Hyperreflexivity Constants of the Bounded -Cocycle Spaces of Group Algebras and C*-Algebras

Abstract

<jats:p>We introduce the concept of strong property <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline2.png" /><jats:tex-math>$(\mathbb{B})$</jats:tex-math></jats:alternatives></jats:inline-formula> with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline3.png" /><jats:tex-math>$(\mathbb{B})$</jats:tex-math></jats:alternatives></jats:inline-formula> with a constant given by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline4.png" /><jats:tex-math>$288\unicode[STIX]{x1D70B}(1+\sqrt{2})$</jats:tex-math></jats:alternatives></jats:inline-formula>. We then use this result to find a concrete upper bound for the hyperreflexivity constant of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline5.png" /><jats:tex-math>${\mathcal{Z}}^{n}(A,X)$</jats:tex-math></jats:alternatives></jats:inline-formula>, the space of bounded <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline6.png" /><jats:tex-math>$n$</jats:tex-math></jats:alternatives></jats:inline-formula>-cocycles from <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline7.png" /><jats:tex-math>$A$</jats:tex-math></jats:alternatives></jats:inline-formula> into <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline8.png" /><jats:tex-math>$X$</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline9.png" /><jats:tex-math>$A$</jats:tex-math></jats:alternatives></jats:inline-formula> is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline10.png" /><jats:tex-math>$X$</jats:tex-math></jats:alternatives></jats:inline-formula> is a Banach <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline11.png" /><jats:tex-math>$A$</jats:tex-math></jats:alternatives></jats:inline-formula>-bimodule for which <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline12.png" /><jats:tex-math>${\mathcal{H}}^{n+1}(A,X)$</jats:tex-math></jats:alternatives></jats:inline-formula> is a Banach space. As another application, we show that for a locally compact amenable group <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline13.png" /><jats:tex-math>$G$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline14.png" /><jats:tex-math>$1&lt;p&lt;\infty$</jats:tex-math></jats:alternatives></jats:inline-formula>, the space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline15.png" /><jats:tex-math>$CV_{P}(G)$</jats:tex-math></jats:alternatives></jats:inline-formula> of convolution operators on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline16.png" /><jats:tex-math>$L^{p}(G)$</jats:tex-math></jats:alternatives></jats:inline-formula> is hyperreflexive with a constant given by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline17.png" /><jats:tex-math>$384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$</jats:tex-math></jats:alternatives></jats:inline-formula>. This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, <jats:italic>Math. Scand.</jats:italic> <jats:bold>50</jats:bold>(1) (1982), 111–122] for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000089_inline18.png" /><jats:tex-math>$p=2$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>

Strongly zero product determined Banach algebras

“…In this section we prove that the group algebra L 1 (G) of each locally compact group G is strongly zero product determined and we provide an estimate of the constants α L 1 (G) and β L 1 (G) . Our estimate of β L 1 (G) improves the one given in [7]. For the basic properties of this important class of Banach algebras we refer the reader to [12,Section 3.3].…”

“…From [11,Corollary 1.3], it follows that each C * -algebra A is strongly zero product determined, has the strong property B, and α A , β A ≤ 8. It is shown in [7] that each group algebra has the strong property B and so (by Corollary 2.2 below) it is also strongly zero product determined. In this paper we give, for each group algebra, a sharper estimate for the constant of the strong property B to the one given in [7,Theorem 3.4].…”

“…It is shown in [7] that each group algebra has the strong property B and so (by Corollary 2.2 below) it is also strongly zero product determined. In this paper we give, for each group algebra, a sharper estimate for the constant of the strong property B to the one given in [7,Theorem 3.4]. Finally, we prove that the algebra A(X) is strongly zero product determined for each Banach space X having property (A) (which is a rather strong approximation property for the space X).…”

“…The inequality |ϕ| zp ≤ M |ϕ| b is always true for some constant M (Proposition 2.1 below). The spirit of this concept first appeared in [5], and was subsequently formulated in [6,7]. This property has proven to be useful to study the hyperreflexivity of the spaces of derivations and continuous cocycles on A (see [6,7,8,9,10]).…”

Strongly zero product determined Banach algebras

“…The inequality |ϕ| zp ≤ M |ϕ| b is always true for some constant M (Proposition 2.1 below). The spirit of this concept first appeared in [6], and was subsequently formulated in [14] and refined in [15]. This property has proven to be useful to study the hyperreflexivity of the spaces of continuous derivations and, more generally, continuous cocycles on A (see [7,8,[13][14][15]).…”