Comparison of topologies on *-algebras of locally measurable operators

We consider the locally measure topology $t(\mathcal{M})$ on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$. We prove that $t(\mathcal{M})$ coincides with the $(o)$-topology on $LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^*=T\}$ if and only if the algebra $\mathcal{M}$ is $\sigma$-finite and a finite algebra. We study relationships between the topology $t(\mathcal{M})$ and various topologies generated by faithful normal semifinite traces on $\mathcal{M}$.Comment: 21 page

Derivations on symmetric quasi-Banach ideals of compact operators

Let $\mathcal{I,J}$ be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J:I}$ be a space of multipliers from $\mathcal{I}$ to $\mathcal{J}$. Obviously, ideals $\mathcal{I}$ and $\mathcal{J}$ are quasi-Banach algebras and it is clear that ideal $\mathcal{J}$ is a bimodule for $\mathcal{I}$. We study the set of all derivations from $\mathcal{I}$ into $\mathcal{J}$. We show that any such derivation is automatically continuous and there exists an operator $a\in\mathcal{J:I}$ such that $\delta(\cdot)=[a,\cdot]$, moreover $\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to J}\leq 2C\|a\|_\mathcal{J:I}$, where $C$ is the modulus of concavity of the quasi-norm $\|\cdot\|_\mathcal{J}$. In the special case, when $\mathcal{I=J=K}(H)$ is a symmetric Banach ideal of compact operators on $H$ our result yields the classical fact that any derivation $\delta$ on $\mathcal{K}(H)$ may be written as $\delta(\cdot)=[a,\cdot]$, where $a$ is some bounded operator on $H$ and $\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to I}\leq 2\|a\|_{\mathcal{B}(H)}$.Comment: 21 page

Orlicz Spaces associated with a Semi-Finite Von Neumann Algebra

In the present paper we introduce a certain class of non commutative Orlicz spaces, associated with arbitrary faithful normal locally-finite weights on a semi-finite von Neumann algebra $M.$ We describe the dual spaces for such Orlicz spaces and, in the case of regular weights, we show that they can be realized as linear subspaces of the algebra of $LS(M)$ of locally measurable operators affiliated with $M.$Comment: 12 page

Warm Microhabitats Drive Both Increased Respiration and Growth Rates of Intertidal Consumers

Rocky intertidal organisms are often exposed to broadly fluctuating temperatures as the tides rise and fall. Many mobile consumers living on the shore are immobile during low tide, and can be exposed to high temperatures on calm, warm days. Rising body temperatures can raise metabolic rates, induce stress responses, and potentially affect growth and survival, but the effects may differ among species with different microhabitat preferences. We measured aerial and aquatic respiration rates of 4 species of Lottia limpets from central California, and estimated critical thermal maxima. In a variety of microhabitats in the field, we tracked body temperatures and measured limpet growth rates on experimental plates colonized by natural microalgae. Limpet species found higher on the shore had lower peak respiration rates during high temperature aerial exposure, and had higher critical thermal maxima. Using our long-term records of field body temperatures, we estimated cumulative respiration to be 5 to 14% higher in warm microhabitats. Growth rates in the field appear to be driven by an interaction between available microalgal food resources, low tide temperature, and limpet species identity, with limpets from warmer microhabitats responding positively to higher food availability and higher low tide temperatures. Stressful conditions in warm microhabitats make up a small portion of the total lifetime of these limpets, but the greater proportion of time spent at non-stressful, but warm, body temperatures may result in enhanced growth compared to limpets living in cooler microhabitats

Weighted ergodic theorems for Banach-Kantorovich lattice Lp(∇^,μ^)L_{p}(\hat{\nabla},\hat{\mu})

In the present paper we prove weighted ergodic theorems and multiparameter weighted ergodic theorems for positive contractions acting on $L_p(\hat{\nabla},\hat{\mu})$. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices.Comment: 11 page