Topologies of compact families on the ideal space of a Banach algebra

• Autorzy: Ferdinand Beckhoff
• Języki publikacji: EN

Abstrakty

EN

Let 𝐾 be a family of compact sets in a Banach algebra A such that 𝐾 is stable with respect to finite unions and contains all finite sets. Then the sets $U(K) := {I ∈ Id(A): I ∩ K = ∅}$, K ∈ 𝐾 define a topology τ(𝐾) on the space Id(A) of closed two-sided ideals of A. 𝐾 is called normal if $I_i → I$ in (Id(A),τ(𝐾)) and x ∈ A╲I imply $lim inf_i∥x + I_i∥ > 0$. (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then Id(A) is second countable. (2) If the family of countable compact sets is normal and A is separable then there is a countable subset S ⊂ A such that for all closed two-sided ideals I we have $\overline{I ∩ S} = I$. Examples are separable C*-algebras, the convolution algebras $L^p(G)$ where 1 ≤ p < ∞ and G is a metrizable compact group, and others; but not all separable Banach algebras share this property.

Kategorie tematyczne

• 46H10: Ideals and subalgebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 118
• Numer: 1
• Strony: 63-75

Daty

wydano

1996

otrzymano

1995-06-05

poprawiono

1995-09-18

Twórcy

autor

Ferdinand Beckhoff

• Mathematisches Institut der Universität Münster, Einsteinstr. 62, 48149 Münster, Germany

Bibliografia

• [1] R. J. Archbold, Topologies for primal ideals, J. London Math. Soc. (2) 36 (1987), 524-542
• [2] F. Beckhoff, Topologies on the space of ideals of a Banach algebra, Studia Math. 115 (1995), 189-205.
• [3] B. Blackadar, Weak expectations and nuclear C*-algebras, Indiana Univ. Math. J. 27 (1978), 1021-1026
• [4] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer, 1970.
• [5] J. L. Kelley, General Topology, Springer, 1955.
• [6] D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), 39-52.