Michael Leinert

Wiener's lemma for twisted convolution and Gabor frames

As a consequence, Ca is invertible and bounded on all ?p(Zd) for 1 < p < oo simultaneously. In this article we study several non-commutative generalizations of Wieners Lemma and their application to Gabor theory. The paper is divided into two parts: the first part (Sections 2 and 3) is devoted to abstract harmonic analysis and extends Wieners Lemma to twisted convolution. The second part (Section 4) is devoted to the theory of Gabor frames, specifically to the design of dual windows with good time-frequency localization. In particular, we solve a conjecture of Janssen, Feichtinger and one of us [17], [18], [9]. These two topics appear to be completely disjoint, but they are not. The solution of the conjectures about Gabor frames is an unexpected application of methods from non-commutative harmonic analysis to application-oriented mathematics. It turns out that the connection between twisted convolution and the Heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis. To be more concrete, we formulate some of our main results first and will deal with the details and the technical background later.

Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.

SYMMETRY OF WEIGHTED

Let G be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra L ω (G) is a symmetric Banach ∗-algebra, if and only if the weight function ω satisfies the GRS-condition. This condition expresses in a precise technical sense that ω grows subexponentially. As a fact of independent interest, we show that groups of (at most) polynomial growth have strict polynomial growth.

L{\uppercase{\footnotesize{L^1}}}

We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) L 1 spaces and use this for the Fourier algebra A(G) of a locally compact group G. In particular we show that if G is an IN-group, then A(G) has the weak fpp if and only if G is compact. We also show that if G is any locally compact group, then A(G) has the fixed point property (fpp) if and only if G is finite. Furthermore if a nonzero closed ideal of A(G) has the fpp, then G must be discrete.

-ALGEBRAS AND THE GRS-CONDITION

A submarkovian C0 semigroup (Tt)t2R+ acting on the scale of complex-valued functions Lp(X;C) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X;E), when E is a Banach space. It is known that, if f 2 Lp(X;C), where 1 < p < 1, then Ttf ! f pointwise almost everywhere. We show that the same holds when f 2 Lp(X;E)

Fixed point property and the Fourier algebra of a locally compact group

Let X be a Banach space and G a locally compact Hausdorff group that acts as a group of isometric linear operators on X. The operation of x E G on X will be denoted by Lx . We study the set Xc of elements ,u E X such that x Lx,u is continuous with respect to the topology on G and the norm-topology on X. The spaces X studied include M(G)*, LUC(G)*, L°°(G)*, VN(G), and VN(G)* . In most cases, characterizations of Xc do not appear to be possible, and we give constructions that illustrate this. We relate properties of Xc to properties of G . For example, if Xc is sufficiently small, then G is compact, or even finite, depending on the case. We give related results and open problems.

POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

If

Continuity of translation in the dual of ^{∞}() and related spaces

G

Convolution dominated operators on compact extensions of abelian groups

is a locally compact group,

ℒ p -Räume

CD(G)