All Questions
Tagged with operator-algebras or oa.operator-algebras
2,152 questions
0
votes
0
answers
76
views
The completely reducible bimodules coming from subfactors
This post is a sequel of: Are all the R-R-bimodules completely reducible?
Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
2
votes
1
answer
391
views
When C(K) is closed in sigma strong topology?
Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in:
$\sigma$-strong
$\sigma$-strong*
...
- 11.3k
0
votes
1
answer
431
views
Affine Homeomorphism between a compact set K and the state space on A(K)
I posted this question a few days back at math.SE but could not get help. So I was kind of forced to ask here. Please excuse me if this question is not suitable here.
Let $V$ be a locally convex ...
- 215
5
votes
1
answer
694
views
Log structure and degeneration
I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem.
I am developping the same technique for quantum geometry.
...
- 53
1
vote
1
answer
90
views
Square of Pierce 1/2 elements
Let $p$ be a nontrivial idempotent in a JB-algebra $A$ with Pierce decomposition $A = A_1 \oplus A_{1/2} \oplus A_0$. Then the projection onto $A_1$ (resp. $A_0$) is given by $U_p$ (resp. $U_{p'}$). ...
- 388
5
votes
0
answers
241
views
Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$
In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group ...
- 1,528
2
votes
0
answers
132
views
Extension of a bounded operator on manifold
I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow.
The problem, if stated in as full generality as ...
- 63
3
votes
2
answers
429
views
Kernel projections in the universal representation.
Let $A \subseteq \mathcal B(\mathcal H)$ be a unital C*-algebra in its universal representation. The GNS representation $\pi_\mu\colon A \rightarrow \mathcal B(\mathcal H_\mu)$ with base state $\mu$ ...
- 1,199
1
vote
0
answers
108
views
Reference request on operator matrices [closed]
I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that
$$Tx = \begin{pmatrix}A & B \\
C & D
\end{pmatrix}\...
- 11
0
votes
0
answers
354
views
abstract algebra for component wise operations on "vectors" or what it might be called
I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations:
- multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
- 254
7
votes
0
answers
189
views
Replacing commutative C*-algebras by simple ones
I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$.
I am particularly ...
- 7,637
3
votes
1
answer
191
views
Passing automorphism group through a representation
This is a very general question-- I'm really after any references to anything similar...
Let $A$ be a $C^*$-algebra equipped with a continuous one-parameter group of automorphisms $(\alpha_t)_{t\in\...
- 18.7k
5
votes
0
answers
154
views
When is an inner derivation a Fredholm operator?
Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by ...
- 767
1
vote
1
answer
478
views
Is exp(rA) = (exp(A))^r for real r and A in a Banach space?
Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
...
- 657
4
votes
0
answers
195
views
The groupoid VN algebra of the transversal to a uniquely ergodic action
I have a uniquely ergodic dynamical system preserving a finite ergodic measure (specifically, I have a nice aperiodic tiling space with an action of $\mathbb{R}^d$). Thus the transformation group von ...
- 1,010
5
votes
2
answers
670
views
Reference for von Neumann algebras coming from a group algebra twisted by a 2-cocycle?
I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, ...
- 363
0
votes
0
answers
85
views
Is $KK^G(\mathbb{C}^n,B)$ countably additive in $B$ and countable?
Let $G$ be a finite discrete groupoid, $A=\mathbb{C}^n$ a finite dimensional, commutative $C^*$-algebra and assume we have given a $G$-action on $A$. Note that the action of $G$ on $\mathbb{C}^n=C_0(\{...
- 58
5
votes
1
answer
410
views
Is the unitary group of $l^2(A)$ with the strict topology contractible?
Let $A$ be a $C^*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary ...
- 7,637
3
votes
0
answers
170
views
Closure of pseudodifferential operators of order 0 on compact manifolds
Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...
- 2,998
17
votes
1
answer
710
views
Standard polynomials applied to matrices
The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\...
- 52.3k
3
votes
0
answers
301
views
What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
user23860
5
votes
2
answers
475
views
Can finitely many hermitian (positive-semidefinite) operators always be embedded into a small dimensional space preserving inner product?
Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which ...
- 363
4
votes
1
answer
270
views
A detail in the construction of the coarse index of a Dirac operator in "Roe: An Index Theorem on Open Manifold, I"
Hi,
I'm currently wreading "Roe: An Index Theorem on Open Manifolds, I, J. Differential Geometry 27 (1988), p. 87-113" and there is a detail in the construction of the coarse index of a Dirac ...
- 2,998
28
votes
0
answers
2k
views
Finite-dimensional subalgebras of $C^\star$-algebras
Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
- 25.5k
6
votes
0
answers
226
views
Bound on number of multiplications required to generate a matrix algebra from generators?
I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all?
Suppose you have ...
- 61
3
votes
1
answer
367
views
algebraic VS topological ergodicity
Let A be a $C^*$-algebra with unit $I$, and G a locally compact (Hausdorff) group. An action $\alpha$ of G on A is a strongly continuous homomorphism of G into Aut(A), the group of *-automorphisms of ...
- 33
6
votes
0
answers
369
views
Paving conjecture for Toeplitz matrices
Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
- 16.2k
1
vote
1
answer
416
views
Projection in Hereditary C* subalgebra
This is actually something in a paper but the author claimed it without proof.
Let x be a positive elment of norm 1 in a $C^*-$algebra A, and Her(x) is the hereditary subalegbra generated by x. Given $...
- 411
5
votes
1
answer
796
views
Orthogonal similarity of matrices
Given $M\in M_n({\mathbb R})$
and $\ell\in{0,\ldots,n-1}$, we define
$$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$
where the indices are understood mod $n$. In particular, $d_0$ is the trace operator.
Let ...
