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

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102 views

Direct proof of “Nuclear implies $C_{red}^*(G) \cong C^*(G)$”

It is well-known that for a discrete group $G$ the following statements are equivalent: $C_{red}^*(G)$ nuclear $C_{red}^*(G) \cong C^*(G)$ canonically i.e. there exists an *-isomorphism between the ...
8
votes
0answers
94 views

What does it mean for a category to admit direct integrals?

Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals. Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
2
votes
0answers
38 views

To what extend can a von Neumann algebra be determined by its projection lattice structure?

Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
3
votes
1answer
69 views

Sequence in *-algebra with different limits for two C*-norms?

The following question looks simple, but the answer is not obvious for me: Let $S$ be a $*$-algebra and $\left\Vert \cdot \right\Vert _{1}$, $\left\Vert \cdot \right\Vert _{2}$ $C^*$-norms on $S$ ...
4
votes
1answer
116 views

Behaviour of direct limit with matrices

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts: Let $(A_n,f_n)$ be a ...
3
votes
0answers
90 views

Kernel of multiplication in noncommutative 2-torus

What is $\Omega^1 (A_{\theta})$, that is, what is the kernel of the multiplication map $m:A_{\theta} \otimes A_{\theta} \to A_{\theta}$ where $A_{\theta}$ is the noncommutative 2-torus with parameter $...
5
votes
0answers
140 views

Are these element in a group algebra of a torsion-free group zero divisors?

Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements‌ can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)? $$1+x+y,\quad 4+x+x^{-1}+y+...
3
votes
1answer
80 views

Is the norm of the Banach space projective tensor product of finite-dimensional C*-algebras a C*-norm?

Let $A$ and $B$ be two finite-dimensional C*-algebras. Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum_{i}\|...
3
votes
0answers
67 views

informative examples for understanding spectral triples

I am at the beginning of my thesis work and I am trying to understand spectral triples. I can recall the definition but I have no informative examples with which to make sense of it. What are some ...
0
votes
1answer
99 views

Classification of finite-dimensional real super C*-algebras

The title says it all. I feel like one should be able to find this somewhere, but every time I try to google, I just get results for "super Lie algebras". Does anybody know a reference? I am not so ...
2
votes
2answers
123 views

Behaviour of Direct limit with quotient and double dual

I am trying to understand direct limit in category of $C^*$ algebras. Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras? Any references or ideas? P....
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vote
1answer
275 views

Topology of state space in von Neumann algebras

What are the sufficient conditions for a von Neumann algebra to have a first countable set of states with respect to the weak * operator topology?
3
votes
1answer
79 views

Non-point spectrum for diagonalisable self-adjoint unbounded operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
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votes
0answers
59 views

Reference sought: maximal ideal space of $\ell_\infty$ modulo sequences that go to 0 along a filter

It is not too hard to show the following. Suppose that $\mathcal{F}$ is a non-principal filter on $\mathbb N$. Denote by $c_0^{\mathcal{F}}$ the subspace of $\ell_\infty$ consisting of sequences that ...
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votes
0answers
56 views

Support size of a zero divisor

Let $G$ and $\mathbb C[G]$ be a torsion free group and its group algebra. Is there a function $f:\mathbb N\rightarrow\mathbb R$, with $\lim_nf(n)=\infty$ such that if $0\neq\alpha,\beta\in\mathbb C[G]$...
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vote
0answers
102 views

Tensor product decomposition of commuting representations

If $\mathscr{X}$ is a Hilbert space, we denote by $\mathrm{GL}(\mathscr{X})$ the group of all bounded operators on $\mathscr{X}$ with bounded inverses. Let $\mathbb{F}_2$ be the free group on two ...
5
votes
1answer
119 views

Unbounded version of continuous functional calculus

For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
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0answers
98 views

An example of non trivial projections in a group von Neumann algebra

Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
3
votes
1answer
137 views

An analytical zero divisor

Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...
9
votes
1answer
122 views

Separability of compact quantum groups

In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...
3
votes
0answers
66 views

Hilbert space separability for spectral triples

A spectral triple $({\cal A},{\cal H},D)$ consists of a unital $*$-algebra ${\cal A}$ represented as bounded operators on a Hilbert space ${\cal H}$, together with an unbounded operator $D$ having ...
4
votes
2answers
184 views

Unitaries in Banach *-algebras

We know that the norm of a unitary in a unital $C^*$-algebra is one. Also, in a unital Banach algebra A, $u \in A$ is defined to be a unitary if $\|u\| = \|u^{-1}\| =1$. I tried to prove it for the ...
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vote
0answers
125 views

Idempotents in Group Algebras

What is known about idempotents in Lie group algebras (such as on the classical Lie groups)? Specifically the self-adjoint ones. Is there anything interesting to say? I haven't been able to find much ...
2
votes
1answer
84 views

Direct proof a property of hyperstonean spaces

First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
2
votes
1answer
80 views

Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$

Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...
2
votes
1answer
54 views

Approximation of unity by projectors

Let $A$ be a $\sigma$-unital $C^*$-algebra and $A_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an ...
3
votes
1answer
94 views

Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
10
votes
1answer
244 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
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0answers
75 views

Conditional Expectation for von Neumann algebra

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...
1
vote
0answers
71 views

Injective differential of linear operators on a Hilbertspace

Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \...
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0answers
59 views

Multiplying positive operators

Consider positive linear operators on Hilbert space that satisfy $A-B\geq 0$ and $C\geq 0$. What is the necessary and sufficient condition for $A^{1/2}C A^{1/2}\geq B^{1/2} C B^{1/2}$? If they all ...
7
votes
0answers
89 views

Is there an adaptation of the theory of standard forms and Tomita-Takesaki theory to the $\mathbb{Z}_{2}$-graded case?

Let $A$ be a von Neumann algebra acting on a Hilbert space $H$, and suppose that $\Omega \in H$ is a cyclic and separating vector for $A$. Then in Tomita-Takesaki theory one defines an unbounded ...
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0answers
71 views

An example of a sequence of finite projections

Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...
2
votes
0answers
79 views

On the set of indices of irreducible depth 3 subfactors

Let $I_n$ be the set of indices of (finite index) irreducible depth $n$ subfactors. Then $I_2 = \mathbb{Z}_{>0}$. Question 1: Is it true that $I_3$ has no accumulation point? If so: ...
3
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0answers
81 views

An example of a particular vn-algebra

Let $A$ be a vn-algebra. Let us suppose $e$ is a finite projection in $A$ and $x$ is an isometry (meaning $x^*x=1$) in $A$ such that $e$ does not commute with $x$. Then $\{q_n=x^nex^{*n}\}$ forms a ...
1
vote
1answer
51 views

The range projection of product of projections

Let $A$ be a von Neumann algebra. Let $p$ be a projection in $A$. Suppose that $e$ is a finite projection. Can we determine all types of vn-algebras in which $p-p\wedge(1-e)$ is a finite projection?...
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0answers
42 views

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 ...
3
votes
1answer
61 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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vote
1answer
43 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,\...
1
vote
1answer
76 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 ...
1
vote
2answers
104 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 $B$ ...
5
votes
1answer
191 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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votes
0answers
153 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 ...
3
votes
0answers
49 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}(\...
6
votes
1answer
149 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 ...
2
votes
0answers
99 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
votes
1answer
82 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 ...
0
votes
1answer
78 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??
2
votes
1answer
123 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 ...
1
vote
1answer
57 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 ...