the B -spline interpolation problem corresponding to a C ∗ -valued sesquilinear form on a Hilbert C ∗ -module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert C ∗ -module is self-dual. Extending a bounded C ∗ -valued sesquilinear form on a Hilbert C ∗ -module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the B -spline interpolation problem to have a solution. Passing to the setting of Hilbert W ∗ -modules, we present our main result by characterizing when the spline interpolation problem for the extended C ∗ -valued sesquilinear to the dual X ′ of the Hilbert W ∗ -module X has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert W ∗ -module X is orthogonally complemented with respect to another C ∗ -inner product on X . Finally, solutions of the B -spline interpolation problem for Hilbert C ∗ -modules over C ∗ -ideals of W ∗ -algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.

We consider semi-group BMO spaces associated with an arbitrary σ -finite von Neumann algebra (M,φ) . We prove that BMO always admits a predual, extending results from the finite case. Consequently, we can prove - in the current setting of BMO - that they are Banach spaces and they interpolate with L p as in the commutative situation, namely [BMO(M), L ∘ p (M) ] 1/q ≈ L ∘ pq (M) . We then study a new class of examples. We introduce the notion of Fourier-Schur multiplier on a compact quantum group and show that such multipliers naturally exist for S U q (2) .

For a unital C ??-algebra A and a subspace B of A , a characterization for a best approximation to an element of A in B is obtained. As an application, a formula for the distance of an element of A from B has been obtained, when a best approximation of that element to B exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert C ??-module to a subspace is obtained.

Let A⊂C and B⊂D be unital inclusions of unital C ∗ -algebras. Let A B A (C,A) (resp. B B B (D,B) ) be the space of all bounded A -bimodule (resp. B -bimodule) linear maps from C (resp. D ) to A (resp. B ). We suppose that A⊂C and B⊂D are strongly Morita equivalent. We shall show that there is an isometric isomorphism f of A B A (C,A) onto B B B (D,B) and we shall study on basic properties about f .

We study subordination of free convolutions. We prove that for free random variables X,Y and a Borel function f the conditional expectation E φ [(z−X−f(X)Y f ∗ (X) ) −1 |X] , is a resolvent again. This result allows explicit calculation of the distribution of X+f(X)Y f ∗ (X) . The main tool is a formula for conditional expectations in terms of Boolean cumulant transforms, generalizing subordination formulas for free additive and multiplicative convolutions.

We explore connections between boundary representations of operator spaces and those of the associated Paulsen systems. Using the notions of finite representation and separating property which we introduced, boundary representations for operator spaces is characterised. We also introduce weak boundary for operator spaces. Rectangular hyperrigidity of operator spaces introduced here is used to establish an analogue of Saskin's theorem in the setting of operator spaces in finite dimensions.