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Register a new userA short proof of Hulanickis Theorem
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Nico Spronk
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We outline a simple proof of Hulanickis theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.
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Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated groups $A$.
We show that Liebs concavity theorem holds more generally for any unitary invariant matrix function $phi:mathbf{H}_+^nrightarrow mathbb{R}_+^n$ that is concave and satisfies Holders inequality. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^ntimesmathbf{H}_+^m$, for any $Kin mathbb{C}^{ntimes m}$ and any $s,p,qin(0,1], p+qleq 1$. This result improves a recent work by Huang for a more specific class of $phi$.
We show that Liebs concavity theorem holds more generally for any unitarily invariant matrix function $phi:mathbf{H}^n_+rightarrow mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^mtimesmathbf{H}_+^n$, for any $Kin mathbb{C}^{mtimes n},sin(0,1],p,qin[0,1], p+qleq 1$.
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilberts 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $varphi_1,ldots,varphi_5 : [,0, 1,]to [,0,1,]$ such that for any continuous function $f: [,0,1,]^2tomathbb R$ there is a continuous function $h: [,0,3,]tomathbb R$ such that for any $x,yin [,0, 1,]$ we have $$f(x,y)=sumlimits_{k=1}^5 hleft(varphi_k(x)+sqrt{2},varphi_k(y)right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
Hanners theorem is a classical theorem in the theory of retracts and extensors in topological spaces, which states that a local ANE is an ANE. While Hanners original proof of the theorem is quite simple for separable spaces, it is rather involved for the general case. We provide a proof which is not only short, but also elementary, relying only on well-known classical point-set topology.
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