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Commutative $AW^*$-algebra are characterized as those $C^*$-algebras such that their space of projections is a complete boolean algebra (see http://en.wikipedia.org/wiki/AW*-algebra).

Von Neumann algebras are characterized as those $C^*$-subalgebras $A$ of $B(H)$ satisfying any of these three equivalent conditions - see section 4.6 in Pedersen's book "Analysis Now" (specifically Theorem 4.6.7 and Proposition 4.6.15):

  • $A$ overlaps with its double commutant $A''$ (where $A'=\{T:ST=TS \, for \, all \, S\in A\}$).

  • $A$ is weakly closed ($A$ is closed in the topology generated by the seminorms $T\mapsto |(Tx,y)|$ for $x,y\in H$).

  • $A$ is $\sigma$-weakly closed ($A$ is closed in the topology generated by by the seminorms $T\mapsto |\sum_n(Tx_n,y_n)|$ for $x_n,y_n\in H$ such that $\sum \|x_n \|,\sum\|y_n\|<\infty$).

Let now for example $A$ be the commutative $AW^*$-algebra given by $[f]_I$ with $f:\mathbb{R}\to\mathbb{C}$ Borel and bounded and $I$ the ideal generated by functions with meager support. It is shown in Tristan Bice's answer to Examples of hyperstonean space that this $AW^*$-algebra does not carry normal positive functionals and thus it is not a Von Neumann algebra. However $A$ is a $C^*$-algebra so (By the GNS-construction) it has an isomorphic copy as a $C^*$-subalgebra of $B(H)$ (It may be the case that $H$ is a non-separable Hilbert space though, but I don't think this matters in what follows). Where does the double commutant Theorem (4.6.7) fails for this $AW^*$-subalgebra of $B(H)$? I'm a bit puzzled - I can expect that there is a problem with $\sigma$-weakly continuous functionals, but I don't see where the problem can arise with weakly continuous functionals.

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The first statement is not quite right - $C([0,1])$ is a commutative C*-algebra but not an AW*-algebra, even though its only projections are the constant 0 and 1 functions, which certainly constitutes a complete Boolean algebra. However, it is true that AW*-algebras are precisely the real rank zero C*-algebras whose projections form a complete lattice. In the commutative case this corresponds to the fact that a topological space is extremally disconnected iff it is totally disconnected and its clopen sets form a complete lattice. I should have mentioned this myself in the previous post you refer to.

To represent an abstract C*-algebra $A$ concretely on a Hilbert space $H$, we usually consider the universal representation $\pi=\bigoplus\pi_\phi$ where $\phi$ ranges over all states on $A$ and $\pi_\phi$ comes from the GNS construction, as you mention. I'm not sure if this answers your question, but you can show that $\pi[A]$ is not a von Neumann subalgebra of $B(H)$, i.e. $\pi[A]\neq\pi[A]''$, not just for the AW*-algebra you had in mind, but in fact for any infinite dimensional C*-algebra $A$. This is because you can identify $\pi[A]\subseteq\pi[A]''$ with $A\subseteq A^{**}$ and C*-algebras are reflexive if and only if they are finite dimensional (see Finite dimensional $C^*$-algebras). So, strange as it may seem, even a von Neumann algebra $A$ can be represented faithfully as a non-weakly closed subalgebra of $\mathcal{B}(H)$, as long as $A$ is infinite dimensional.

