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Register a new userOn weakly complete group algebras of compact groups
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We study group algebras for compact groups in the category of real and complex weakly complete vector spaces. We also show that the group algebra is a quotient of the weakly complete universal enveloping algebra of the Lie algebra of the compact group. We relate this to Tannaka duality, and to functorial properties of the group algebra. We determine the structure of the group algebra in terms of the irreducible representation, both in the real and the complex case. The particular case of a compact abelian group is worked out in detail.
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A weakly complete vector space over $mathbb{K}=mathbb{R}$ or $mathbb{K}=mathbb{C}$ is isomorphic to $mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $expcolon Ato A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $mathbb{K}[G]$ is associated functorially so that the functor $Gmapsto mathbb{K}[G]$ is left adjoint to $Amapsto A^{-1}$. The group algebra $mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $mathcal{L}(G)$ in $A_{rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,mathbb{R})$. The dual of the group algebra $mathbb{R}[G]$ is the Hopf algebra ${mathcal R}(G,mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${mathcal R}(G,mathbb{R})leftrightarrow mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.
Let $G$ be a compact group. For $1leq pleqinfty$ we introduce a class of Banach function algebras $mathrm{A}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in cite{forrestss1}. In the case $p ot=2$ we find that $mathrm{A}^p(G)cong mathrm{A}^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups. These algebras admit natural operator space structures, and also weighte
We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.
This paper concerns the study of regular Fourier hypergroups through multipliers of their associated Fourier algebras. We establish hypergroup analogues of well-known characterizations of group amenability, introduce a notion of weak amenability for hypergroups, and show that every discrete commutative hypergroup is weakly amenable with constant 1. Using similar techniques, we provide a sufficient condition for amenability of hypergroup Fourier algebras, which, as an immediate application, answers one direction of a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564, 2009] on the amenability of $ZL^1(G)$ for compact groups $G$. In the final section we consider Fourier algebras of hypergroups arising from compact quantum groups $mathbb{G}$, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact $mathbb{G}$ of Kac type.
Let $G$ be a compact connected Lie group. The question of when a weighted Fourier algebra on $G$ is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on $G$ with the order of growth strictly bigger than the half of the dimension of the group. The case of SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.
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