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Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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On direct integral of states of von Neumann algebras

Suppose we consider a direct integral of GNS states of a measure space in von Neumann algebra, get the new state by direct integral. Does the GNS represenation of the state breaks down into direct ...
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1answer
47 views

Quasinilpotent operator in finite von Neumann algebra

If the trace of all positive powers of a $n \times n$ complex matrix is $0$, then the matrix must be nilpotent. https://math.stackexchange.com/questions/159167/traces-of-all-positive-powers-of-a-...
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1answer
40 views

Subprojections of the sum of mutually orthogonal Abelian projections

Let $f_i$, $i=1,\dotsc,n$, be mutually orthogonal Abelian projections in a von Numann algebra, and let $e\leq\sum f_i$. Is it true that there exist mutually orthogonal Abelian projections $e_j$, $j=1,\...
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1answer
66 views

Compatibility of the absolute value with the integration

Let $H$ be a separable Hilbert space and $L^{1}(H)$ be the space of trace class operators on $H$. Let $f:\mathbb{R}\to L^{1}(H)$ be a measurable function that is $t\to \operatorname{tr}(f(t))$ is ...
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2answers
95 views

Upper triangular $2\times2$-matrices over a Baer *-ring

Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices $\left(\begin{array}{cc} a_1& a_2 \\ 0 & a_4 \end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...
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0answers
105 views

Banach algebras with trivial center [migrated]

Let $A$ be a Banach algebra. The center of $A$, denoted by $Z(A)$, is the set of elements of $A$ that commute with all elements of $A$. Please give some examples of Banach algebras with trivial center....
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1answer
166 views

Model theory of Banach algebras

Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra: $$(\forall x) (\forall y) (\forall \varepsilon &...
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0answers
140 views

$C(X) \otimes A \cong C(X, A)$

The following appears to be something of a folk theorem. Let $X$ be a compact Hausdorff space, and let $A$ be a C*-algebra. Then, the C*-algebra $C(X) \otimes A$ is $*$-isomorphic to the C*-algebra ...
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0answers
46 views

Characterizing fullness of a von Neumann algebra by the topology of its bimodules

Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map: $$ \mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
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1answer
131 views

Trivializing unitary cocycles in abelian von Neumann algebras that are uniformly close to the trivial one

Suppose $M$ is an abelian von Neumann algebra, carrying a (point-ultraweakly) continuous action $G\curvearrowright M$ of a locally compact, second-countable group. Let $y: G\to\cal U(M)$ be an ...
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0answers
88 views

The Hahn-Hellinger Theorem [closed]

People can tell the question is not up to the mark, or research level question, but I felt without understanding the Hahn-Hellinger Theorem properly there is no point talking von Neumann algebras for ...
2
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1answer
74 views

Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation

Suppose $X$ and $Y$ are compact metric spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...
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1answer
70 views

Isomorphism of preduals implies isomorphism of the $W^*$-algebras or not?

Let $M$ and $N$ are two von Neumann algebras such that their preduals $M_{∗}$ and $N_{∗}$ are isomorphic in the sense of Banach spaces, does it imply M and N are $∗$-isomorphic or not??
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1answer
116 views

Description of (completely) bounded operator

I am somewhat a beginner in the field of operator algebras and was wondering about the following: Let $T$ be a linear map between the space of bounded operators $B(H)$ on some Hilbert space and $S$ a ...
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1answer
53 views

Clarification on predual on existence of separating vector

We know predual of a von Neumann algebra $M$ as a Banach space is independent of Hilbert space where the $M$ is represented. Now the question is if we represent $M$ in $B(\mathcal{H})$, where $M$ has ...
7
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0answers
146 views

A robust version of Schur's lemma?

Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this: Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
3
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1answer
105 views

How rich the group of unitary elements in a von Neumann algebra to get “Murray-von Neumann” equivalence?

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be ...
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1answer
129 views

About separability of von Neumann algebras [closed]

Is a von Neumann algebra always separable in the $\sigma$-weak topology? If not, give a counterexample. Under what conditions will it be separable?
7
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1answer
134 views

Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?

The question is a special case of a previous question. Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection ...
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2answers
160 views

Is $C^{\infty}(M)$ a projective Frechet $C^{\infty}(N)$-module for a smooth map $M\to N$ between compact smooth manifolds?

Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\...
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0answers
70 views

References for hyperfinite factors

Can I have references of hyperfine $II_1$ factors where I can get structural properties to be studied and more characterizations.
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2answers
132 views

On topology in von Neumann algebras

Suppose $M$ is von Neumann algebra, $A$ is $*$-algebra in $M$, further if $(A)_1$, the unit ball of $A$ is strong operator closed, does it implies $A$ is von Neumann algebra? I started proving this ...
17
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2answers
420 views

Almost isometric linear maps

Say that a linear map $\varphi : B(\mathcal H) \rightarrow B(\mathcal H)$ is a $\epsilon$-almost isometric if $$ 1 - \epsilon \leq \|\varphi(a)\| \leq 1+\epsilon, \quad \forall a\in B(\mathcal H), \...
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0answers
111 views

Schröder–Bernstein for representations of operator algebras

This is claimed in a Wikipedia Article: If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...
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1answer
115 views

Extending maps from dense $*$-algebras of $C^*$-algebras

Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...
3
votes
1answer
155 views

About some positive elements in a group von Neumann algebra

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
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1answer
98 views

On predual of von Neumann algebra

Suppose $T_{n}$ converging $T$ in vN algebra $M$ in weak operator topology, can we conclude $||T_{n}||$ is uniformly bounded? Another question if a linear functional $\varphi$ is continuous in unit ...
2
votes
1answer
229 views

Does this sequence contain a nonnegative number?