Alternatively, you could consider the atomic representation, where $\phi$ above ranges only over pure states, which is still faithful on $A$. But here too we can show that $\pi[A]\neq\pi[A]''$ whenever $A$ is infinite dimensional, by even constructing a projection $p\in\pi[A]''\setminus\pi[A]$ (which is perhaps more what you are looking for?). To see this, first note that it suffices to consider unital $A$ - otherwise $1\in\pi[A]''\setminus\pi[A]$. Now take a maximal commutative C*-subalgebra $B$ of $A$, which necessarily contains the unit and is also infinite dimensional. Identify $B$ with $C(X)$, where $X$ is the space of all pure states on $B$ with the weak*-topology. As $B$ is unital, $X$ is compact, and as $B$ is infinite dimensional, $X$ is infinite. But infinite discrete spaces are not compact so $X$ must contain a non-isolated point $\phi$, which can be extended to a pure state on all of $A$. Thus we have $v\in H$ with $\phi(a)=\langle\pi(a)v,v\rangle$, for all $a\in A$ (and $T\mapsto\langle Tv,v\rangle$ is certainly a weakly continuous functional on $\mathcal{B}(H)$). As $\{\pi(b):b\in B^1_+\text{ and }\phi(b)=0\}$ is directed, it has a supremum $p\in\pi[B]''$ with $\langle pv,v\rangle=0$. But $B$ is commutative so $\pi[B]\subseteq\pi[B]'$ and hence $\pi[B]''\subseteq\pi[B]'$. Thus if we had $q\in A$ with $\pi(q)=p$ then we would have $q\in B'$ and hence $q\in B$ by maximality. As $b\leq q$, for all $b\in B^1_+$ with $\phi(b)=0$, we must have $q=1$, as $\phi$ is not isolated. But $1=\phi(1)=\langle\pi(1)v,v\rangle=\langle\pi(q)v,v\rangle=\langle pv,v\rangle=0$, a contradiction. Thus $p\notin\pi[A]$.

To ensure that $\pi[A]=\pi[A]''$, we must restrict to a different class of states on $A$, namely the normal states. And if there are not enough of these (e.g. none in Dixmier's example) then $\pi$ will no longer be faithful.

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  • $\begingroup$ Thanks, very informative answer. Is there a simple definition of normal state or where do I find one? I've also come up with the following idea which I haven't written up since I do not know how much correct it was, however you might immediately say if it's a good direction. If $A=C(X)$ with $X$ extremely disconnected we can consider $D\subset X$ has a base for an Hilbert space $H_D$ and then assign a state (or projection $p_U:H_D\to H_D$ to any clopen set $U$ of $X$ by letting $p_U(x)=1$ off $x\in D\cap U$ and then extend by linearity. $\endgroup$ – matteo viale Apr 9 '15 at 16:14
  • $\begingroup$ Now the family $p_U$ describes the projections in the representation of $A$ and the double commutant of $A$ will instead include all possible projections of $H_D$ whose range is contained in spaces generated by unit vectors in an arbitrary subset of D. In this way the double commutant of $A$ is given by all projections whose range is a subspace of H_D generated by unit vectors in D which corresponds if I understand correctly thing to a hyperstonean space given by the ``atomic'' measures on $D$ (even if it is not totally clear to me what is an atomic measure on $D$ if $D$ is uncountable). $\endgroup$ – matteo viale Apr 9 '15 at 16:17
  • $\begingroup$ Your example (by Dixmier) of an $A=C(X)$ in your answer cited in my question gives an extremely disconnected $X$ which has a countable subset $D$. So in this case the double commutant of $A$ would really be $l^\infty(D)$ with the counting measure on $D$. Is this correct? $\endgroup$ – matteo viale Apr 9 '15 at 16:26
  • $\begingroup$ A state $\phi$ on a monotone complete C*-algebra $A$ is normal iff it preserves bounded directed supremums $\bigvee$, i.e. iff $\phi(\bigvee X)=\bigvee\phi(X)$ for any bounded directed subset $X$ of $A_\mathrm{sa}$ (see Pedersen's "C*-algebras and their Automorphism Groups" 3.9.2). And yes, I think you're on the right track with your other thoughts but again you don't need to restrict to commutative AW*-algebras/extremally disconnected spaces. $\endgroup$ – Tristan Bice Apr 13 '15 at 19:47
  • $\begingroup$ Specifically, for any subset $D$ of a compact Hausdorff space $X$, we have a representation $\pi$ of the C*-algebra $C(X)$ on the Hilbert space $l^2(D)$ given by $\pi(f)e_d=f(d)e_d$ (where $e_d$ is the canonical basis vector corresponding to $d\in D$, i.e. $e_d(d)=1$ while $e_d(x)=0$ otherwise). Then $\pi[A]''$ can be identified with $l^\infty(D)$, so it's projections are characteristic functions of arbitrary subsets of $D$, while the projections of $\pi[A]$ are precisely the characteristic functions of clopen subsets of $D$. $\endgroup$ – Tristan Bice Apr 13 '15 at 19:50

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