• 대한수학회보
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• 제31권2호
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• pp.173-180
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• 1994

Let H be a separable, infinite dimensional, compled Hilbert space and let L(H) be the algebra of all bounded linear operators on H. A dual algebra is a subalgebra of L(H) that contains the identity operator $I_{H}$ and is closed in the ultraweak topology on L(H). Note that the ultraweak operator topology coincides with the wea $k^{*}$ topology on L(H)(see [3]). Bercovici-Foias-Pearcy [3] studied the problem of solving systems of simultaneous equations in the predual of a dual algebra. The theory of dual algebras has been applied to the topics of invariant subspaces, dilation theory and reflexibity (see [1],[2],[3],[5],[6]), and is deeply related with properties ( $A_{m,n}$). Jung-Lee-Lee [7] introduced n-separating sets for subalgebras and proved the relationship between n-separating sets and properties ( $A_{m,n}$). In this paper we will study the relationship between direct sum and properties ( $A_{m,n}$). In particular, using some results of [7] we obtain relationship between n-separating sets and direct sum of von Neumann algebras.ras.s.ras.

• 대한수학회보
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• 제22권1호
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• pp.7-10
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• 1985

For each a.mem.S and f.mem.m(S), denote by $l_{a}$ f(s)=f(as) for all s.mem.S. If A is a norm closed left translation invariant subalgebra of m(S) (i.e. $l_{a}$ f.mem.A whenever f.mem.A and a.mem.S) containing 1, the constant ont function on S and .phi..mem. $A^{*}$, the dual of A, then .phi. is a mean on A if .phi.(f).geq.0 for f.geq.0 and .phi.(1) = 1, .phi. is multiplicative if .phi. (fg)=.phi.(f).phi.(g) for all f, g.mem.A; .phi. is left invariant if .phi.(1sf)=.phi.(f) for all s.mem.S and f.mem.A. It is well known that the set of multiplicative means on m(S) is precisely .betha.S, the Stone-Cech compactification of S[7]. A subalgebra of m(S) is (extremely) left amenable, denoted by (ELA)LA if it is nom closed, left translation invariant containing contants and has a multiplicative left invariant mean (LIM). A semigroup S is (ELA) LA, if m(S) is (ELA)LA. A subset E.contnd.S is left thick (T. Mitchell, [4]) if for any finite subser F.contnd.S, there exists s.mem.S such that $F_{s}$ .contnd.E or equivalently, the family { $s^{-1}$ E : s.mem.S} has finite intersection property.y.

• 대한수학회지
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• 제46권3호
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• pp.657-673
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• 2009

The fixed point algebra $C^*(E)^{\gamma}$ of a gauge action $\gamma$ on a graph $C^*$-algebra $C^*(E)$ and its AF subalgebras $C^*(E)^{\gamma}_{\upsilon}$ associated to each vertex v do play an important role for the study of dynamical properties of $C^*(E)$. In this paper, we consider the stability of $C^*(E)^{\gamma}$ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that $C^*(E)^{\gamma}$ is stably isomorphic to a graph $C^*$-algebra $C^*(E_{\mathbb{Z}}\;{\times}\;E)$ which we observe being stable. We first give an explicit isomorphism from $C^*(E)^{\gamma}$ to a full hereditary $C^*$-subalgebra of $C^*(E_{\mathbb{N}}\;{\times}\;E)({\subset}\;C^*(E_{\mathbb{Z}}\;{\times}\;E))$ and then show that $C^*(E_{\mathbb{N}}\;{\times}\;E)$ is stable whenever $C^*(E)^{\gamma}$ is so. Thus $C^*(E)^{\gamma}$ cannot be stable if $C^*(E_{\mathbb{N}}\;{\times}\;E)$ admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue $\lambda$ > 1. The AF algebras $C^*(E)^{\gamma}_{\upsilon}$ are shown to be nonstable whenever E is irreducible. Several examples are discussed.