I am learning operator algebras and the book (Kadison-Ringrose, vol 1) which I'm following just introduced "factor". The definition is as follows:
(Factor) It is a von Neumann algebra whose center is $\Bbb{C}I$
Clearly $\Bbb{C}I$ is a factor and I've proved that whole $\mathcal{B}(\mathcal{H})$ is a factor too. But I cannot find enough example of factor. I know that for an von Neumann algebra $\mathfrak{A}$ the matrix algebra $M_n(\mathfrak{A})$ is also a von Neumann algebra of operators on $\mathcal{B}(\mathcal{H}\oplus\dots\oplus\mathcal{H})$. But we know the center of $M_n(\mathfrak{A})$ is given by: let's denote $Z(-)$ for "center" $$ Z(M_n(\mathfrak{A})) = M_n(Z(\mathfrak{A})) $$
So if $\mathfrak{A}$ is a factor then so is $M_n(\mathfrak{A})$. But it also doesn't help me to construct an example of factor because I have to verify whether $\mathfrak{A}$ is a factor or not!
So my question is:
Let $\mathcal{H}$ is given Hilbert space. What are some examples of non trivial factors in $\mathcal{B}(\mathcal{H})$?
Thanks.
There are plenty sort of "standard" ones that you'll see if you continue to study vN-algebras. First off, one can take group von Neumann algebras $L(G)$, which is the SOT-closure of the image of $\mathbb{C}G$ under the left regular representation $\lambda: G \to B(\ell^2(G))$. This is a factor if and only if the group is ICC (every non-trivial conjugacy class is infinite). So groups like $L(\mathbb{F}_2)$ and $L(S_{\infty})$ have the property ($S_{\infty}$ is the set of finitely supported bijections $\mathbb{N} \to \mathbb{N}$). These are non-isomorphic because the former is non-amenable, while the latter is. These are examples of $II_1$ factors (i.e., they have a tracial state)
Some other non-trivial ones are Powers' factors $R_\lambda$, $0 < \lambda < 1$. These are hyperfinite factors which are infinite tensor products of 2x2 matrix algebras (I won't get into the details of what infinite tensor products are, but essentially you need to have a vN-algebra and a state in mind for each one). For $0 < \lambda < 1$, you take $$R_\lambda = \otimes_1^{\infty} (M_2,\phi), $$ where $$\phi(x) = \text{tr}(hx), h = \begin{pmatrix} \frac{1}{\lambda + 1} & 0 \\ 0 & \frac{\lambda}{\lambda + 1} \end{pmatrix}. $$ These ones are hyperfinite type $III_\lambda$ factors. More general (at first glance) factors of this form are Araki-Woods infinite tensor products of finite type I.
If $\lambda = 1$ you just get the hyperfinite $II_1$ factor, which can be also realized as the SOT-closure of the image of the CAR algebra $A = M_{2^{\infty}} = \otimes_{\mathbb{N}} M_2$ (this is just a UHF algebra) under the GNS representation of associated to the unique trace. (amazingly, this is isomorphic to $L(S_{\infty})$).
Maybe they get into it in later volumes, I wouldn't know, but there is a ton of literature around this stuff. One of the standard sources is Takesaki's series of books Theory of operator algebras.