# Tagged Questions

A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a Hilbert space, closed in the weak operator topology. Also called a $W^*$-algebra, and may be regarded as a non-commutative generalization of $L^\infty$ space. These algebras are extensively used in knot theory, ...

5 views

### Group von Neumann algebras

I have a question about group von Neumann algebras structure. If $L(G)$ is a subset of $L(H)$, can we find a subgroup $G_1$ of $H$ such that $L(G_1)$ be isomorphic to $L(G)$? I appreciate any help.
27 views

### Lemma V.1.25 (page 298) from the book of Takesaki (Vol 1).

I have a problem with the following lemma (Takesaki Vol 1-page 298): Lemma 1.25. If $e$ is an abelian projection in a von Neumann algebra $\mathcal M$, then for any projection $f\in\mathcal M$ ...
23 views

### amalgamated free product of von Neumann algebras

If $G$ and $H$ are two discrete groups and $L(G)$ and $L(H)$ be their group von Neumann algebras and $A$ be their common *-subalgebra, what can we say about their amalgamated free product under $A$, i....
32 views

### What is the dual space of a von Neumann algebra?

What is the dual space of a von Neumann algebra $\mathcal{M}$? Does it have any specific form? Or just $\mathcal{M}^*$.
35 views

### Is $p \vee q \leq p+q$ for $p,q$ projections?

I am wondering if $p \vee q \leq p+q$ for $p,q$ projections acting on some Hilbert space $H$. In particular, I wonder if the set of finite trace projections is upwards directed with the usual ...
32 views

### Uncountable sequence $R_1 \subset R_2 \subset …$ of von Neumann algebras acting on separable $H$?

Can there exist an uncountable sequence $R_1 \subset R_2 ...$ of von Neumann algebras all acting on the same separable Hilbert space $H$, with a "limit" algebra $R$ such that $R_\alpha \subset R$ for ...
17 views

### Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
44 views

### $A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
24 views

41 views

### if $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$?

Suppose that $A_n,A$ are self-adjoint operators in $B(H)$. If $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$? From Proposition. 2.3.2 of Pederson'book, I know the result holds in ...
70 views

### Trace and norm bounded sequence of positive elements has convergent subsequence in hyperfinite $II_1$ factor

Let $A$ is a hyperfinite $\operatorname{II_1}$ factor and $x_n \in A$ is some sequence of positive elements such that $||x_n||$ convergent and $\operatorname{Tr}(x_n^2) = 1$ (where $\operatorname{Tr}$ ...
41 views

### Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
Given a Hilbert space $\mathcal{H}$. Consider a von Neumann algebra: $$M\subseteq\mathcal{B}(\mathcal{H}):\quad M=M''$$ Suppose a cyclic vector: $$\Omega\in\mathcal{H}:\quad\overline{\mathcal{M}\... 1answer 24 views ### Strong convergence of Spectral Projection Let H be a Hilbert space and B(H) be the space of all bounded linear operators on H. Assume that \{A_n\in B(H)\}_n strongly converges to A. E^{|A|}(1,\infty) is a spectral projection of ... 1answer 20 views ### separable weakly dense subalgebra of a von Neumann algebra This is an exercise of "A short course on spectral theory" by William Arveson. For any von Neumann algebra \mathcal{M} on a Hilbert space H, I need to show there exists a unital C^\ast-... 0answers 16 views ### Characterization of noncommutative L^2-spaces as ordered vector spaces If M is a von Neumann algebra and \tau\colon M_+\to[0,\infty] is a normal, semi-finite, faithful trace, the associated GNS Hilbert space is the completion of \{x\in M\mid \tau(x^\ast x)<\infty\... 1answer 30 views ### predual of von Neumann algebra I know that each separable finite von Neumann has separable predual. Can some one give me an example of a non-separable finite von Neumann algebra with separable predual? 0answers 17 views ### Automorphisms of group von Neumann algebras I study group von Neumann algebras L(G), and I extremely want to know about automorophism (groups) of these algebras. Is there any good reference about this? I appreciate of everybody help. 0answers 46 views ### Elements of C(K)^{**}, do they have a name? Let K be a compact (Hausdorff) space, and let C(K) be the Banach algebra of contunous functions on K (with the usual \sup-norm). The enveloping von Neumann algebra of C(K) is its second dual ... 1answer 23 views ### Types of W*- algebras Searching to find a reference for "homogeneous type I_{{\aleph}_0} W*-algebra", I was not successful. Please guide me. Thanks in advance. 1answer 56 views ### What is the motivation for Murray-von Neumann equivalence Definition: If p,q are projections in a C^*-algebra A, we say that they are Murray-von Neumann equivalent, and we write p\sim q, if there exist u\in A such that p=u^*u and q=uu^*. I ... 1answer 105 views ### Is there an example of a non von Neumann algebra with this property? What is an example of a C^{*} subalgebra A of B(H) such that A contains the identity I_{H} and satisfies the following properties: 1) For every T\in A, The orthogonal projection \... 1answer 25 views ### When set of projections is closed under multiplication Let \mathcal A\subset B(H) be an unital C^* algebra of operators on a Hilbert space H. Let's denote by \mathcal P the set of projections in \mathcal A, that is \mathcal P:=\{a\in \mathcal A ... 0answers 51 views ### Is the algebra of universally integrable functions a von Neumann algebra? I would like to continue this discussion. Let X be a compact space. Let us call a function f:X\to {\mathbb C} universally integrable if it is integrable with respect to each regular Borel ... 1answer 30 views ### Are Baer Rings worth studying for significant modern progress in vNa's? I am wondering if Kaplansky's "Ring of Operators" is worth studying if I'm interested in functional analysis (more specifically von Neumann algebras). Yes any vNa is a B. ring but my question is more ... 1answer 15 views ### Do a von Neumann Algebra's subalgebras have unique “complements”? Let A \subset B \subset C be von Neumann algebras (and more specifically factors, if it helps) either all type II or all type III, acting on the same separable Hilbert space. Call A' \subset B ... 0answers 21 views ### Representations of SU_q(n) I am searching for a classification of all irreducibel representations of the quantum group SU_q(n) for general n. Can someone give referenced or some statements about this? Moreover does one has ... 0answers 35 views ### Atomic W*- algebra Let A be a C*-algebra. Put z: = \sup\{ e\in A^{**} ;\text{ e is a minimal projection}\}. Easily can see z is a central projection. Set M:= A^{**}z. 1) Is M an atomic W*-algebra in general? ... 1answer 57 views ### Approximate unit for a certain C*-algebra Let A be a C*-algebra and p a projection in A^{**}. To prove p is the smallest unit for B: = \{a\in A; pap=a\}, suppose \{u_i\} is an approximate unit for B. It's easy to see q: = w^*-\... 0answers 91 views ### Comparing two positive linear functionals when support(\psi)\leq support(\phi) The following point has been addressed in the significant paper "order ideal in C*-algebras and its dual (By E. Effros - Lemma 4.1). The reference is Dixmier's book (1957 French). Unfortunately, I ... 1answer 35 views ### Atomic W*-algebras I am looking for any information concerning atomic W*-algebras. Def. A W*-algebra M is called atomic if for any projection p in M there is a minimal projection e\in M with e\leq p. Q. ... 1answer 40 views ### A certain *-isomorphism let A be a C*-algebra and z\in A^{**} the supremum of all the minimal projections in A^{**}. How can show * - homomorphism A\to zA\subset zA^{**} is injective? 1answer 39 views ### Two approaches on the support of functionals I am writing concerning support of functionals on C*-algebras. I feel there are two different definitions for this notion. Let A be a C*-algebra and \phi be a positive linear functional on A. ... 1answer 40 views ### A review of an example given by C. A. Akemann I am witting concerning the example II.6 given in the paper "The General Stone-Weierstrass Problem" by C. A. Akemann: Let me review this example. Let H_i be the two dimensional Hilbert space \... 1answer 31 views ### Jones construction projections Let M be a von Neumann algebra with faithful normal normalized trace tr. Let \{ e_i | i=1,2,\dots \} be projections in M such that: e_ie_{i \pm 1}e_i=\tau e_i  for some \tau \leq 1 e_ie_j=... 0answers 40 views ### The left kernel of a positive linear functional and its w^*-extention Let A be a C*-algebra and \phi be a positive linear functional on A. We let \tilde{\phi} be its unique w^*-continuous extension on A^{**}. It is supposed to focus on the left kernel of \... 1answer 36 views ### A remark on projections in von Neumann algebras Let M be a von Neumann algebra and e,f be projections in M. For a given central projection z\in M, is the following true?$$z(e\vee f)=ze\vee zf$$1answer 50 views ### The point-wise closure of the space of continuous functions Let X be a locally compact space and consider C_0(X). We denote b(X), by the set of all bounded functions on X. It is easy to be checked that b(X) may be considered as a C*-sub algebra of ... 0answers 28 views ### Multiplicity free representation contain irreducible representation (for type I representation)? While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central ... 0answers 16 views ### Which open projections are support of closed left ideals in a C*-algebra Let A be a C*-algebra. Let \phi be a positive linear functional on A. The support of \phi is defined by the smallest projection q\in A^{**} with \phi(q)=||\phi|| and in the literature ... 1answer 34 views ### Support of a faithful representation Let A be a C*-algebra. Any representation \pi:A\to B(H) is uniquely extended to a w^*-continuous representation \tilde{\pi}:A^{**}\to B(H). Q: I am looking for an example of a faithful ... 0answers 55 views ### Is the bilinear map M\times M^*\to M^* jointly continuous? Let M be a W*-algebra and consider the following map:$$\gamma: M\times M^*\to M^*: (a,f)\to af where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with ...