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
7
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0answers
86 views

Properly outer automorphisms on type II$_1$ von Neumann algebras

Let $M$ be a von Neumann algebra with separable predual. Let us assume that $M$ is of type II$_1$, meaning that it is finite but has no type I part. Let $\tau$ be a faithful normal tracial state on $M$...
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1answer
220 views

finite dimensional C*-algebras

Let $A$ be a C*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional. Q. Is $A$ finite dimensional?
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1answer
158 views

Convergence of nuclear operators

Let $H$ be a separable infinite-dimensional real Hilbert space. We consider operators in $H.$ Nuclear norm of a nuclear operator is the sum of its singular values. A nuclear, positive and self-...
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1answer
153 views

is the conditional expectation faithful?

Let $G$ be locally compact group and let $H$ be a open subgroup in $G$. Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\colon ...
5
votes
1answer
142 views

Generator of $K_0(C_0(\mathbb{C}))$

$\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}$ I know from Bott-periodicity that $K_0(C_0(\mathbb{C}))\simeq \Z$, is there any easy way to compute an explicit generator of $K_0(C_0(\mathbb{C}...
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0answers
63 views

“Adding” a projection to a von Neumann algebra

This is a question about what happens when you "add" a new projection $p$ to a von Neumann algebra $\mathcal{R}$ to generate a larger v.N. algebra $(\mathcal{R} \cup \{p\})''$. Suppose that $\mathcal{...
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votes
0answers
38 views

Non-existence of projections in crossed product

If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such ...
3
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1answer
148 views

Pure infiniteness of tensor product $C^\ast$-algebras

I found the following theorem (Proposition 4.5) in the paper of E. Kirchberg and M. Rørdam: Non-simple purely infinite $C^\ast$-algebras. Amer. J. Math. 122(3) (2000), 637-666 (MR1759891, jstor, ...
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1answer
200 views

A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO I am sorry if this question is elementary: What is a precise definition of a contractible Banach ...
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2answers
273 views

Is the ideal property of $X^{**}$ inheritable to $X$?

Let $X$ be an operator space such that there is a weak$^*$-continuous complete isometry $\phi$ from its second dual $X^{**}$ into a $W^*$-algebra $M$ in which $\phi(X^{**})$ is a (necessarily weak$^*$-...
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0answers
106 views

Type III factor examples?

How to prove the crossed product of $G$ and von Neumann algebra $M$, where $G$ is locally compact group acting on $M$ via free ergodic action and $M$ is type $II_{\infty}$ factor, is type $III$ factor,...
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1answer
121 views

On projection theory for inseparable Hilbert spaces

How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
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0answers
50 views

What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?

When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in ...
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0answers
63 views

Projections in properly infinite factor

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let $e$ be a projection in a properly infinite factor. Is it always true that $e\sim 1$ or $1-e\...
4
votes
1answer
170 views

Relation between maximal and reduced group $C^*$-algebras

Let $G$ be a Lie group and $C_r^*(G)$ and $C^*(G)$ be its reduced and maximal group $C^*$-algebras respectively. The left-regular representation of a group $G$ induces a surjective map $$\lambda_G:C^...
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votes
2answers
160 views

Point spectrum of a positive invertible operator

Let $G$ be a l.c. group and $f$ belong to $C_c(G)$, the space of continuous functions with compact support. Define an operator$T_f$ on $L^2(G)$ by $T_f(g)=f*g$ (the convolution product). If $T_f$ is ...
3
votes
1answer
218 views

Is every nontrivial idempotent in the Cuntz algebra, a commutator element?

Is it true to say that every nontrivial idempotent in the Cuntz algebra $\mathcal{O}(n)$ is a commutator element?(Or a linear combination of commutator elements?)
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1answer
127 views

A particular separation example

Q1. Does there exist a separable Banach space $X$ satisfying in the following property? 1- $X^*$ is non separable. 2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ ...
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1answer
159 views

Simple $C^*$ algebras whose all commutator elements have scalar square

Is there a simple $C^*$ algebra $A$, not isomorphic to $M_2(\mathbb{C})$, such that for every commutator element $x=ab-ba$, $x^2$ is an scalar element?
7
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0answers
249 views

Open projections and Murray-von Neumann equivalence

Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
4
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1answer
248 views

A generalization of unsolvable equation $ab-ba=1$ in a Banach algebra

It is well known that the equation $$(*)\;\;\;\;ab-ba=1$$ is unsolvable in a Banach algebra. I search for some reasonable generalization of this equation in higher variable for investigation ...
9
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1answer
270 views

Comparing two $\sigma$-algebras on $B(\ell^1)$

Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow $$w-\lim T_i=T \Longleftrightarrow \...