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Register a new userProjections in $L^1(G)$; the unimodular case
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We consider the issue of describing all self-adjoint idempotents (projections) in $L^1(G)$ when $G$ is a unimodular locally compact group. The approach is to take advantage of known facts concerning subspaces of the Fourier-Stieltjes and Fourier algebras of $G$ and the topology of the dual space of $G$. We obtain an explicit description of any projection in $L^1(G)$ which happens to also lie in the coefficient space of a finite direct sum of irreducible representations. This leads to a complete description of all projections in $L^1(G)$ for $G$ belonging to a class of groups that includes $SL(2,R)$ and all almost connected nilpotent locally compact groups.
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We obtain a slow exponential growth estimate for the spherical principal series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of Cowlings strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy rho_s (s in [0, 1]) of the spherical principal series which is required for the first authors program for proving the Baum--Connes conjecture with coefficients for Sp(n,1).
In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$ |I_alpha F|_{dot{B}^{0,1}_{d/(d-alpha),1}(mathbb{R}^d;mathbb{R}^k)} leq C |F |_{L^1(mathbb{R}^d;mathbb{R}^k)} $$ for all $F in L^1(mathbb{R}^d;mathbb{R}^k)$ which satisfy a first order cocancelling differential constraint. We show how this implies endpoint Besov-Lorentz estimates for Hodge systems with $L^1$ data via fractional integration for exterior derivatives.
Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]sqsubseteq [q]$ iff for every primitive ideal $mathfrak p$ of $A$ either $p/mathfrak ppreceq q/mathfrak ppreceq (1- q)/mathfrak p$ or $p/mathfrak psucceq q/mathfrak p succeq (1-q)/mathfrak p.$ We prove that $p$ is central iff $[p]$ is $sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]mapsto [p]^Game$ that moves $p$ towards the center. The number $n(p) $ of $Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]sqsubseteq [q]Rightarrow n(p)leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliotts classification, the categorical equivalence $Gamma$ between MV-algebras and unital $ell$-groups, and L os ultraproduct theorem for first-order logic.
We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are able to easily give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $|P| < 2/sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.
Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for their existence. Representations of the generalized entropic projections are obtained: they are the ``measure component of some extended entropy minimization problem.
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