Abstract
Let G be a locally compact group equipped with a normalized Haar measure , A(G) the Fourier algebraof G and V N(G) the von Neumann algebra generated by the left regular representation of G. In this paper, we introduce the space V N(G;A) associated with the Fourier algebra A(G;A) for vector-valued functions on G, where A is a H-algebra. Some basic properties are discussed in the category of Banach space, and alsoin the category of operator space.
Highlights
The theory of rings of operators called today von Neumann algebra was first introduced and developed by Murray and von Neumann in 1936 [10], with the aim of developing a suitable mathematical framework for quantum mechanics
One can assign to a locally compact group G an operator algebra such that representations of the algebra are related to representations of the group
Any space constructed in this way is called group algebra
Summary
The theory of rings of operators called today von Neumann algebra was first introduced and developed by Murray and von Neumann in 1936 [10], with the aim of developing a suitable mathematical framework for quantum mechanics. Let Cc(G) be the space of complex-valued continuous functions on G with compact support; this acts on L2(G) by left convolution, and forms a ∗-subalgebra of Hom(L2(G)): {Λf: L2(G) ∋ g ⟼ f ∗ g ∈ L2(G), f ∈ Cc(G)} , , which closure is Cr∗(G), the reduced group C∗-algebra. For a locally compact group G, we denote by C∗(G, A) the (vector-valued) C∗-algebra of G, which is G the C∗-envelopping algebra of L1(G, A), i.e. the completion of Cc(G, A) with respect to the largest C∗-norm. The (vector-valued) reduced group C∗-algebra Cr∗(G, A) is the closure of the space T(G, A), with respect to the operator norm on B(L2(G, A)). Definition 4.4 Let G be a locally compact group, A an H∗-algebra and Cb(G, A) the space of all bounded continuous functions from G to A.
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