Positive Type and Positive Definite Functions on Matrix Valued Group Algebras
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Results in Mathematics
Positive Type and Positive Definite Functions on Matrix Valued Group Algebras Ali Jabbari Dedicated to Professor Ali Ebadian
Abstract. Let G be a locally compact group equipped with the left Haar measure mG , Mn be an n × n matrix with entries in C and let L1 (G, Mn ) be the Banach algebra respect to the convolution products ∗ and ∗ that consists all Mn -valued functions on G. We define the left and right positive type functions on (L1 (G, Mn ), ∗) and (L1 (G, Mn ), ∗ ). Moreover, analogues to complex valued case, we construct two Hilbert spaces by the right and left positive type functions on (L1 (G, Mn ), ∗) and (L1 (G, Mn ), ∗ ) and we characterize the right and left positive type functions on (L1 (G, Mn ), ∗) and (L1 (G, Mn ), ∗ ). We also define positive definite functions on (L1 (G, Mn ), ∗) and (L1 (G, Mn ), ∗ ) and we show that any continuous right (left) positive function is positive definite and vice versa. Mathematics Subject Classification. 43A35. Keywords. Locally compact group, matrix valued function, positive definite function, positive type function.
1. Introduction Positive type and positive definite functions on locally compact groups are important objects. These subjects relations with the Fourier analysis appeared in the works of Eymard and Godement [10,13]. Moreover, for more information about the fundamental role of positive definite functions on locally compact groups we refer to [16,17,19,20]. Moreover, for some interesting results related to positive definite functions on (free generated) locally compact, see [1–3,7,8, 14]. The notion of vector-valued positive definite functions studied by Pedersen 0123456789().: V,-vol
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Results Math
[18], where the motivation for his study is to give a description for the positive cone of the dual space of the full crossed product of an action of a locally compact group G on a C ∗ -algebra. Vector-valued positive definite functions are very concrete and so in this paper we characterize these functions that their values are in Mn . Banach spaces (algebras) with values in the matrix Mn introduced and investigated by Chu [4] and some notable results in these spaces (algebras) are given in [5,6,11]. In the next section, we give some definitions and notations of matrix valued Banach spaces (algebras). In Sect. 3, we define the right and left positive type functions on L1 (G, Mn ) and characterize these functions, where we denote the set of these functions by Pr (G, Mn ) and P (G, Mn ), respectively. In Sect. 4, we define two Hilbert structures on L1 (G, Mn ) and we show that for a unimodular locally compact group G, values of any ϕ ∈ Pr (G, Mn ) lie in an n × n diagonal matrix. Similarly, we show that for a locally compact group G, values of any ϕ ∈ P (G, Mn ) lie in an n × n diagonal matrix. In Sect. 5, we define positive definite functions on (L1 (G, Mn ), ∗) and (L1 (G, Mn ), ∗ ). We prove that for a (unimodular) locally compact group G, any continuous (ϕ ∈ Pr (G, Mn )) ϕ ∈ P (G, Mn ) is positive definite a
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