There are several well-known fundamental theorems in Gabor analysis that are naturally connected to group representation theory and theory of operator algebras.While some of these connections between time-frequency analysis and operator algebras were established by Jon von Neumann in 1930s, they have been extensively investigated more recently mainly due to the developments of wavelet/Gabor theory, or more generally, the theory of frames in the last two decades.In this article, we will discuss some of the main results we obtained in the last few years together with some new results, exposition and open problems.We will be mainly focused on the results that were originated from time-frequency analysis but reflect intrinsic connections with group representation theory.In particular, we give a detailed account on an abstract version of the duality principle in time-frequency analysis for group representations, and its connections with some open problems in the theory of operator algebras.
Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H.Let M⊆B(H) be a von Neumann algebra and .jpg){kind=small}
" the star partial order in M, that is, for A, B∈M, then we say that A.jpg){kind=small}
B if A*A=A*B and AA*=BA*.It is proved that the supremum and infimum of a subset in M with respect to the star partial order are the same as in B(H).Moreover, we give the representation of a star partial order-hereditary subspace in M, that is, a W* closed nonzero subspace A in M is star partial order-hereditary, which means that for any A∈M and B∈A, A∈A whenever A.jpg){kind=small}
B, if and only if there is a unique pair of nonzero projections E and F which have the same central carrier in M such that A=EMF.
Let U=Tri(A, M, B) be a triangular algebra.In this paper, under mild assumptions, we prove that if δ:U→U is a linear map satisfying δ([[U, V], W])=[[δ(U), V], W]+[[U, δ(V)], W]+[[U, V], δ(W)], for any U, V, W∈U with UV=UW=0(resp.U
V=UW=0), then δ(U)=Φ(U)+h(U) for any U∈U, where Φ:U→U is a derivation, h:U→Z(U) is a linear map vanishing at second commutators with UV=UW=0(resp.UV=UW=0).
Let H be a complex Hilbert space with dim H>2 and A be a selfadjoint standard operator algebra on H.For given positive integer k≥1, the k-skew commutator of operators A and B on H is defined as *[A, B]k=*[A, *[A, B]k-1], where *[A, B]0=B, *[A, B]1=AB-BA*.Assume k≥4 and Φ is a map on A with range containing all rank one projections.It is shown that, *[Φ(A), Φ(B)]k=*[A, B]k holds for all A, B∈A if and only if Φ(A)=A for all A∈A, or Φ(A)=-A for all A∈A.The latter case does not occur if k is even.
We show that when the cb-distance dcb(V, W) between two W*-ternary ring of operators V and W is small, the distance between their linking von Neumann algebras R(V) and R(W) is also small.We show that W*-ternary ring of operators close to injective W*-ternary ring of operators is injective again.We also consider similar properties for W*-ternary ring of operators with property Γ or McDuff property.
We have proved that the algebraic action determined by some elements in the integer group ring of the free group on two generators is ergodic and computed the Fuglede-Kadison determinant of specific element in the Heisenberg group factor.
We introduce some work on Hardy-Sobolev spaces and Fock spaces and their operators and operator algebras, including discussing the boundedness, compactness, Fredholmness, index theory, spectrum and essential spectrum, norm and essential norm and Schatten-p class of some special kinds of operators on these two types of space, and studying the corresponding C*-algebra generated by them.
We present the recent work on the noncommutative Fourier transform for subfactors and locally compact quantum groups, a Survey.We give a short introduction to subfactors and locally compact quantum groups and their properties;the Hausdorff-Young inequality and its extremal functions;Young's inequality and its extremal pairs;uncertainty principles and its minimizers;a noncommutative sum set theorem.
We investigate the existence of quasi-invariant subspaces with arbitrary index.We first give a general criterion.As applications, we show that both the Focktype spaces Fp(C)={f∈Hol(C):(1)/(π)∫C|f(z)|pe-|z|2dA(z)< +∞, 1≤p< +∞} and the Hilbert space H={f∈Hol(C):f∈Hol(C):(1)/(π)∫C|f(z)|2e-|z|dA(z)<+∞} have quasiinvariant subspaces with arbitary index.
We give an equivalent characterization for the α-comparison property of C*-algebras:any simple unital stably finite C*-algebra A has the α-comparison property, if and only if, for any <a>, <b>∈W(A), α·dτ(a) <dτ(b)(∀τ∈QT(A)) implies that <a>≤<b> holds in W(A).Using this characterization, we prove the following results:C*-algebras with α-comparison property have weak comparison;C*-algebras with α-comparison property for α=m+1 have strong tracial m-comparison of positive elements;Z-stability strict comparison α-comparison property for α=m+1 strong tracial m-comparison weak comparison and local weak comparison all agree for the C*-algebras satisfying the conditions given by Kirchberg-Rørdam;if α:=inf{α'∈(1, ∞)|A has the α-comparison property}<∞, then A has the α-comparison property.
We introduce the concept of p-completely bounded frames for p-operator spaces.We prove that a separable p-operator space X has a p-completely bounded frame if and only if it has the p-completely bounded approximation property if and only if it can be p-completely complementedly embedded into a p-operator space with a pcompletely bounded basis.For a non-separable p-operator space with the p-completely bounded approximation property, we prove that its separable subspace always can be p-completely isomorphically embedded into a p-operator space with a p-completely bounded frame.
We introduce compact quantum metric spaces and quantum Gromov-Hausdorff distance defined by Rieffel and quantum Gromov-Hausdorff propinquity recently defined by Latrémolière, and discuss the question of how matrix algebras converge to the sphere in both quantum distances, respectively.
We show that Voiculescu's topological entropy of the canonical automorphism of the C*-algebra arising from the asymptotic equivalence on every irreducible zero-dimensional Smale space is equal to the topological entropy of the original topological dynamics.For the related C*-dynamical system, we have the "variational principle" with respect to the CNT-entropy and the topological entropy, and also show that the state defined by the Bowen measure of the Smale space is the unique equilibrium state of the canonical automorphism.
The famous work of Murray and von Neumann about decomposing W*-algebras into different types(which is known as the classification theory of W*-algebras) is based on the study of projections in W*-algebras.Different from W*-algebras(which are generated by projections), a C*-algebra may contain no non-zero projection.Therefore, we cannot transport the classification theory of Murray and von Neumann directly to C*-algebras.In our recent works, we have developed two classifying(or decomposition) schemes of C*-algebras using the properties of their open projections and properties of their positive elements, respectively.In this note, after a briefing of our two classifying schemes of C*-algebras, we introduce a more general classification framework that, on top of giving many other possible schemes, can be used to obtain, compare and refine the two classification schemes mentioned above.
Let R be a ring, M be an R-bimodule, m and n be two fixed nonnegative integers with m+n=0.If an additive mapping δ from R into M satisfies(m+n)δ(A2)=2mAδ(A)+2nδ(A)A for every A in R, then δ is called an(m, n)-Jordan derivation.In this paper, we prove that if R is a unital ring and M is a unital Rbimodule with a left(right) separating set generated algebraically by all idempotents in R, then every(m, n)-Jordan derivation from R into M is identical with zero whenever m, n>0 and m=n.We also show that if A and B be two unital rings, M is a faithful unital(A, B)-bimodule(N is a faithful unital(B, A)-bimodule), m, n>0 and m=n, U=[NABM] is a |mn(m-n)(m+n)|-torsion-free generalized matrix ring, then every(m, n)-Jordan derivation from U into itself is equal to zero.