- 52.3k
4
votes
2
answers
498
views
The state space of the stabilization of a C*-algebra
Given a $C^*$-algebra $A$, I wonder up to what extent we can describe the state space of the stabilization $A\otimes K$ of $A$ in terms of the state space of $A$. Of course, the "tensor-product" ...
2
votes
2
answers
270
views
Homomorphisms preserving constant functions
Assume we have a homomorphism $\phi: C(S^{1},M_{n}(\mathbb{C}))\rightarrow C(S^{1},M_{m}(\mathbb{C}))$ where $n$ divides $m$. Under what conditions does $\phi$ send constant functions to constant ...
- 169
1
vote
0
answers
57
views
Is 6 the smallest index for an irreducible subfactor to have a principal graph with a multiplicity >1 edge?
The irreducible subfactor $(R^{S_3} \subset R)$, of index $6$, admits a principal graph with a multiplicity $2$ edge because the group $S_3$ admits an irreducible complex representation of dimension $...
3
votes
1
answer
338
views
Ultraproduct of n-dimensional Banach spaces and algebras
Hi, I am interested in the following question:
Fix $n$.
Let $M_n$ be matrix algebra over the field of complex numbers with normalized trace $tr_n$. Let $M_n^{\omega}$ be an ultrapover of $M_n$, ...
6
votes
1
answer
456
views
Stable orthogonalization procedure
At a high level, my question is the following: given a set of $k$ vectors in Euclidean space which are pairwise "almost orthogonal", can one find a set of $k$ orthogonal vectors which are pairwise ...
- 63
0
votes
1
answer
156
views
Calculation of L2-dimension
For a group $G$, can we calculate $dim^{(2)}_{\mathcal{N}G}(\ell^2 G)$, where $\mathcal{N}G$ is the von Neumann algebra of $G$ and $\ell^2 G$ is the Hilbert space on $G$? I want to see whether this is ...
- 537
4
votes
1
answer
1k
views
When can a partial isometry $u$ in $\mathcal B(H \otimes K)$ be extended to a unitary in $1 \otimes \mathcal B(K)$?
Let $H$ and $K$ be Hilbert spaces, and let $u$ be a partial isometry in $\mathcal{B}(H \otimes K)$ between projections $p_0 = u^\ast u$ and $p_1 = u u^\ast$ such that $p_0, p_1 \leq 1 \otimes (1-q)$ ...
- 1,199
0
votes
1
answer
274
views
Dual notion of a local homeomorphism between topological spaces for C*-algebras
Given two locally compact topological spaces $X$ and $Y$, and a local homeomorphism $f : X \to Y$, Gelfand duality gives us a homomorhism $Cf : C_0(Y) \to C_0(X)$. How does the fact that $f$ is a ...
- 3
2
votes
1
answer
362
views
Quasinilpotent example [duplicate]
Possible Duplicate:
Quasinilpotent operator
Do you know any example of a quasinilpotent operator such that every its power is non-compact?
Of course direct sum of nilpotent operators(or Volterra ...
- 21
2
votes
0
answers
149
views
Planar algebraic translation of a subfactor property
Let $N \subset M$ be an irreducible finite depth and finite index subfactor.
$M$ is a completely reducible (algebraic) $N$-$N$ bimodule, it decomposes into irreducibles as follows :
$$M=\bigoplus_{...
2
votes
3
answers
1k
views
Norm on quotient algebra of a tensor algebra
Suppose you have a finite dimensional real Hilbert space $V$ and you form the tensor algebra
$$T(V) = \mathbb{R} \oplus V \oplus (V\otimes V) \oplus (V\otimes V \otimes V) \oplus \cdots$$
where the ...
- 223
8
votes
1
answer
431
views
Injectivity for bimodules and Hochschild cohomology
Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...
user2412
1
vote
1
answer
239
views
Functoriality of the Group-Measure -space construction
Let $G$ be a discrete group. Consider the action of $G$ on itself
a) by left multiplication,
b) by conjugation.
Under which conditions on group homomorphisms is the Group-...
- 2,630
3
votes
0
answers
455
views
Morphism of von Neumann Algebras
Hello,
Is there a counterexample to the following statement:
let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ?
( a $W^* $-...
- 663
1
vote
0
answers
112
views
Existence of orthogonal projections generating Von Neumann algebras
Let $V$ and $V'$ be Abelian Von Neumann algebras of projections on some Hilbert space $H$, and let $V_{1}$ and $V_{2}$ be minimal sub-algebras of $V$ and $V'$ generated by projections $P_{1} \in V$ ...
- 631
2
votes
1
answer
199
views
Uniqueness of free complements
Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ ...
- 1,411
1
vote
0
answers
232
views
From positive definite function to Følner sequence ----- a question on amenability and nuclearity
We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
- 11
2
votes
1
answer
317
views
Continuity of Borel measurable Gleason frame functions
Gleason's theorem (Journal of Mathematics and Mechanics, Vol. 6, No. 6, 1957) classifies measures on the closed subspaces of a separable Hilbert space. A key lemma toward the proof of the theorem ...
1
vote
2
answers
335
views
Bounded operators on direct limit of direct sums of spaces of cusp forms
Consider $S_{2k} (\Gamma_0 (N))$ and let $S(N)$ denote the direct limit of the finite direct sums of the $S_{2k}$. Since each $S_{2k} (\Gamma_0 (N))$ is also a Hilbert space w.r.t. the Petersson inner ...
- 15.4k
1
vote
0
answers
266
views
Nuclear Space problem
I need to show that if X is compact,then C(X) is nuclear.Also is the condition X is metrisable
necessary. I am at present attending a conference "Recent Aadvances in Operator Theory". This
problem ...
- 2,106
2
votes
0
answers
166
views
How simplify the pentagonal equation from two fusion rings?
A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...