A short note on hereditary Mazur intersection property
In this note, we prove that hereditary Mazur intersection property (MIP) does not imply Fr\'echet smoothness using an example by Borwein and Fabian (1993).
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arXiv subjects
Deepak Gothwal
Publications and source records attributed to Deepak Gothwal.
Semi denting points and related notions in Banach spaces
In this work, we study semi denting points and related notions in Banach spaces. We observe that $X$ has the Radon-Nikod\'ym Property if and only if every closed bounded convex set has a semi denting point. We also study the stability properties of semi denting, semi PC, and semi SCS points, as well as their $w^*$-analogues in Banach spaces, with respect to $l_p$-sums ( $1\leq p \leq \infty$), ideals, and projective tensor products.
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Some Geometric Aspects Related to Lim's Condition
In their seminal work, Lau and Mah (1986) study $w^*$-normal structure in the space of operators $\mathcal{L}(H)$, on a Hilbert space $H$, using a geometric property of the dual unit ball called Lim's condition. In this paper, we study a weaker form of Lim's condition, which we call property ($\ddagger$), for $C^\ast$-algebras, uniform algebras, and $L^1$-predual spaces. In the case of a $C^\ast$-algebra, we prove that property $(\ddagger)$ is equivalent to Lim's condition and consequently, we obtain a geometric characterization of $C^*$-algebras which are $c_0$-direct sum of finite-dimensional operator spaces. For a uniform algebra, we extend a result of Lau and Mah to show that property $(\ddagger)$ implies that the space is finite-dimensional. In the case of an $L^1$-predual space, we show that this condition implies $k$-smoothness of the norm in the sense considered in Lin and Rao (2007).
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(Asymptotic) uniform smoothness, ball separation and residuality results
In this article, we discuss a ball separation characterisation of asymptotically uniformly smooth (AUS) norms. We use this characterisation to prove the residuality of the set of equivalent AUS norms. We discuss similar residuality results for uniformly smooth norms and norms with uniform Mazur intersection property (UMIP).
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More on the (Uniform) Mazur Intersection Property
In this paper, we introduce two moduli of w*-semidenting points and characterise the Mazur Intersection Property (MIP) and the Uniform MIP (UMIP) in terms of these moduli. We show that a property slightly stronger than UMIP already implies uniform convexity of the dual. This may lead to a possible approach towards answering the long standing open question whether the UMIP implies the existence of an equivalent uniformly convex renorming. We also obtain the condition for stability of the UMIP under $\ell_p$-sums.
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The Generalised (Uniform) Mazur Intersection Property
Given a family $\mathcal{C}$ of closed bounded convex sets in a Banach space $X$, we say that $X$ has the $\mathcal{C}$-MIP if every $C \in \mathcal{C}$ is the intersection of the closed balls containing it. In this paper, we introduce a stronger version of the $\mathcal{C}$-MIP and show that it is a more satisfactory generalisation of the MIP inasmuch as one can obtain complete analogues of various characterisations of the MIP. We also introduce uniform versions of the (strong) $\mathcal{C}$-MIP and characterise them analogously. Even in this case, the strong $\mathcal{C}$-UMIP appears to have richer characterisations than the $\mathcal{C}$-UMIP.
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On uniform Mazur intersection property
In this paper, we show that a Banach space $X$ has the Uniform Mazur Intersection Property (UMIP) if and only if every $f \in S(X^*)$ is uniformly w*-semidenting point of $B(X^*)$. We also prove analogous results for uniform w*-MIP.
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