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 $\|\cdot\|_2$-Norm that *do* occur regularly 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.

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