We study several classes of Banach bimodules over a II$_1$ factor $M$, endowed with topologies that make them ``smooth'' with respect to $L^p$-norms implemented by the trace $\tau$ on $M$. In particular, letting $M\subset \B= \B(L^2M)$, and $2\leq p < \infty$, we first consider the space $\B(p)$, obtained as the completion of $\B$ in the norm $|||T|||_p $ defined as the supremum of $|\varphi(T)|$ over all linear functionals $\varphi \in \B^*$ such that $|\varphi(xYz)| \leq \|x\|_p \|Y\| \|z\|_p$, for all $x,y \in M$ and $Y \in \B$. We continue by examining the subspace $\K(p)\subset \B(p)$, obtained as the closure in $\B(p)$ of the space of compact operators $\K(L^2M)$. We then construct the space $\K_p\subset \B$ of operators that are $||| \ |||_p$-limits of bounded sequences of operators in $\K(L^2M)$. We prove that $\K_p$ are all equal to the \textit{$\tau$-rank-completion} of $\K(L^2M)$ in $\B$, denoted as $\text{q}\K_M$, which is the subspace of all operators $K \in \B(L^2M)$ such that there exists a sequence operators $K_n \in \K(L^2M)$ and projections $p_n\in \mathcal{P}(M)$ with $\lim_n \|p_n(K-K_n)p_n\|= 0$ and $\lim_n\tau(1-p_n)=0$. We conclude by proving that any separable II$_1$ factor $M$ admits non-inner derivations into $\text{q}\K_M$, but that any derivation $\delta:M \rightarrow \text{q}\K_M$ is a pointwise limit in $\tau$-rank-topology of inner derivations. This gives us an example of a ``non-vanishing" theory of 1-cohomology for II$_1$ factors $M$.

The work in this thesis, which is based to a large extent on a joint paper with Professor

Popa [GP14], provides a systematic study of derivations of von Neumann algebras M with values in Banach M -bimodules that satisfy a combination of smoothness and operatorial conditions. A typical example of such a smooth bimodule for M is the ideal of compact operators K(H) on the Hilbert space H on which M acts, a case that was fully analyzed in [JP72; Pop84], where it was shown that any derivation of M into K(H) is inner, i.e. it is implemented by an element in K(H).

However, we show that the class of smooth subbimodules of B(H) is much larger. We then provide a far reaching generalizations of the results in [JP72, Pop84], by showing that given any norm-closed smooth M -subbimodule B0 B(H), any derivation of M into B0

is inner. Moreover, we prove that if the von Neumann algebra M is either properly infnite, or it is finite but has \good decomposability" properties (e.g, if it contains a diuse amenable wq-normal von Neumann subalgebra), then given any abstract Banach M -bimodule B_0 satisfying the operatorial and smoothness conditions, all derivations of M into B0 are inner.

In this thesis, we establish a sufficient condition for an amenable von Neumann algebra to be a maximal amenable subalgebra of an amalgamated free product von Neumann algebra. In particular, if $P$ is a diffuse maximal amenable von Neumann subalgebra of a finite von Neumann algebra $N_1$, and $B$ is a von Neumann subalgebra of $N_1$ with the property that no corner of $P$ embeds into $B$ inside $N_1$ in the sense of Popa's intertwining by bimodules, then we conclude that $P$ is a maximal amenable subalgebra of the amalgamated free product of $N_1$ and $N_2$ over $B$, where $N_2$ is another finite von Neumann algebra containing $B$. To this end, we utilize Popa's asymptotic orthogonality property. We also observe several special cases in which this intertwining condition holds, and we note a connection to the Pimsner-Popa index in the case when we take $P=N_1$ to be amenable.

The purpose of this dissertation is to study some new notions of equivalence

of measure preserving group actions on probability spaces. These

include approximate versions of conjugacy and orbit equivalence (OE) as

well as a notion of weak OE, which is engendered by the notion of weak

conjugacy that was recently introduced by A. Kechris. We study the relationship

between our new notions and the classical ones. For instance,

we show that approximate conjugacy is the same as conjugacy for actions

of groups with Kazhdan's property (T) whereas for groups with infinite

amenable quotients these notions are very much distinct. We also use

results of Monod-Shalom, Kida, and Chifan-Kida to deduce superrigidity

results within the paradigm of approximate conjugacy and OE. Moreover,

we show that a number of invariants and properties are preserved by weak

OE, including strong ergodicity. This allows us to deduce that any nonamenable

group without Kazhdan's property (T) has at least two weakly

orbit inequivalent actions. This dissertation is largely based on the author's

joint work with Professor Sorin Popa in the paper [AP15].

We use malleable deformations combined with spectral gap rigidity theory, in the framework

of Popas deformation/rigidity theory to prove unique tensor product decomposition results

for II1 factors arising as tensor product of wreath product factors and free group factors. We

also obtain a similar result regarding measure equivalence decomposition of direct products

of such groups.

We use deformation-rigidity theory in the von Neumann algebra framework to study probability measure preserving actions by wreath product groups and their associated von Neumann

algebras. In particular, we single out large families of wreath product groups satisfying various types of orbit equivalence (OE) rigidity.