For a locally compact group G and an arbitrary subset J of [1, ∞], we introduce IL J (G) as a subspace of p∈J L p (G) with some norm to make it a Banach space. Then, for some special choice of J, we investigate some topological and algebraic properties of IL J (G) as a Banach algebra under a convolution product.2010 Mathematics subject classification: primary 43A07; secondary 43A15, 43A20.

Let (X, µ) be a measure space. For p, q ∈ (0, ∞] and arbitrary subsets P, Q of (0, ∞], we introduce and characterize some intersections of Lorentz spaces, denoted by IL p,Q (X, µ), IL J,q (X, µ) and IL J,Q (X, µ).2000 Mathematics Subject Classification. Primary 43A15, Secondary 43A20.

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