First, let me preface this by saying that I am fairly new to the wide field of (finite) von Neumann algebras. In my studies of $L^2$-invariants, I am mostly concerned with Group von Neumann algebras, but it will be very helpful to gain some more insight into general finite von Neumann algebras.


Let $\mathcal A$ be a von Neumann algebra, i.e a unital, weakly closed $*$-subalgebra of $B(\mathcal H)$, the space of bounded operators over some Hilbert space $\mathcal H$, and assume additionally that $\mathcal A$ is equipped with a *finite*, *positive*, *faithful* and *normal* trace $\phi: \mathcal A \to \mathbb C$, such that, wlog, $\phi(1) = 1$. Denote by $||.||_1$ the operator norm on $B(\mathcal H)$. 

Then, $\phi$ determines an inner product on $\mathcal A$ given by $\langle a,b \rangle := \phi(ab^*)$ for any pair $a,b \in \mathcal A$. Denote the induced *tracial norm* on $\mathcal A$ by $||.||_\phi$ and by $l^2(\mathcal A)$ the Hilbert space completion of $\mathcal A$ with respect to $\langle \;,\; \rangle$. Assume that $l^2(\mathcal A)$ is infinite-dimensional and seperable, and denote by $||.||_2$ the operator norm on $B(l^2(\mathcal A))$. Then left-multiplication by elements in $\mathcal A$ identifies elements of $\mathcal A$ with certain bounded operators over $l^2(\mathcal A)$, giving rise to the well-known *left-regular representation* $L: \mathcal A \hookrightarrow B(l^2(\mathcal A))$, an embedding satisfying $||a||_\phi \leq ||L(a)||_2 \leq ||a||_1$ for all $a \in \mathcal A$.

Of course, in most circumstances, one will not have an inequality of the form $||L(a)||_2 \leq C||a||_\phi$ for all $a \in \mathcal A$ and for some $C > 0$, but there are certain relations between the tracial norm and the $||.||_2$-Norm that *do* occur regularily and are of very special use to me. Let me list three properties.

(1) There exists an *orthonormal unitary system* of $\mathcal A$, i.e, an orthonormal base $(b_i)_{i = 1}^\infty \subset \mathcal A$ of $l^2(\mathcal A)$, such that $L(b_i) \in B(l^2(\mathcal A))$ is a unitary operator.

(2) There exists an orthonormal base $(b_i)_{i = 1}^\infty \subset \mathcal A$ of $l^2(\mathcal A)$ such that $||L(b_i)||_2 = ||b_i||_\phi = 1$.

(3) There exists an orthonormal base $(b_i)_{i = 1}^\infty \subset \mathcal A$ of $l^2(\mathcal A)$, such that $\sup_{i \in \mathbb N}||L(b_i)||_2 < \infty$.

Of course $(1) \Rightarrow (2) \Rightarrow (3)$ and all von Group von Neumann algebras $\mathcal A = \mathcal N(G)$ with $G$ a countably infinite group already trivially satisfy $(1)$ (with the obvious choice for $\phi$ and orthonormal base).

*Do there exist finite von Neumann algebras that satisfy (2), but not (1), and/or (3), but not (2) ? Do there exist finite von Neumann algebras that do not even satisfy (3). If so, are they classified (up to some equivalence) ?*

*Do there exist finite von Neumann algebras that satisfy (1), but are not Group von Neumann algebras ?*

Any help is appreciated.