A note on p-completely bounded homomorphisms of the Fig脿-Talamanca-Herz algebras

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• 作者：Monica Ilie¹ ; ^{milie@lakeheadu.ca" class="auth_mail}
• 关键词：Figà ; -Talamanca&ndash ; Herz algebras ; p-Operator spaces ; p-Completely contractive maps ; Piecewise affine maps ; Amenable groups
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：1 November 2014
• 年：2014
• 卷：419
• 期：1
• 页码：273-284
• 全文大小：313 K

文摘

For a locally compact group G   and X14004181&_mathId=si1.gif&_user=111111111&_pii=S0022247X14004181&_rdoc=1&_issn=0022247X&md5=3acd0b0a59832d107e743a92afdca9ad" title="Click to view the MathML source">1<p<∞mi>pmi><∞ let A_p(G)mi>Ami><mi>pmi>(<mi>Gmi>) be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)mi>Ami>(<mi>Gmi>) (p=2mi>pmi>=2). It is shown that for any amenable group H  , a proper affine map 14209f12fcdb85" title="Click to view the MathML source">伪:Y⊂H→Gmi>伪mi>:<mi>Ymi>⊂<mi>Hmi>→<mi>Gmi> induces a p  -completely contractive algebra homomorphism 蠒_伪:A_p(G)→A_p(H)mi>蠒mi><mi>伪mi>:<mi>Ami><mi>pmi>(<mi>Gmi>)→<mi>Ami><mi>pmi>(<mi>Hmi>) by setting 蠒_伪(u)=u鈭樜?/span>mi>蠒mi><mi>伪mi>(<mi>umi>)=<mi>umi>鈭?/mo><mi>伪mi> on Y   and 蠒_伪(u)=0mi>蠒mi><mi>伪mi>(<mi>umi>)=0 off of Y  . Moreover, we show that if both G   and H   are amenable then any p  -completely contractive algebra homomorphism 蠒:A_p(G)→A_p(H)mi>蠒mi>:<mi>Ami><mi>pmi>(<mi>Gmi>)→<mi>Ami><mi>pmi>(<mi>Hmi>) is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2mi>pmi>=2) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.