We introduce a new category of Banach algebras, l 1 -Munn algebras which we use as a tool in the study of semigroup algebras. Then we characterize amenable l 1 -Munn algebras and also semisimple ones in this category. Applying these results to the semigroup algebras provides some characterizations of amenable semigroup algebras. We also provide a counter example to a conjecture of Duncan and Paterson.1999 Academic Press

We introduce the notions of approximate Connes-amenability and approximate strong Connes-amenability for dual Banach algebras. Then we characterize these two types of algebras in terms of approximate normal virtual diagonals and approximate σW C−virtual diagonals. We investigate these properties for von Neumann algebras and measure algebras of locally compact groups. In particular we show that a von Neumann algebra is approximately Connes-amenable if and only if it has an approximate normal virtual diagonal. This is the "approximate" analog of the main result of Effros in [E. G. Effros, Amenability and virtual diagonals for von Neumann algebras, J. Funct. Anal. 78 (1988), 137-153].We show that in general the concepts of approximate Connes-ameanbility and Connes-ameanbility are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of A * * implies approximate Connes-amenability or approximate amenability of A are also discussed.2010 Mathematics Subject Classification. Primary 46H25, 46H20; Secondary 46H35.

In this paper we investigate algebraic structure of quasi operator spaces and quasi operator systems. We call a subspace of a unital complex * -algebra A [resp. self-adjoint subspace of A containing 1 A ] a quasi operator space [resp. quasi operator system]. Our keys in this investigation are the bounded subalgebra A 0 of A, a C * -seminorm on A 0 , and a new notion of algebraic bound.Our main goal is to show that many of fundamental results concerning operator systems are of algebraic nature. We obtain an algebraic characterization of operator systems which improves a classic result due to Choi-Effros. Moreover we show that A 0 is the largest T * -subalgebra of A. Then we extend the classical relationships among positivity, boundedness, complete positivity and complete boundedness to quasi operator systems. Schwarz inequality for 2-positive maps and Smith's theorem are also extended to quasi operator spaces.Several examples are provided to show the relations and distinctions of various classes of spaces under consideration, with their classic analogs and also to show up to what extent our conclusions are true.

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate n-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs.Our main result is that under some mild conditions on a given Banach algebra A, if its second dual A * * is (2n − 1)-weakly [respectively approximately/ approximately weakly/ approximately n-weakly] amenable, then so is A. Also if A is approximately (n + 2)-weakly amenable, then it is approximately n-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability.

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