Some dense linear subspaces of extended little Lipschitz algebras (Q429068)

The authors generalise a result by Bade, Curtis and Dales, regarding a sufficient condition for density of a subspace \(P\) of a little Lipschitz algebra \(\text{lip}(X,\alpha)\) in this algebra, \(X\) being a compact metric space and \(\alpha>0\). Actually, in a compact metric space \((X,d)\), for a compact set \(K\) and for \(\alpha>0\), the authors construct two function algebras, viz. the extended Lipschitz algebra \(\text{Lip}(X,K,\alpha)\) and the extended little Lipschitz algebra \(\text{lip}(X,K,\alpha)\) of order \(\alpha\) with respect to \(K\), consisting of those continuous complex-valued functions \(f\) on \(X\) such that \(f\big|_K\in\text{Lip}(K,\alpha)\) and \(f\big|_K\in\text{lip}(K,\alpha)\), respectively; here \(\text{Lip}(K,\alpha)\) is the set of all complex-valued functions \(f\) on \(K\) such that NEWLINE\[NEWLINEp_{\alpha,K}(f):=\sup\left\{\frac{|f(x)-f(y)|}{d^\alpha(x,y)}:x,y\in K,x\neq y\right\}<\inftyNEWLINE\]NEWLINE and \(\text{lip}(K,\alpha)\) is the set of all those complex-valued functions \(f\) on \(K\) for which NEWLINE\[NEWLINE\displaystyle\lim_{d(x,y)\to0}\tfrac{|f(x)-f(y)|}{d^\alpha(x,y)}=0.NEWLINE\]NEWLINE With the help of the norm NEWLINE\[NEWLINE\| f\|_{\alpha,K}:=\| f\|_X+p_{\alpha,K}(f),NEWLINE\]NEWLINE for \(f\in\text{Lip}(X,K,\alpha)\) these extended Lipschitz algebras become Banach algebras. The authors compare these two algebras for different values of \(\alpha\) and show that both are uniformly dense in \(C(X)\), the Banach algebra of all complex-valued continuous functions on the compact metric space \((X,d)\).NEWLINENEWLINEThe authors establish a linear isometry \(T_{X,K}\) from the extended Lipschitz algebra \(\text{Lip}(X,K,\alpha)\) (for \(0<\alpha<1\)) into the Banach space \(C^b\big(W(X,K)\big)\) of all bounded continuous complex-valued functions on the locally compact Hausdorff space \(W(X,K)\), with the norm given by NEWLINE\[NEWLINE\| |h|\|:=\| h|_X\|_X+\| h|_{V(K)}\|_{V(K)},NEWLINE\]NEWLINE for \(f\in C^b\big(W(X,K)\big)\) \big[here, \(W(X,K):=X\cup V(K)\), where \(V(K):=(K\times K)\setminus\{(x,y)\in K\times K:x=y\}\)\big]. The authors also show that \(T_{X,K}\big(\text{Lip}(X,K,\alpha)\big)\) is a closed linear subspace of the Banach space \(\Big(C_0\big(W(X,K)\big),\||.|\|\Big)\), consisting of all those \(f\in C^b\big(W(X,K)\big)\) vanishing at infinity. Then they show that, ``for each \(\Phi\in\big(\text{ lip}(X,K,\alpha),\|.\|_{\alpha,K}\big)^\ast\), there exists \(\mu\in M\big(W(X,K)\big)\) such that NEWLINE\[NEWLINE\Phi(f)=\displaystyle\int_{W(X,K)}T_{X,K}(f)d\mu\quad \big(f\in\text{ lip}(X,K,\alpha)\big)NEWLINE\]NEWLINE and \(\| |\mu|\|=\|\Phi\|\), where \(M\big(W(X,K)\big)\) is the linear space of all complex regular Borel measures on \(W(X,K)\) and \(\| |\mu|\|:=\max\Big\{|\mu|(X),|\mu|\big(V(K)\big)\Big\}\), for \(\mu\in M\big(W(X,K)\big)\).''NEWLINENEWLINELastly, the authors find a sufficient condition for a linear subspace of \(\text{lip}(X,K,\alpha)\) to be dense in this algebra. In particular, they show that \(\text{Lip}(X,K,1)\) is dense in \(\big(\text{lip}(X,K,\alpha),||\cdot||_{\alpha,K}\big)\).